**5. Proposed Plug-and-Play Protection Scheme**

The previous section presented two challenges for the selectivity of non-unit protection schemes. These challenges can be tackled by utilizing communication, but communication will likely slow down fault detection. Furthermore, (directional) thresholds could be designed to prevent unnecessary tripping, but doing so would require knowledge about the system's topology and parameters. Therefore, in order to achieve selective protection for plug-and-play low voltage dc grids, an alternative approach is proposed.

#### *5.1. Proposed SSCB Topology to Delay Fault Propagation*

It is proposed to append the SSCB topology with an RC damper on each terminal, as is shown in Figure 14. The purpose of the dampers' capacitance is to temporarily provide a low impedance path for fault currents and commutated inductive currents, delaying their propagation. Consequently, before a fault current can flow on the non-faulted side of the SSCB the damper capacitor on the non-faulted side of the SSCB must be discharged through the current limiting inductance. Similarly, a (commutated) current must first charge the damper capacitor before current can flow in the current limiting inductance. Therefore, the propagation of these currents is delayed, providing time for the SSCBs in the faulted areas to clear the fault and smoothing the commutation of inductive currents. However, if just a capacitance was added, high frequency oscillations with low damping could occur between the damper capacitors through the current limiting inductance, since the on-state resistance of the switches is small. Therefore, resistances are added to the dampers in order to attenuate these oscillations.

**Figure 14.** Proposed solid-state circuit breaker topology with added RC dampers.

In the proposed topology, the RC dampers together with the current limiting inductance essentially form a low-pass LCR filter. The inductance, resistance and capacitance of this LCR filter are 2*LCB*, 2*Rd* + 2*RCB* and 1 2*Cd* respectively. Making a loop inside the SSCB, the sum of the voltages over the damper capacitors, damper resistors, on-state resistances and current limiting inductances must be zero. Therefore, the differential equation for the inductor current is given by

$$2L\_{\mathbb{C}B}\frac{\partial}{\partial t}I(t) + (2R\_d + 2R\_{\mathbb{C}B})I(t) + \frac{2}{\mathbb{C}\_d}\int I(t)dt = 0. \tag{3}$$

Differentiating this equation, and dividing by 2*LCB* yields

$$\frac{\partial^2}{\partial t^2} I(t) + \frac{R\_d + R\_{CB}}{L\_{CB}} \frac{\partial}{\partial t} I(t) + \frac{1}{L\_{CB} \mathbb{C}\_d} I(t) = 0. \tag{4}$$

Applying the Laplace transform on this equation results in

$$s^2I(s) + \frac{R\_d + R\_{CB}}{L\_{CB}}sI(s) + \frac{1}{L\_{CB}C\_d}I(s) = I'(0),\tag{5}$$

where *s* is a complex variable representing attenuation and frequency in the Laplace domain (*s* = *σ* + *jω*), and *I*- (0) is the initial condition for the first derivative of the current. Consequently, the transfer function of this system is given by

$$H(s) = \frac{I(s)}{I'(0)} = \frac{1}{s^2 + \frac{R\_d + R\_{CB}}{L\_{CB}}s + \frac{1}{L\_{CB}C\_d}}.\tag{6}$$

The resonant frequency *fr* and attenuation frequency *α* of this standard second-order system are

$$f\_{\mathcal{I}} = \frac{1}{2\pi\sqrt{L\_{\mathcal{C}B}C\_d}},\tag{7}$$

$$
\alpha = \frac{\mathcal{R}\_d + \mathcal{R}\_{\mathcal{C}B}}{4\pi L\_{\mathcal{C}B}},
\tag{8}
$$

which will be used later in this section to provide design guidelines for the damper parameters.

Note that, the higher damper capacitor, the lower the resonant frequency of the SSCB's LCR circuit and the longer the SSCB will delay the propagation of fault currents. From a different perspective, a higher damper capacitance can provide the energy for the fault current for a longer time.

An additional benefit of the damper capacitance is that it delays and smoothes the commutation of an (inductive) current. This is illustrated by simulating the inductor current in the circuit from Figure 15. For the simulations the grid voltage *Udc* is 350 V, the on-resistance *RCB* is 0.1 Ω, the SSCB's

inductance *LCB* is 1 μH, and the damper resistance *Rd* is 2 Ω. The simulation results for the current in the SSCB's inductors, when the current *Io* is stepped up from 0 to 10 A at *t* = 0, are given in Figure 16.

**Figure 15.** Circuit that is used to show the effect of the RC dampers on the commutation of an (inductive) current.

**Figure 16.** Simulation results for the inductor current in the circuit of Figure 15 for different damper capacitances.

The simulation results show that the damper capacitance absorbs the forced current, delaying the current from flowing inside the SSCB. It also illustrates that, at lower damper capacitances, current overshoot and underdamped high frequency oscillations can occur.
