Methods applied for solving power flow problems may exhibit varying levels of convergence depending on the electrical parameter under analysis used for traversing the topological matrix, whether through a summation of currents or a summation of powers, converging from the Jacobian of the load flow as a sensitivity model. For this reason, the effectiveness of convergence is assessed in the face of the extreme penetration of distributed resources, resulting in percentage differences in injected power that may not become apparent during simulations of their incremental of scaling factors (SF) for the distributed resources into the system.

Over time, power system simulation has evolved through numerical techniques based on efficient developments of power flow computational algorithms [13]. According to [14] efficient algorithms for load flow calculation have been developed and the large availability of computational resources has been exploited to simulate power system simulation using numerical techniques. An argument is presented that traditional load flow methods, Gauss Seidel and Newton Raphson, which are perfectly suitable for load flow analysis in transmission systems, are not suitable for distribution networks, due to factors such as the presence of unbalanced loads, the radiality of the network and the physical condition of the R/X ratio, where the resistance of the lines is comparable in magnitude with the reactance, and this ratio is significant.

For power flow studies in the transmission system, a single-phase representation of the three-phase system is used, which is assumed to be a balanced network in most cases. However, in the case of the distribution system, due to unbalanced loads, the radial structure of the network, and non-transposed conductors, scenarios of an unbalanced system are often simulated [15]. This is why there is a growing need to conduct a three-phase power flow analysis for distribution systems.

In this context, in the future, power flow analysis in distribution systems will require solving unbalanced three-phase power flow for scenarios involving meshed network configurations and radial operation of the distribution network integrated with active distributed resources, depending on the load distribution.

Taking into consideration the general aspects studied and after conducting a literature review for reference systems that integrate Distributed Generation (DG), a study was carried out on reference systems for the integration of renewable energy DG into the medium-voltage grid [17]. The flexibility of the reference systems presented by CIGRE for America and Europe allows for the management of the specific characteristics of each continental region within the same interconnection topology, while maintaining realistic distribution aspects and focusing attention on potential faults during operation.

By using the Benchmarking methodology, it can be concluded that the reference medium-voltage rural distribution network, when considered as a reference point for studies on DERs integration, retains the characteristics of a real network, the data are shared in [17]. The topology is shown in Figure 1.

Figure 1 Based in [17].

The following considerations are presented in the case studies performed:

DERs units were modelled as PQ nodes, operating at unity constant power factor. It is assumed that reactive power (Q) is supplied from the infinite bus.

To achieve power flows with acceptable voltages on each bus, tap changers are essential for voltage stability.

In the operational flexibility of the topology, two main cases have been defined: first. a case without DERs on the existing network. Second, a case using DERs-load flow. From these two main cases, subcases and analysis scenarios are generated based on the type of test applied, as mentioned, and the desired outcome in the increase of distributed generation, as described in Figure 2.

Figure 2 Source: own elaboration.

A sensitivity analysis is then performed to establish which method fails first at convergence, looking at the number of iterations, the loadability of the transformer supplying power to the feeder in each zone, varying discretely for each zone (F1 and F2) scaling factors, depending on how close to non-convergence each method is. In general, the test follows the steps of the flowchart sensitivity analysis towards non-convergence (CSA) in Figure 3.

Figure 3 Source: own elaboration.

Generally, the algorithm finds the convergence limit for one method and proceeds to the next non-convergence (NC) limit. This way, a percentage convergence difference is defined between the two compared methods, as follows in equation 1:

For this analytical case, both methods correspond to the nonlinear-iterative Newton-Raphson method:

Method 1 corresponds to power summation, and method 2 is aimed at current summation.

Assumptions:

The convergence difference always exists.

It is important to carefully maintain comparability between topologically comparable scenarios of the network to perform an operational assessment of the power flow results. This includes considering the base state (zero state) of the network and the level of power injection from distributed resources.

As shown in Figure 4, the N-R method using current equations requires fewer iterations to reach the result and, in its trend, a lower number of iterations than the N-R method using power equations (classic).

Figure 4 Source: own elaboration.

Table 1 summarizes the main difference values at convergence for given scaling factor conditions (NC: Not convergence):

Table 1

Configuration	Zone Fl (Mesh)	Zona F2 (Radial)	Zona Fl (Mesh)	Zona F2 (Radial)	Percentage difference (%)	Remarks
Method	N-R Classic_Power Eq.		Is-R Current Equations.
Same scaling factor	2.31	2.31	2.36	2.36	2.12
Different scaling factor	1	3.16	1	3.3	4.24
	2.31	1	2.36	1	2.12	Note -The Kmit is fixed and there is certainty that it cannot be moved any further.
	2.29	3.16	2.36	3.16	2.97
	2.31	3.11	2.31	3.3	5.8
	NC	NC	2.36	3.3		For a value of 3.3 in the radial circuit method 1 never converges, as its maximum is 3.16.

Source: own elaboration.

For a half-meshed or weakly meshed characteristic, both methods use the same number of iterations before reaching their non-convergent limit, managing to increase the load up to 2.36 times with a current equation method compared to 2.31 times for the classic algorithm, approximately a 2.1% percentage difference in load variation. This tolerance contrasts with the 4.2% derived from the same variables for the radial circuit, which implies that the current equation algorithm accepts greater load variation in both radial and meshed cases. It's worth mentioning that the percentage difference for different scaling factors can reach up to 5.8% between the two analyzed methods.

The results are presented below for the most meshed case including the progressive penentration of resources distributed to the network in Figures 5 and 6.

Figure 5 Source: own elaboration.

Figure 6 Source: own elaboration.

In this result, the rightward shift is not so crucial, which is basically due to the availability of DG active power and the progressive increase of the scaling factors. What is observed is that the N-R method using power equations no longer converges in the same number of iterations before non-convergence, while the N-R method using current equations does. This allows us to conclude that the convergence associated with an increase in distributed DG in a distribution network is better tolerated by the method 2, which follows the topology, under these study conditions.

Reconfiguring a radial to meshed circuit or the connections between the two feeders varies the R/X constant of the system. However, as the effect of R/X variation was reviewed, this requires a significant constraint of the differences that can arise between one-line configuration or another from its physical parameters, section by section.

At this point, a progressive increase of positive sequence resistance was performed on the system type lines, in order to review the convergence behavior, plotted in Figure 7.

Figure 7 Source: own elaboration.

A resistive increase has two effects, an increase in network losses and thus an increase in network loadability. However, in a complementary way, the inyected DER surplus covers this congestion. This balance caused by the injected power allows the non-convergence to be more flexible up to 3 times in case 2A, before its limit, which is simulated in a network or feeder without distributed resources, as in case 1B.

When a new feeder line or tie-line is added to the system, the grid sensitivity to voltage and load responses changes significantly in the local area of influence, and this is evidenced by the reliability of the system to evacuate DERs power injection to neighbouring nodes of the application point. This last results in a decongestion of the feeder head and a redistribution of power to consumption areas.