As we already touched on during our previous discussion on Steady State Initialization, behavior is represented by both the equations contained in a model as well as the initial conditions given to the state variables in the model. In Modelica, the initial conditions are computed by combining the normal equations (present in equation sections) with any initial equations (present in initial equation sections).
One of the first sources of confusion for new users is understanding how many initial conditions are required. The answer to this question is simple. In order to have a well-posed initialization problem (one where we don’t have too many or too few initial equations), we need to have the same number of equations in the initial equation sections as we have states in our system. Note, we can get away with having too few, because tools can augment the initial equations we provide with additional ones until the problem is well-posed, but we cannot solve a problem where we have too many initial equations.
Of course, saying the number of initial equations has to be equal to
the number of states answers one question, but quickly creates another,
i.e., how do we determine how many states there are? For the
models we’ve seen in this chapter, the answer is quite simple. The
states in each of our examples so far are the variables that appear
der(...) operator. In other words, every variable that
we differentiated in those examples is a state.
Ordinary Differential Equations¶
It is important to note that it will not always be the case that every variable that we differentiate will be a state. In this chapter, all the models we have seen so far are ordinary differential equations (ODEs). When dealing with ODEs, every differentiated variable is a state, which, in turn, means that you need an initial equation for each of these differentiated variables. But in subsequent chapters we will eventually run across examples that are so-called differential-algebraic equations (DAEs). In those cases, only some of the differentiated variables can be considered states.
As it turns out, understanding initialization doesn’t really require us to get into a detailed discussion about DAEs. In practice, all Modelica tools perform something called “index reduction”. While the index reduction algorithms themselves are fairly complicated (so we won’t get into those now), the effect is quite simple. Index reduction transforms the DAEs into ODEs. In other words, Modelica compilers will transform whatever DAE problem contained in our Modelica code into this relatively easy to explain ODE form.
So let’s side-step the discussion about DAEs and index reduction and just pick up our discussion of initialization assuming our problem has already been reduced to an ODE. In this case, the only thing we really need to understand is that initialization is required for all states in the model and that our model will have the following general ODE form:
where is the current simulation time, are the values of the states in our system at time , are the values of any external inputs to our system at time .
Note that the arrow over a variable simply indicates that it is a vector, not a scalar. Also note that the only variable that appears differentiated in this problem is . This is how we know that represents the states in the system. One final thing to note about this system is that neither function, nor , depends on .
If you think about it, both and are external to our system. We don’t compute them or control them. The reason that we call the state of our system is that it the only information (from within our system) needed to compute and (which, in turn, are the only things we need to compute in order to arrive at a solution).
Getting back to the topic of initialization, during a normal time step we will solve for by integrating to compute . In other words:
This all works as long as there was a previous time step. When there wasn’t a previous time step, then the value of that we plug into our equations has to be the very first value of in our simulation. In other words, our initial conditions.
One might imagine that we would specify our initial conditions by adding an equation like this:
where is the start time of our simulation and is an explicit specification of the initial values. Providing explicit values for states is a very common case when specifying initial conditions. So we definitely need to be able to handle this case. But this approach won’t work for the cases we showed in Steady State Initialization. There we didn’t provide explicit initial values for states. Instead, we provided initial values for . So how can we capture both of these cases?
The answer is to assume that at the start of our simulation we need to solve a problem that looks like this:
Note the introduction of a new function, . This new function represents any equations we have placed in initial equation sections. The fact that takes both and as arguments allows us to express a wide range of initial conditions. To define explicit initial values for states, we could define as:
But we could also express our desire to start with a steady state solution by defining as:
And, of course, we could mix these different forms or use a wide range of other forms on a per state basis to describe our initial conditions. So when writing initial equations, all you need to keep in mind is that they need to be of the general form shown above and that you cannot have more of them than you have states in your system.
As we’ve demonstrated in this chapter, the initial equation construct in Modelica allows us to express many ways to initialize our system. In the end, all of them will compute the initial values for the states in our system. But we are given tremendous latitude in describing exactly how those values will be computed.
This is an area where Modelica excels. Initialization is given first class treatment in Modelica and this flexibility pays off in many real world applications.