# AC Microgrids Protection: A Digital Coordinated Adaptive Scheme
## 3 Relays Coordination
### DOCR Principle and Setting  
In order to minimize the overall clearing time for different faults in the online procedure, the following objective function for coordinating the DOCR is employed. Based on  
\begin{table}
\begin{tabular}{c c|c} \hline \multicolumn{3}{c}{**Electrical Parameters**} \\ \hline Parameters & Symbol & Value \\ \hline DC Source Voltage & \(V_{DC}\) & 650 V \\ \hline Nominal voltage magnitude & \(V_{MG}\) & 325 V \\ \hline Nominal Frequency & \(f\) & 50 Hz \\ \hline Switching Frequency & \(f_{s}\) & 10 kHz \\ \hline Capacitance of LCL filter & \(C_{f}\) & 25 μF \\ \hline Inductances of LCL filter & \(L_{i}/L_{o}\) & 1.8 mH \\ \hline Load 1 and Load 4 & \(Z_{1}\) and \(Z_{4}\) & 43 f 0.3 h \\ \hline Load 2 and Load 3 & \(Z_{2}\) and \(Z_{3}\) & 124 \(\Omega\), 0.1 h \\ \hline Line 11 & \(Z_{11}\) & 0.4 \(\Omega\), 3.6 mH \\ \hline Line 12 & \(Z_{12}\) & 0.8 \(\Omega\), 1.8 mH \\ \hline Line 23 & \(Z_{23}\) & 0.4 \(\Omega\), 1.2 mH \\ \hline Line 34 & \(Z_{34}\) & 0.8 \(\Omega\), 3.6 mH \\ \hline \multicolumn{3}{c}{**Inner loop coefficients and other control parameters**} \\ \hline Control Parameters & DGU: 1 and 3 & DGU: 2 \\ \hline \(P-\omega\) droop coefficient & 0.001 rad/W.s & 0.002 rad/W.s \\ \hline \(Q-v\) droop coefficient & 0.005 V/Var & 0.01 V/Varr \\ \hline Current proportional/resonance terms & 1000/0.5 & 1000/0.5 \\ \hline Voltage proportional/resonance terms & 120/0.05 & 120/0.05 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of the test system.  
the linear or nonlinear modeling, different optimization tools can be used. The expressed optimization problem is a function of two variables, i.e., the pickup current (\(I_{p}\)) and time dial setting (\(TDS\)).  
\[\min_{\{TDS_{i}^{pm},TDS_{i}^{blk},I_{p_{i}}\}}\quad\sum^{(c,i,j)} \left(t_{i}^{pm}(I_{j,c}^{SC})\right)+\sum^{K}_{k=0}\left(t_{i}^{bk}(I_{k,c}^{ SC})\right) \tag{1}\]  
subject to the IEC 60255 standard inverse time relays characteristic, expressed as follows:  
\[t_{ij}=TDS_{i}\frac{A}{\left(t_{i,j}^{SC}/I_{p_{i}}\right)^{B}-1} \tag{2}\]  
where \(I^{SC}\) is the short circuit current, \(i\) and \(j\) are the relay and fault location indices, and \(k\) stands for the backup relay numbers. Superscripts \(pm\) and \(bk\) stand for primary and backup protection, and \(c\) determines the operation mode of the MG, such that for grid connected has a zero value, while it is set to one for islanded mode. \(t_{ij}\) is the operating time of relay \(i\) when a fault is happened in location \(j\). \(A\) and \(B\) are relay's tripping curve coefficients, whose values are 0.14 and 0.02, respectively, [43]. In addition, the expressed objective function should fulfill the following constraint in both of the grid connected and islanded modes.  
\[t_{i}^{bk}(I_{SC}^{k,c})-t_{i}^{pm}(I_{SC}^{j,c})\geq CTI\quad \forall\;i,\;j \tag{3}\]  
where \(CTI\) stands for the minimum coordination time interval required for discriminating between \(pm\) and \(bk\) for the same fault in \(j\), recommended to be \(CTI\in\) [0.2-0.5] in most of the contexts, as [43]. In the same way, for the pickup setting and TDS of DOCRs, following constraints should be considered.  
\[1.1\cdot I_{i}^{L} \leq I_{p_{i}}^{bk}\leq I_{SC}^{j} \tag{4}\] \[1.1\cdot I_{i}^{L} \leq I_{p_{i}}^{pm}\leq I_{SC}^{j}\] (5) \[TDS_{\min} \leq TDS_{i}^{bk}\leq TDS_{\max}\] (6) \[TDS_{\min} \leq TDS_{i}^{pm}\leq TDS_{\max} \tag{7}\]  
where \(I_{i}^{L}\) is the maximum load current sensed by relay \(i\). \(TDS_{\min}\) and \(TDS_{\max}\) stand for the minimum and maximum values for \(pm\) and \(bk\) relay's TDS.