# State Residualisation and Kron Reduction for Model Order Reduction of Energy Systems
## 3 Model Order Reduction by Combination of the State Residualisation and Kron Reduction Technique
### State Residualisation Technique
#### 3.1.2 State Categorisation  
Adopting the state residualisation method proposed by [25], the residualisation of state variables for a nonlinear model is linked to discarding some poles in the linearised model around its operating point and preserving the other poles. By acknowledging links between the states and poles of a linearised system, it is possible to choose which states to residualise in the nonlinear system. This process, illustrated in Figure 3a, is described step by step in the following.  
Figure 3: (a) Synoptic describing the state categorisation [25], and (b) synoptic describing the modal state residualisation [25].  
1. Linearisation. The non-linear DAE model in (20) is linearised around a steady-state operating point [30], resulting in a linear time invariant system that can be represented in state space as in (43). \[\left\{\begin{array}{l}\frac{dx}{dt}=Ax+Bu\\ y=Cx+D\end{array},\;x\in\mathbb{R}^{N_{d}},u\in\mathbb{R}^{p},y\in\mathbb{R}^{q}.\right.\] (43)
2. Participation Factor. Participation factors are a linear algebraic tool, introduced in [31], which quantify the links between the eigenvalues of a linear system and its state variables. More precisely, the participation factor \(p_{ki}\) is a complex number whose modulus gives the participation of the state \(x_{k}\) in the eigenvalue \(\lambda_{i}\)(and vice versa) and indicates how much the state variable influences the considered pole. The participation factor is calculated as in (44): \[p_{ki}=u_{i}(k)y_{i}(k),\] (44) where \(u_{i}(k)\) and \(y_{i}(k)\) are the \(k\)th entry of the left and right eigenvector \(u_{i}\) and \(y_{i}\) of \(A\) associated with eigenvalue \(\lambda_{i}\). \(u_{i}\) and \(y_{i}\) can be defined as (45). \[\begin{array}{l}Ay_{i}=\lambda_{i}y_{i},\\ u_{i}^{T}A=\lambda_{i}u_{i}^{T}.\end{array}\] (45)  
The left and right eigenvectors of A should be normalised, such that the sum of all the participation factors of all states for an eigenvalue is equal to 1 (for example, \(\sum_{k=1}^{N_{d}}p_{ki}=1\), associated with eigenvalue \(\lambda_{i}\)). Subsequently, the degree of participation of each state to an eigenvalue can be compared.
3. Groups of States and Eigenvalues. With \(p_{ki}\) obtained, the states that participate the most in each eigenvalue can be grouped together, with the rest discarded. Thus, \(N_{d}\) groups (states that do not belong to any group can be set to zero, as they do not significantly contribute to any eigenvalue) are obtained, as there are \(N_{d}\) eigenvalues. This process can be achieved by ordering \(p_{ki}\) in descending order and summing the first \(j\) terms larger than a threshold, \(\epsilon_{part}\).  
In a coupled system, a state variable may participate in several eigenvalues, which means that some of the previously created groups can be merged to form new groups, where each new group is associated with one or more eigenvalues. After this process, fewer than \(N_{d}\) groups are normally left. The state variables in the same group can be residualised together, or not at all (using an optimisation method describe in the next subsection). In other words, the associated poles will be discarded together, or not.  
Based on the above approach, the criterion \(\epsilon_{part}\) indicates the precision of the state categorisation. If \(\epsilon_{part}\) is high, interactions between the states are well recognised, but since the number of groups is reduced, the possibilities for reductions are limited. Alternatively, if \(\epsilon_{part}\) is low, the reduction possibilities are improved but the precision with which the poles are retained will be poor. The choice of this criterion is empirical, but different values can easily be tested. This paper demonstrated that the choice of \(\epsilon_{part}\) can be based on the type of event.