## 17 Non-Traveling Wave (NTW Method Based on Electromagnetic Time Reversal (EMTR):  
Electromagnetic time reversal (EMTR) method for fault estimation in MT--HVdc offers significant advantages, such as tolerance of low sampling rate, non-requirement of accurate fault arrival time, and superiority compared to transient method for high resistance faults [114; 115; 116; 117; 118; 119; 120; 121]. This method is based on reversing the direction of time (\(t\)), given as:  
\[t\to T-t,\ te\ [0,T]\]  
where \(T\) is time delay. Figure 27: Non-traveling wave (NTW) method based on transient values in multi-terminal high-voltage direct current (MT–HVdc) system. In this method, electric fields are recorded at terminals (VSCs). Recorded electric fields are flipped over time and re-transmission of electric fields is achieved from sensing terminals and reflected back to the sensing terminals are observed [114; 117]. Mathematically, electric field \(E(t)\) is given as:  
\[E_{VSC1}(L_{VSC1},\ t)=\ \frac{\varnothing\Big{(}t-\left(\frac{\left\|L_{VSC1}-L_{F} \right\|}{\overline{v}}\right)\Big{)}}{\left\|L_{VSC1}-\ L_{F}\right\|} \tag{127}\]  
\[E_{VSC2}(L_{VSC2},\ t)=\ \frac{\varnothing\Big{(}t-\left(\frac{\left\|L_{VSC2}-L_{F} \right\|}{\overline{v}}\right)\Big{)}}{\left\|L_{VSC2}-\ L_{F}\right\|} \tag{128}\]  
where \(\varnothing(\mathrm{t})\) is the electric potential depending on the temporal-spatial behavior of the fault current, \(v\) is the propagation speed of the electromagnetic field, \(L\) is the length of HVdc line between two VSC stations i.e., \(L_{\mathrm{sc}1}=0,\ L_{\mathrm{sc}2}=L\) as shown in Figure 28 [114; 115; 116]. Thus, time reversed electric fields are:  
\[E_{VSC1}(L_{VSC1},\ T-t)=\ \frac{\varnothing\Big{(}T-\left(t-\left(\frac{\left\|L_ {VSC1}-L_{F}\right\|}{\overline{v}}\right)\right)\Big{)}}{\left\|L_{VSC1}-\ L_{F} \right\|} \tag{129}\]  
\[E_{VSC2}(L_{VSC2},\ T-\ t)=\ \frac{\varnothing\Big{(}T-\left(t-\left(\frac{\left\|L_ {VSC2}-L_{F}\right\|}{\overline{v}}\right)\right)\Big{)}}{\left\|L_{VSC2}-\ L_{F} \right\|} \tag{130}\]  
Re-transmission of electric fields is accomplished and total reflected field is [117]:  
\[ \begin{array}{l}E_{X}\ =\ \frac{\varnothing\Big{(}T-\left(t-\left(\frac{ \left\|L_{VSC1}-L_{F}\right\|}{\overline{v}}\right)\right)-\left(\frac{\left\|L _{X}-L_{F}\right\|}{\overline{v}}\right)\Big{)}}{\left\|L_{VSC1}-L_{F}\right\| \left\|L_{X}-L_{F}\right\|}\\ +\frac{\varnothing\Big{(}T-\left(t-\left(\frac{\left\|L_{VSC2}-L_{F}\right\|}{ \overline{v}}\right)\right)-\left(\frac{\left\|L_{X}-L_{F}\right\|}{ \overline{v}}\right)\Big{)}}{\left\|L_{VSC2}-L_{F}\right\|}\end{array}  \tag{131}\]  
If a fault is set at the point where total electric field is calculated then, \(L_{x}=\ L_{F}\), given as:  
\[E_{X,max}=\ \varnothing(T-t)\left(\frac{1}{\left\|L_{F}-L_{VSC1}\right\|^{2}}+ \ \frac{1}{\left\|L_{F}-L_{VSC2}\right\|^{2}}\right) \tag{132}\]  
There exists a positive correlation between electric field and currents given by:  
\[\left|I\right|=\ \oint\limits_{\mathcal{C}}\sigma Eds \tag{133}\]  
Figure 28: Electromagnetic time reversal (EMTR) based model of high-voltage direct current (HVdc) transmission system. Therefore, currents of HVdc systems are measured and converted into 1--mode and 0--mode components. For bipolar HVdc system, modal components are given by:  
\[i_{VSCI,\,0}(t)=\ \frac{i_{VSCIP}(t)+\ i_{VSCIN}(t)}{\sqrt{2}} \tag{134}\]  
\[i_{VSCI,\,0}(t)=\ \frac{i_{VSC2P}(t)+\ i_{VSC2N}(t)}{\sqrt{2}} \tag{135}\]  
\[i_{VSCI,\,1}(t)=\ \frac{i_{VSC1P}(t)-\ i_{VSC1N}(t)}{\sqrt{2}} \tag{136}\]  
\[i_{VSC2,\,1}(t)=\ \frac{i_{VSC2P}(t)-\ i_{VSC2N}(t)}{\sqrt{2}} \tag{137}\]  
Effects of nonlinear voltage source are eliminated by 0--mode components of current because of opposite direction.