Component Models

In this section, we’ll summarize how component models are different from the previous models we’ve created. This discussion will be broken into two parts. The first part will focus on acausal modeling and how it provides a framework for schematic-based, component-oriented modeling where conservation equations are automatically generated and enforced. The second part will provide an overview of how the topics in this chapter impact, mostly syntactically, the definition of component models.

However, before we dive into that discussion, it is worth taking some time to talk about terminology. In this chapter, we’ve created two different types of models. The first type represent individual effects (e.g., resistors, capacitors, springs, dampers). The other type represent more complex assemblies (e.g., circuits, mechanisms).

Before we discuss some of the differences between these different types of models, let’s introduce some terminology so we can refer to them precisely. A component model is a model that is used to encapsulate equations into a reusable form. By creating such a model, an instance of the component can be used in place of the equations it contains. A subsystem model is a model that is composed of components or other subsystems. In other words, it doesn’t (generally) include equations. Instead, it represents an assembly of other components. Typically, these subsystem models are created by dragging, dropping and connecting component and other subsystem models schematically. While component models are “flat” (they don’t contain other components or subsystems, only equations), subsystem models are hierarchical.

We’ll often refer to a subsystem model as a system model. A system model is a model that we expect to simulate. In simulating it, the Modelica compiler will traverse the hierarchy of the model and note all the variables and equations present throughout the hierarchy. These are the variables and equations that will be used in the simulation. Of course, in order for there to be a unique solution, the system model (like any non-partial model), must be balanced.

Note that a subsystem model can include equations. There is no rule against it in Modelica. But most of the time models tend to be composed either of equations or other components/subsystems. It is actually a good idea to avoid putting equations in models containing subcomponents or subsystems because doing so means that some information about the model will be “invisible” when looking at a diagram of the subsystem. One possible exception to this could be the use of initial equation sections in subsystems.

With that discussion of terminology out of the way, let’s dive into discussions about component models.

Acausal Modeling

We’ll start with a discussion about acausal modeling. We touched on this topic very briefly in the chapter on Connectors. Here we will provide a more comprehensive discussion about acausal modeling.

Composability

There are two very big advantages to acausal modeling. The first is composability. In this context, composability means the ability to drag, drop and connect component instances in nearly any configuration we wish without having to concern ourselves with “compatibility”. This is because acausal connectors are designed around the idea of physical compatibility, not causal compatibility. This is possible because acausal connector definitions focus on physical information exchanged, not the direction that information flows. The result is that we can create component models around the idea of physical interactions without requiring any a priori knowledge about the nature (i.e., directionality) of the information exchange.

But there are other implications to this composability. Not only can we easily create systems by dragging, dropping and connecting components, but we can also easily reconfigure them. Replacing a voltage source in an electrical circuit with a current source can have a profound impact on the mathematical representation of that system (e.g., if the system is represented as a block diagram). But such a change has no significant impact when using an acausal approach. Although the underlying mathematical representation still changes, sometimes profoundly, there is no impact on the user, because that representation is generated automatically as part of the compilation process.

Finally, another aspect of composability is in the support for multi-domain systems. In fact, Modelica not only supports different engineering domains (electrical, thermal, hydraulic), it supports multiple modeling formalisms. Model developers have created libraries for block diagrams, state charts, petri nets, etc. Instead of requiring special tools or editors in each case, all of these different domains and formalisms can be freely combined in Modelica as appropriate.

Accounting

Connectors

The other advantage of acausal modeling is the amount of automatic “accounting” performed with this approach. To understand exactly what accounting is performed, let’s consider the following rotational connector definitions from the Modelica Standard Library:

connector Flange_a "1-dim. rotational flange of a shaft (filled square icon)"
  Modelica.SIunits.Angle phi "Absolute rotation angle of flange";
  flow Modelica.SIunits.Torque tau "Cut torque in the flange";
  annotation(Icon(/* Filled gray circle */));
end Flange_a;

connector Flange_b "1-dim. rotational flange of a shaft (filled square icon)"
  Modelica.SIunits.Angle phi "Absolute rotation angle of flange";
  flow Modelica.SIunits.Torque tau "Cut torque in the flange";
  annotation(Icon(/* Gray circular outline */));
end Flange_b;

As we’ve discussed previously, an acausal connector includes two different types of variables, across variables and through variables. The through variable is indicated by the presence of the flow qualifier. In the case of the Rotational connector, the across variable is phi, the angular position, and the through variable is tau, the torque.

Sign Conventions

Also recall from our previous discussion that Modelica models should observe the following convention: a positive value for the flow variable on a connector represents the flow of that quantity into the component that the connector is connected to. This is an important sign convention not only because it make sure all the accounting is correct, but it also helps with composability as well by allowing (inherently symmetric) components like springs, dampers, etc. to be flipped over and still function identically.

Connection Sets

Before we can get into the details of the accounting performed by the compiler, we need to introduce the concept of a connection set. To demonstrate what a connection set is, consider the following schematic:

Note that there are 8 connections in this model:

equation
  connect(ground.flange_a, damper2.flange_b);
  connect(ground.flange_a, spring2.flange_b);
  connect(damper2.flange_a, inertia2.flange_b);
  connect(spring2.flange_a, inertia2.flange_b);
  connect(inertia2.flange_a, damper1.flange_b);
  connect(inertia2.flange_a, spring1.flange_b);
  connect(damper1.flange_a, inertia1.flange_b);
  connect(spring1.flange_a, inertia1.flange_b);

If two connect statements have one connector in common, they belong to the same connection set. If a connector is not connected to any other connectors, then it belongs to a connection set that includes only itself. Using this rule, we can organize the connectors into connection sets as follows:

  • Connection Set #1
    • ground.flange_a
    • damper2.flange_b
    • spring2.flange_b
  • Connection Set #2
    • damper2.flange_a
    • spring2.flange_a
    • inertia2.flange_b
  • Connection Set #3
    • inertia2.flange_a
    • damper1.flange_b
    • spring1.flange_b
  • Connection Set #4
    • inertia1.flange_b
    • damper1.flange_a
    • spring1.flange_a
  • Connection Set #5
    • inertia1.flange_a

Note that these connection sets appear from right to left in the diagram. It may be useful to take the time to match the connectors in the diagram with those listed in the connection sets to understand what a connection set intuitively is. Note that the flange_a connectors are filled circles whereas the flange_b ones are only outlined.

Generated Equations

This is where the “accounting” starts. For each connection set, special equations are automatically generated. The first set of automatic equations are related to the across variables. We need to impose the constraint, mathematically speaking, that all across variables must have the same value. Furthermore, we also introduce an equation that states that the sum of all through variables in the connection set must sum to zero.

In the case of the connection sets above, the following equations will be automatically generated:

// Connection Set #1
//   Equality Equations:
ground.flange_a.phi = damper2.flange_b.phi;
damper2.flange_b.phi = spring2.flange_b.phi;
//   Conservation Equation:
ground.flange_a.tau + damper2.flange_b.tau + spring2.flange_b.tau = 0;

// Connection Set #2
//   Equality Equations:
damper2.flange_a.phi = spring2.flange_a.phi;
spring2.flange_a.phi = inertia2.flange_b.phi;
//   Conservation Equation:
damper2.flange_a.tau + spring2.flange_a.tau + inertia2.flange_b.tau = 0;

// Connection Set #3
//   Equality Equations:
inertia2.flange_a.phi = damper1.flange_b.phi;
damper1.flange_b.phi = spring1.flange_b.phi;
//   Conservation Equation:
inertia2.flange_a.tau + damper1.flange_b.tau + spring1.flange_b.tau = 0;

// Connection Set #4
//   Equality Equations:
inertia1.flange_b.phi = damper1.flange_a.phi;
damper1.flange_a.phi = spring1.flange_a.phi;
//   Conservation Equation:
inertia1.flange_b.tau + damper1.flange_a.tau + spring1.flange_a.tau = 0;

// Connection Set #5
//   Equality Equations: NONE
//   Conservation Equation:
inertia1.flange_a.tau = 0;

Note that for an empty connection set (i.e., Connection Set #5), there is only one across variable in the set, so no equality equations are generated. The conservation equation is still generated but it contains only one term. So it amounts to a statement that nothing can flow out of an unconnected connector. This makes intuitive physical sense as well.

What does all this mean physically? In the case of an electrical connection, this implies that each connection can be treated as a “perfect short” between the connectors. In the case of a mechanical system, connections are treated as perfectly rigid shafts with zero inertia. The bottom line is that a connection means that the across variables on each connector will be equal and that any conserved quantity that leaves one component must enter another one. Nothing can get lost or stored between components.

Conservation

There are two important consequences to these equations. The first is that the flow variable is automatically conserved. Typical flow variables are current, torque, mass flow rate, etc. Since these are all the time derivative of a conserved quantity (i.e., charge, angular momentum and mass, respectively), such equations are automatically conserving these quantities.

But something else is being implicitly conserved as well. Specifically, we can ensure that energy is conserved as well. For all of these domains, the power flow through a connector can be represented by the product of the through variable and either the across variable or a derivative of the across variable. As a result, for each domain we can easily derive a power conservation equation from the equations automatically generated for the connection set. From our example above, we know that in the first connection set we have the following equations:

ground.flange_a.phi = damper2.flange_b.phi;
damper2.flange_b.phi = spring2.flange_b.phi;
ground.flange_a.tau + damper2.flange_b.tau + spring2.flange_b.tau = 0;

If we multiply the last equation by der(ground.flange_a.phi), the angular velocity of the ground.flange_a connector, we get:

der(ground.flange_a.phi)*ground.flange_a.tau
+ der(ground.flange_a.phi)*damper2.flange_b.tau
+ der(ground.flange_a.phi)*spring2.flange_b.tau = 0;

However, we also know that all the across variables in the connection set are equal. As a result, their derivatives must also be equal. This means that we can substitute any one of them for another. Making two such substitutions yields:

der(ground.flange_a.phi)*ground.flange_a.tau
+ der(damper2.flange_b.phi)*damper2.flange_b.tau
+ der(spring2.flange_b.phi)*spring2.flange_b.tau = 0;

The first term in the equation above is the power flowing into the ground component through flange_a. The second term is the power flowing into the damper2 component through flange_b. The last term is the power flowing into the spring2 component through flange_b. Since these represent the power flowing through all connectors in the connection set, this implies that power is conserved by that connection set (i.e., all power that flows out of one component must flow into another, nothing is lost or stored).

Balanced Components

If we look carefully at the previous discussion on equations generated involving acausal variables in connection sets, we’ll see something very interesting. But to see it, we first need to review a few things we’ve learned about connectors and connector sets:

  1. A connection can only belong to one connection set.
  2. As we learned in our previous discussion on Acausal Variables, for every through variable in a connector (i.e., a variable declared with the flow qualifier), there must be a matching across variable (i.e., a variable without any qualifier).
  3. The number of equations generated in a connection set is equal to the number of connectors in the connection set multiplied by the number of through-across pairs in the connector.

Remember that acausal variables come in pairs. Equations for half of those variables (one per pair) will be generated automatically via connections. That means the remaining half of the equations must come from the component models themselves.

Keep in mind that this discussion is focused only on acausal variables in connectors. We also need to take into account two other cases:

  1. Variables declared within a component model (as opposed to on a connector).
  2. Causal variables on connectors (i.e., those qualified by either input or output).

Modelica requires that any non-partial model be balanced. But what does that mean? It means that the component should provide the proper number of equations (no more and no less than necessary). The question is how to compute the number of equations required?

We already have a start based on our discussion about acausal variables. Since half of the equations needed for acausal variables come from generated equations, the other half must come from within component models containing these connectors. Specifically, the component must provide one equation for every through-across pair in each of its connectors. In addition, it should also provide one equation for every variable on its connectors that has the output qualifier (note, the component does not have to provide equations for any variables on its connectors with the input qualifier). The rationale here is that a component can assume that all input signals are known (specified externally) and that it is responsible for computing any output signals it advertises. Finally, any (non-parameter) variable declared within the component must also have an equation.

In summary, the number of equations that a component must provide is the sum of:

  1. The number of through-across pairs across all its connectors
  2. The number of non-parameter variables declared in the component model.
  3. The number of output variables across all its connectors.

Note that these equations can (and frequently do) originate in a partial model that is inherited.

If the number of equations provided by a component equals the number of equations required, then the component model is said to be balanced.

Component Definitions

In this chapter we’ve discussed how to create component models. Fundamentally, nothing has changed since we first discussed what a Model Definition should include. But it is worth emphasizing a few things about component models.

Blocks

First, in the discussion on Block Diagram Components we introduced the idea of a block. A block is a special kind of model where the connectors contain only input and output signals.

Conditional Variables/Connectors

Another thing we saw in our discussion of the Optional Ground Connector was the ability to make a declaration conditional. The expression on which the conditional declaration depends cannot change as a function of time (i.e., the variable cannot appear and disappear during the simulation). Instead, it must be a function of parameters and constants so that the compiler or simulation runtime can determine whether the variable should be present prior to simulation. As we saw, the syntax for such a declaration is:

VariableType variableName(/* modifications /*) if conditional_expression;

In other words, by including the if keyword and a conditional expression immediately after the name of the variable (and any modifications that are applied to the variable), we can make the declaration of that variable conditional. When the conditional expression is true, the conditional variable will be present. When it is false, it will not be present.

Model Limitations

assert

To understand how to enforce model limitations, we must first explain the assert function. The syntax of a call to the assert function is:

assert(conditional_expression, "Explanation of failure", assertLevel);

where conditional_expression is an expression that yields either true or false. A value of false indicates a failure of the assertion. We’ll discuss the consequences of that momentarily. The second argument must be a String that describes the reason that the assertion failed. The last argument, assertLevel, is of type AssertionLevel (which was defined in our previous discussion on enumerations). This last argument is optional and has the default value of AssertionalLevel.error.

Now that we know how to use the assert function, let’s examine the consequences of assertions during simulation to understand why they are important.

Defining Model Limitations

When creating a component model (or any model, for that matter), it is useful to incorporate any limitations on the equations in a model by including them directly in the model. This is done by adding assert calls in either the equation or algorithm section. As their name implies, these assertions assert that certain conditions must always be true.

If the equations within a model are only accurate or applicable under certain conditions, it is essential that these conditions be included in the model via assertions. Otherwise, the model may silently yield an incorrect solution. If not uncovered, this could lead to bad decisions based on model solutions. If it is uncovered, it will undermine the trust people have in the models. So always try to capture such model limitations.

It is worth taking a moment to understand what impact such an assertion has during simulation. Part of the simulation process is the generation of so-called candidate solutions. These solutions may, or may not, end up being actual solutions. They are usually generated as the underlying solvers propose solutions and then check to make sure that the solutions are accurate to within some numerical tolerance. Those candidate solutions that are found to be inaccurate are typically refined in some way until a sufficiently accurate solution is found.

If a candidate solution violates an assertion, then it is automatically considered to be inaccurate. The violated assertion will automatically trigger the refinement process in an attempt to find a solution that is more accurate and, hopefully, doesn’t violate the assertion. However, if these refinement processes lead to a solution that is sufficiently accurate (i.e., satisfies the accuracy requirements to within the acceptable tolerance), but that solution still violates any assertions in the system, then the simulation environment will do one of two things. If the level argument in the assert call is AssertionLevel.error then the simulation is terminated. If, on the other hand, the level argument is AssertionLevel.warning, then the assertion description will be used to generate a warning message to the user. How this message is delivered is specific to each simulation environment. Recall that the default value for the level argument (if none is provided in the call to assert) is AssertionLevel.error.