## 4 Fault Analysis
### External Fault Analysis  
For the considered MC converter, it is of interest to consider external faults on both AC and DC ports. These faults have a different effect on converter operation, depending on port functionality. If the faulted port is connected to a controllable energy source, the voltage reduction due to the fault makes it impossible to transfer the power required by higher-level control without exceeding the converter current limits. As a consequence, the power balance in the internal bus is momentarily lost, resulting in a voltage deviation, which will then be compensated by the converter control by acting on the power exchange on the healthy ports. If the faulted port is connected to a load or non-controllable energy source, then the fault will make it impossible to maintain the correct voltage without exceeding the converter current limits. As a consequence, voltage regulation is lost. In both cases, however, the converter current control is maintained over the whole transient, ensuring the converter operation's safety. This is obtained by virtue of the stability properties of sliding mode controllers [59, 60, 61, 62]. Indeed, the condition that allows the sliding condition to be reached in finite time and maintains sliding condition thereafter is expressed by (4) for each PEBB realizing an AC port, and by (10) for any module realizing a DC port. During normal operating conditions, (4) and (10) are usually satisfied for a correctly designed system under a non-restrictive hypothesis, namely, when capacitor voltages and current references are ascribable to suitable, limited, Lipschitz-continuous functions. This is taken for granted in the case of capacitor voltages, yet two significant exceptions must be considered for currents, namely, external faults and load step-changes, as the line current \(i_{load\,i}\) at the \(i\)-th port appears in power references (17). Between these two occurrences, the fault is of course the worst case, as not only can an uncontrolled transient be experienced by the system, but it may also not be possible to regulate the port voltage or exchange the required power after the transient is extinguished, depending on the value of the fault impedance. As a worst case, let us hence consider the case of a bolted fault. For a DC port, let us consider, with no loss of generality, a bolted pole-to-midpoint fault. A pole-to-pole fault can be treated similarly due to the converter's symmetric structure. Additionally, let us neglect, in the mathematical analysis, the diodes connected in parallel to each DC port output capacitor, which as mentioned, are used to prevent the port voltage from changing its sign, which would result in loss of controllability and possible converter damage. In case of a bolted pole-to-midpoint fault, the \(i\)-th DC port, the fault causes an uncontrollable current in the faulted line as the converter output capacitor rapidly discharges through the line and the fault. Denote the line resistance and inductance as \(R_{line\,i}\) and \(L_{line\,i}\), respectively, and assume, for simplicity of notation, that the fault occurs at time \(t=0\), the MC fault current contribution can be calculated as a simple second order RLC transient. With reference to Figure 3 for variables, the MC DC fault current contribution is generally obtained as:  
\[i_{load\,i}(t)=H_{1i}e^{\kappa_{1i}t}+H_{2i}e^{\kappa_{2i}t}+I_{Max\,i}, \tag{30}\]  
where \(H_{1i}\), \(H_{2i}\) are constants depending on the circuit initial conditions, \(s_{1i}\), \(s_{2i}\) are constants defined as:  
\[\left\{ \begin{array}{l}s_{1i}=-\alpha+\sqrt{\alpha^{2}-\omega_{0}^{2}}\\ s_{2i}=-\alpha-\sqrt{\alpha^{2}-\omega_{0}^{2}}\end{array} \right. \tag{31}\]  
where \(\alpha=R_{line\,i}/(2L_{line\,i})\) and \(\omega_{0}=1/\sqrt{(L_{line\,i}C_{i})}\) and \(I_{Max\,i}\) is the constant converter current limit set by the converter control, as mentioned at the end of Section 3.2, which is to be considered the fault steady-state value and, hence, the transient steady-state solution.