## 3 Results
Next, based on Equations (1)-(6), 3D graphic surfaces of network parameter non-homogeneity (Figures 4-6) were built for the scheme in Figure 3, which demonstrate the step-by-step development of the cascading failure process. Figure 3: The test scheme of the distribution network with voltage levels of 10, 35, and 110 kV, integrating DG facilities (40 nodes). Analysis of Figure 4 allows us to conclude that in Step 7, the cascading failure process ended with a loss of voltage stability at node 9 at \(U_{\text{cr}}\) < 0.7 \(U_{\text{rated}}\)[77; 78]. In this study, one of the criteria for the transition of the mode to an unacceptable area a\(\backslash\)z the violation of the conditions for ensuring static voltage stability (voltage avalanche). This was possible in circuit-mode situations when the voltage in one or several network nodes dropped below the threshold. Figure 4: A 3D graphic surface of the non-homogeneity of parameters of the distribution network with voltage levels of 10, 35, and 110 kV (40 nodes). Figure 5: A 3D graphic surface of the non-homogeneity of parameters for the distribution network when connecting distributed generation facilities and loads (40 nodes). Figure 6: A 3D graphic surface of parameter non-homogeneity for the distribution network with voltage levels of 10, 35, and 110 kV in the repair scheme (40 nodes). the critical value. In the calculations, it was assumed that the violation of static voltage stability led to the disconnection of the load in the corresponding EPS nodes. Figure 5 illustrates a 3D graphic surface of parameter non-homogeneity for the EPS network (40 nodes), in which we have the following:  
* Initially, the prevention of a cascading failure process by connecting the DG facilities (nodes 26 and 38--20 MW each; nodes 39, 40--10 MW each) was modeled in the first step. As a result, the current in branch 18-23 went down from 271 A (102.2%) to 249 A (92.9%), which made it possible to prevent its shutdown. * Power flows in the branches of the distribution network with a voltage of 10 and 35 kV were created by additionally connecting DG facilities and loads in both types of networks at a value of 60 MW each. In the 10 kV network, they were distributed as follows: node 32 (generator)--5 MW; node 36 (generator)--10 MW; node 33 (load)--15 MW; node 34 (load)--20 MW; node 35 (load)--5 MW; node 37 (load)--5 MW. In the 35 kV network, these values were as follows: node 28 (generator)--10 MW; node 29 (generator)--10 MW; node 30 (generator)--15 MW; node 31 (generator)--10 MW; node 27 (load)--15 MW. Steady states were calculated for the repair scheme of the network, in which eleven components were taken out for repair, including six in the 110 kV network, two in the 10 kV network, two in the 35 kV network, and one 110/10 kV power transformer. In this case, all the operating parameters were within the region of admissible values, as shown in Figure 6.  
Next, the topology and operating conditions were analyzed for the zero-power output from renewable energy facilities due to a sharp change in meteorological conditions under the normal operation and repair scheme of the network. As a result, in the normal 10 kV, 35 kV distribution network scheme, an overload occurred in branch 33-34, and a loss of voltage stability occurred at node 34, at \(U<U_{\text{cr}}\). In this case, the protection device disconnected the overloaded branch 33-34, which sharply increased the current load across the entire adjacent network [79]. Thus, in branch 31-34, the current exceeded the maximum admissible value by 6.18 times (\(I_{\text{limit}}=1920\) A; \(I_{\text{actual}}=11\),876 A). In the repair 10 kV, 35 kV distribution network scheme, an overload occurred in branches 33-34 and 36-37, while a loss of voltage stability was observed at node 31. After the disconnection of the overloaded branches by the protection devices, the maximum current surge was recorded in branch 31-34, which exceeded \(I_{\text{limit}}\) by a factor of 25.