Normal trees

Before we introduce the algorithm for constructing the state-space model in , we introduce the first step from the system graph to the state-space model: the normal tree. It is a subgraph of the system’s linear graph.

In the following, we will consider a connected graph with \(E\) edges, of which \(S\) are sources. There are \(2 E - S\) unknown across- and through-variables, so that’s how many equations we need. We have \(E-S\) elemental equations and for the rest we will write continuity and compatibility equations. \(N\) is the number of nodes.

The following rules must be respected.

1. There can be no loops.

2. Every node must be connected.

Form a normal tree with the following steps.

1. Include all nodes.
2. Include all across-variable sources.
3. Include as many as possible A-type elements.
4. Include as many as possible D-type elements.
5. Include as many as possible T-type elements.

We call those edges in the normal tree its branches and those not, the links.

A-type elements not in and T-type elements in the normal tree are called dependent energy storage elements. All other A- and T-types are independent energy storage elements. The energy in these can be independently controlled.

In order to avoid an artificial excess in state variables and construct what is called a controllable model, whenever A-types in series (sharing one node) or T-types in parallel (sharing two nodes) appear, we should combine them to form equivalent elements in accordance with the formulas $$\begin{align} C_e &= \frac{1} {\sum_i 1/C_i} \quad \text{or} \\ L_e &= \frac{1} {\sum_i 1/L_i}. \end{align}$$

There are special names for power-flow variables associated with an element, depending on whether the element is a branch or link. Primary variables are: across-variables on branches and through-variables on links. Secondary variables are: through-variables on branches and across-variables on links.