## 4 Fault Analysis
### External Fault Analysis
When denoting the initial condition for the converter fault current contribution as follows:  
\[\left\{ \begin{array}{l}i_{load\,i}(0)=I_{0}\\ \frac{d}{dt}i_{load\,i}(0)=\frac{v_{i}(0)-R_{line\,i}i_{load\,i}(0)}{L_{line \,i}}\end{array} \right., \tag{32}\]  
the constants \(H_{1i}\), \(H_{2i}\) are obtained as:  
\[\left\{ \begin{array}{l}H_{1i}=\frac{v_{i}(0)-R_{line\,i}i_{load\,i}(0)-(I_{ Max\,i}-i_{load\,i}(0))L_{line\,i}s_{2i}}{L_{line\,i}(s_{1i}-s_{2i})}\\ H_{2i}=-\frac{v_{i}(0)-R_{line\,i}i_{load\,i}(0)-(I_{Max\,i}-i_{load\,i}(0)) L_{line\,i}s_{1i}}{L_{line\,i}(s_{1i}-s_{2i})}\end{array} \right. \tag{33}\]  
Equations (30)-(33) characterize the fault response of a DC port, but can be easily adapted to AC ports, for each phase, as follows. In this case, (30) and (31) can be used as is, which is the general solution for an RLC circuit transient. The term \(I_{Max\,i}\) appearing in (30) maintains the same meaning, but must be considered as a sinusoidal function (\(I_{Max\,i}\) (\(t\)) = \(I_{Max\,i}\) sin(\(\omega t\) + \(\varphi\))), the amplitude of which is equal to the maximum current contribution set by the converter control. The initial conditions are again defined by (32), yet the solution for the constants \(H_{1i}\), \(H_{2i}\) must now consider the sinusoidal steady-state solution, resulting in:  
\[\left\{ \begin{array}{l}H_{1i}=\frac{v_{i}(0)-R_{line}i_{loadi}(0)-(I_{Maxi}(0 )-i_{loadi}(0))L_{line}i_{2i}-L_{line}i\frac{d}{2i}I_{maxi}(0)}{L_{line}(i_{1i} -s_{2i})}\\ H_{2i}=-\frac{v_{i}(0)-R_{line}i_{loadi}(0)-(I_{Maxi}(0)-i_{loadi}(0))L_{line }i_{1i}-L_{line}i\frac{d}{2i}I_{maxi}(0)}{L_{line}(i_{1i}-s_{2i})}\end{array} \right. \tag{34}\]  
However, as is also the case with bolted faults, current references can be approximated with a Lipschitz-continuous function, except from a limited time. Indeed, it is straightforward to prove that the fault current (30) can exhibit, during the transient phase only, a time derivative higher than the one enforced by the converter current control, the upper bound of which, neglecting resistive terms and external voltage, can be approximated as:  
\[\frac{di_{max}}{dt}=\frac{v_{D}}{L} \tag{35}\]  
As a consequence, during the transient phase, voltage regulation and power exchange may not be performed correctly. However, after the transient is extinguished, (4) and (10) are satisfied for any port. As a consequence, while the potential to control the port voltage/power depends on the fault impedance value, the current sliding condition is lost at the instant corresponding to the step-change, but then reached again in finite time. As a consequence, a pole-to-pole fault on one port does not violate (4) and (10) except from a very limited time, so that output currents are always kept under control and no faults on one output port can affect other ports or the internal DC bus. Additionally, under pole-to-pole fault conditions, which (4) and (10) have satisfied for voltage values smaller than the rated ones, the converter current control is maintained, so that the converter can limit the fault current to a predetermined, bearable value, such that the converter is self-protected during faults. In these regards, the most obvious choice is to keep the current constant and equal to its maximum value until the fault is resolved and voltage can be recovered. If the fault is not resolved in a predetermined time (reasonably, a few seconds) the current reference is set to zero for safety reasons. Nevertheless, if the converter control is integrated in the system protection, any fault current value can be selected. When considering pole-to-earth faults, different problems arise depending on the state of active parts towards earth. Usually, low-voltage distribution systems are not exercised when isolated from earth, except in special applications or in regions where effective earthing is problematic, such that it is of interest to briefly consider the possible earthing solution for a system including the considered MC.