# State Residualisation and Kron Reduction for Model Order Reduction of Energy Systems
## 1 Introduction  
With the increasing integration of renewable energy sources, high-voltage long-distance transmission, DC grids, and power electronic (PE) converter-interfaced loads, very high PE shares are anticipated in future power systems [1]. Existing power converters are mostly controlled as \"grid-following\", using a phase-locked loop (PLL) to synchronise the voltage angle and frequency at the connection point, in order to precisely follow the active and reactive power setpoints [2]. Since grid-following converters (GFLs) rely on first synchronising to the grid voltage and then realise controlled and stable outputs, when their share reaches a point they can no longer remain synchronised [3, 4]. Therefore, the concept of \"grid-forming\" is proposed. The key difference with grid-following converters is that grid-forming converters (GFMs) independently create their own internal voltage angle and frequency without relying on using PLL, by, for example, using an independent external power reference plus power feedback. One of the simplest GFMs is mimicked ideal AC voltage sources, which work by fixing their frequency at the rated one. Thus, GFMs can maintain their voltage angle and provide an immediate response to voltage and frequency disturbances, and naturally, inherit the abilities of synchronous generators (SGs), such as inertial response, offering blackstart capability and providing a voltage reference for GFLs. Consequently, GFMs are often seen as a replacement for SGs in future power grids. As PE converters have fast switching dynamics spanning a wide frequency range, traditional phasor approximation (RMS) simulation programs, which neglect the fast electromagnetic dynamics of power lines as they are faster than the electromechanical dynamics of SGs, may not be suitable to investigate the stability of (future) power systems and validate new converter control laws [5]. Electromagnetic (EMT) models can accurately represent a power system [6], but they involve a larger number of variables, which leads to high simulation computation times and complex analysis. Model order reduction (MOR) can be useful to accelerate simulations and simplify analysis [7; 8]. Several MOR methods exist in the literature [9], including modal truncation [10; 11], balanced truncation [12; 13; 14], proper orthogonal decomposition (POD) [15], proper generalised decomposition (PGD) [16], or Krylov methods [17; 18]. These methods are seen to be accurate, and have been widely used in the power system community [19; 20], but, as they rely on a state projection on a new basis, they do not preserve the physical structure of the system if directly applied [21]. As a result, analysis is not straightforward with these methods. Indeed, the loss of the physical meaning of the reduced models makes it difficult to analyse system behaviour. Instead, it is desirable that the models rely upon the physical variables and parameters of the system, to simplify later analysis and exchanges between transmission system operators (TSOs), and to make it possible to conveniently evaluate new control laws or tuning processes. Moreover, they need to capture fast transients with good accuracy, particularly overcurrents, to ensure converter integrity. A parametric MOR method is proposed in [22], where the specified parameters are explicitly preserved in the reduced mathematical model that is obtained via balanced truncation. This parametric MOR method is particularly useful for studying the impacts of those preserved parameters on system behaviour at lower costs and comparing to the original full EMT model. However, physical variables are not preserved in this method. The MOR method in [23] consists of the aggregation of the same type of parallel connected GFMs, the Kron reduction of the network, and the singular perturbation of the aggregated converters.