Probabilistic Estimation of During-Fault Voltages of Unbalanced Active Distribution: Methods and Tools

Abstract

In low-voltage (LV) distribution networks, system operating conditions are always unbalanced due to the unpredictability of the load demand in each phase, coupled with a potentially asymmetrical network structure due to different phase conductors’ sizes and lengths. The widespread diffusion of distributed generators (DGs) among network users has significantly contributed to reducing the overall load of the electrical system, but at the cost of making voltages slightly more unbalanced. In this article, an LV distribution test network equipped with several single-phase DGs has been considered, and all During-Fault Voltages (DFVs) have been studied, according to each possible type of short circuit. To provide a measure of the asymmetry of unsymmetrical voltage dips, three different indices based on the symmetrical components of the voltages have been considered; moreover, the Monte Carlo simulation (MCS) method has allowed for studying faults and asymmetries in a probabilistic manner. Through the probability density functions (pdfs) of the DFVs, it has been possible to assess the impact of single-phase DGs on the asymmetry of bus voltages due to short-circuits.

1. Introduction

To ensure the reliable operation of electrical systems, distribution networks are typically sized and regulated to achieve highly symmetrical and balanced voltages and currents in three-phase configurations. However, deviations from these ideal conditions give rise to Power Quality (PQ) problems. Focusing on voltage quality at the point of delivery, these PQ issues include disturbances classified as events (voltage dips, voltage swells, interruptions) and variations (RMS variations, frequency variations, unbalances [1]). Voltage dips are recognized as one of the most critical disturbances due to the huge economic damage they can cause to industrial premises or advanced users like data centers [2,3,4,5]. The causes can range from the insertion of transformers into the network to the starting of large induction motors, as well as short circuits on the distribution network. Unbalanced voltage dips due to unbalanced faults are of particular interest in the literature as they are the most frequent, and their occurrence may cause disruption of industrial equipment and damage.

In the study of three-phase networks, the models and methods used assume the perfect symmetry of phase voltages and currents. However, a much more realistic approach is to adopt unbalanced models in phase coordinates, starting from the three-phase load flow, which returns the electrical state of the system for each bus, including magnitude and angle. Moreover, the widespread adoption of single-phase DG units by end users of the network further increases the levels of voltage unbalance in pre-fault conditions.

The classification and the quantification of occurrence and severity are expressed through the well-known site and system indices, such as VDA (Amplitude Voltage Dip), SAVDA (System Average Voltage Dip Amplitude), and SARFI (System Average RMS Variation Frequency Index) [6,7,8]. In the literature, the study of voltage dips has been carried out in countless ways. Several approaches have focused on voltage dips at the nodes of the electric systems [9,10,11,12,13]. Methods proposed in [9,10] both used real-world data on voltage dips to predict the frequency of future occurrences. The methodology in [9] involved correlating a large quantity of measured dips with the MV busbars of electrical stations. Conversely, the authors of [10] collected their data from 10 PQ monitors that were specifically placed on buses operating at 10 kV, 35 kV, and 110 kV. The prediction of voltage sag amplitude was also faced by methods based on data fusion [11]. In [12,13], the problem of ascertaining the system where the fault originates from a dip measured in buses of an interconnected electric system was solved by proposing methods that use only the measured residual voltage and the time of start. The study of voltage dips occurring along the lines was also conducted in [14,15]. In [14], the extension of the Fault Position Method (FPM) was proposed for actual transmission systems to solve real problems with acceptable accuracy and reduced computational times; in [15], a method for estimating the voltage sag performance at non-monitored buses using the simulated data collected at a limited number of metering points obtained simulating faults both at buses and along the lines.

Probabilistic approaches were the most adequate methodologies for considering several uncertainties, such as those deriving from the fault impedance, the fault location, and the pre-fault conditions of the network.

Ref. [16] presents a method for evaluating the equivalent internal potential of certain large DG units due to possible faults in specific nodes of the examined electrical system. The low-voltage-ride-through is the typical fault response behavior of photovoltaic systems, which proves to be quite stochastic: this characteristic is reflected in the probable values of short-circuit currents, especially in the case of large-scale systems.

In [17], a probabilistic approach was applied to calculate the pdf of SARFI 50-70-90 by randomly generating the fault location of the network, the type of fault, and the phases involved. This study led to a potential application for PQ in terms of network robustness analysis, probabilistically evaluating its performance with respect to voltage dips.

In [18], a probabilistic approach was used for a balanced three-phase network under short-circuit conditions equipped with two main DGs, one directly connected to the network and the other connected to the network through a converter. In this study, the line impedance, the position of the three-phase fault, the network loads, and the power injected into the system by the DGs were considered as the input variables sampled randomly, according to their own pdf. The results made it possible to calculate the pdf of the magnitudes of the fault currents circulating in the different components, providing a possible aid in the choice and design of protective devices.

Papers [19,20] also employ a probabilistic approach regarding the independent input variables and the impact of the DGs’ low-voltage-ride-through on the network.

In [21], the same random input variables described in [18,19] are considered, again, through the MCS method. Specifically, Ref. [21] studies the pdf of the zero-sequence fault current for line-to-ground faults in typical low-grounding systems, varying between multiple possible scenarios referring to the pdfs of the input random variables.

Ref. [22] shows a probabilistic short circuit analysis conducted on a test system, where the random samples of the input variables were the load power demand and the fault location: there, it does not consist of a specified bus, but along a particular line of the grid. Furthermore, two MCSs have been performed and compared, where one contained a linearized version of the equations of three-phase load flow. Results show different pdfs for different types of faults, such as line-to-ground, three-phase-to-ground, and phase-to-phase faults, with negligible errors between the two MCSs.

In [23], an energy index that can lead to the most sensitive pivotal lines of the grid is shown, and it can describe the sensitivity of the buses to the short circuits occurring in the network by assessing the severity of voltage dips and swells.

All the previous references and, in general, the literature itself mostly refer to balanced electrical networks, and they usually refer to studies about the fault location, the impact and the contribution of the DGs, and the computation of the short-circuit currents, which may vary depending on the pre-fault voltages of the grid: all these studies certainly consider the stochastic nature of the faults, but they focus mainly on their effects on the network. The contribution of this paper extends the work presented in [24] to a more probabilistic and inclusive perspective. It delves deeper into the behavior of the proposed indices, assessing their utility in evaluating the imbalances of DFVs with respect to all possible short circuits that may occur in the network. In other words, we aim to determine whether these indices can be possibly used to compare the DFVs of two buses, observing whether they are more unbalanced when a fault occurs on one bus rather than another, assuming the same type of short circuit occurs. The proposed analysis presents a data-driven approach, leveraging the substantial data generated by Monte Carlo simulations. A key strength of this analysis is that its foundation lies entirely in data generated from the system’s model. This characteristic will be instrumental in conducting a future correlation analysis between various operating conditions and the resulting MD, MI, and MO outcomes. Eventually, the probabilistic perspective allows us to evaluate whether these indices can also account for the stochastic effects of DGs, as [25] shows, normalizing the variation in any variable of interest, in the presence or absence of DGs. Eventually, comparing this paper with the preceding paper [24], we expanded our analysis by including additional fault cases to secure a wider range of results. Unlike our previous paper in [24], where faults were exclusively at bus #7, we now consider multiple fault positions, including the node where a DG unit is installed.

The remaining part of this paper is organized as follows: Section 2 describes the main method and modeling of the probabilistic analysis that has been conducted, describing the main steps of the MCS and the contribution of the DGs in fault conditions; Section 2 also describes new indices to apply to the DFV; Section 3 shows the numerical applications, on the test system CIGRE [26].

2. Methods

In this Section, the main methods and models used to conduct the analysis in fault conditions of a three-phase unbalanced system are reported. In particular, the Fault Position Method (FPM) in its three-phase version is described in Section 2.1. The iterative procedure used in [24] to consider the contribution of the DGs during faults is recalled in Section 2.2. The probabilistic MCS—method implemented in this study—is presented in Section 2.3. Finally, the proposed method for assessing the DFV imbalances is presented in Section 2.4.

2.1. Three-Phase Fault Position Method

Consider a general passive three-phase distribution network, unbalanced in loads and structure. For a short-circuit at a bus k, let us assume the following [22]:

-

${\left[{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{1}}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{2}}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{3}}\right]}_{\mathrm{d}\mathrm{f}}^{\mathrm{T}}$ and ${\left[{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{1}}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{2}}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{3}}\right]}_{\mathrm{p}\mathrm{f}}^{\mathrm{T}}$ are, respectively, the (3 × 1) vector of during-fault phase-voltages of a given bus k and the pre-fault phase-voltages of the same bus, for the first, second, and third phases;

-

${\left[{\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{1}}{\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{2}}{\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{3}}\right]}_{\mathrm{d}\mathrm{f}}^{\mathrm{T}}$ is the (3 × 1) vector of short-circuit currents at the bus k;

-

${\left[{\dot{\mathbf{Z}}}_{\mathbf{k}\mathbf{k}}\right]}_{\mathrm{s}\mathrm{c}}$ is the (3 × 3) short-circuit impedance submatrix for the bus k, which can be obtained by inverting the admittance matrix of the whole network and is defined as follows:

$${\left[\begin{array}{ccc}{\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{11}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{12}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{13}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{21}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{22}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{23}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{31}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{32}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{33}}\end{array}\right]}_{\mathrm{s}\mathrm{c}}$$

(1)

The diagonal terms of (1) represent the equivalent circuit impedance of the network seen by the bus k, and the other terms represent the electromagnetic coupling impedances of one phase to each other, referring to the same bus.

For a three-phase unbalanced network, it always results in the following:

$${\left[\begin{array}{c}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{d}\mathrm{f}}={\left[\begin{array}{c}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{p}\mathrm{f}}+{\left[\begin{array}{ccc}{\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{11}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{12}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{13}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{21}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{22}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{23}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{31}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{32}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{33}}\end{array}\right]}_{\mathrm{s}\mathrm{c}}{\left[\begin{array}{c}{\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{f}}$$

(2)

In this paper, the following types of short-circuits are considered: three-phase-to-ground fault and line-to-ground fault. Starting from Equation (2), for LLLG faults, it results in the following:

$${\left[\begin{array}{c}{\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{g}}=-{{\left[{\left[\begin{array}{ccc}{\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{11}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{12}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{13}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{21}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{22}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{23}}\\ {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{31}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{32}}& {\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{33}}\end{array}\right]}_{\mathrm{s}\mathrm{c}}\right]}^{-\mathbf{1}}\left[\begin{array}{c}{\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{V}}}_{\mathbf{k}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{p}\mathrm{f}}$$

(3)

For LG faults, it results in the following:

$${\overline{\mathbf{I}}}_{\mathbf{k},\mathbf{l}\mathbf{g}}^{\mathbf{n}}=-{\displaystyle \frac{{\overline{\mathbf{V}}}_{\mathbf{k},\mathbf{p}\mathbf{f}}^{\mathbf{n}}}{{\dot{\mathbf{z}}}_{\mathbf{k}\mathbf{k}}^{\mathbf{n}\mathbf{n}}}}$$

(4)

where n represents the considered faulted phase (n = 1, 2, 3).

Once the value of the short-circuit currents is known, the DFV vector for any type of fault can be obtained as follows:

$${\left[\begin{array}{c}{\overline{\mathbf{V}}}_{\mathbf{1}}^{\mathbf{1}}\\ {\overline{\mathbf{V}}}_{\mathbf{1}}^{\mathbf{2}}\\ .\\ .\\ {\overline{\mathbf{V}}}_{\mathbf{N}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{d}\mathrm{f}}={\left[\begin{array}{c}{\overline{\mathbf{V}}}_{\mathbf{1}}^{\mathbf{1}}\\ {\overline{\mathbf{V}}}_{\mathbf{1}}^{\mathbf{2}}\\ .\\ .\\ {\overline{\mathbf{V}}}_{\mathbf{N}}^{\mathbf{3}}\end{array}\right]}_{\mathrm{p}\mathrm{f}}+{\left[\begin{array}{cccc}{\dot{\mathbf{z}}}_{\mathbf{11}}^{\mathbf{11}}& {\dot{\mathbf{z}}}_{\mathbf{11}}^{\mathbf{12}}& ..& {\dot{\mathbf{z}}}_{\mathbf{1}\mathbf{N}}^{\mathbf{13}}\\ .& .& ..& .\\ .& .& ..& .\\ .& .& ..& .\\ {\dot{\mathbf{z}}}_{\mathbf{N}\mathbf{1}}^{\mathbf{31}}& .& ..& {\dot{\mathbf{z}}}_{\mathbf{N}\mathbf{N}}^{\mathbf{33}}\end{array}\right]}_{\mathbf{s}\mathbf{c}}{\left[\begin{array}{c}\mathbf{0}\\ .\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{1}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{2}}\\ {\overline{\mathbf{I}}}_{\mathbf{k}}^{\mathbf{3}}\\ .\\ \mathbf{0}\end{array}\right]}_{\mathrm{f}}$$

(5)

By positioning the fault in every bus of the grid, Equation (5) leads to the DFV matrix whose (i,k) element represents the voltage on bus i when a fault occurs on bus k. Equation (5) allows for evaluating the performance of the network not only in terms of voltage dips but also in terms of voltage swells.

2.2. Three-Phase Fault Position Method with Distributed Generators

The model of the inverter has not been developed in this paper; instead, it has been considered that during fault events, the inverter supports the grid with a maximum of 1.5 times the rated AC current of the converter itself, according to the standards released by the local distribution system operator. To consider this behavior, a DG can be represented by two models:

• The reactance model, where the DG is represented by an equivalent reactance whose value becomes updated with an iterative procedure, based on the DFV at the DG supply terminal. The admittance matrix of the network must be updated at every iteration;
• Current generator model, where the DG is represented by an equivalent ideal current source. The DG unit is modeled by a current generator whose value is computed by applying an iterative procedure. At each iteration, the value of the injected current is modified as a function of the calculated DFV at the DG supply terminal to guarantee the reference active power. The admittance matrix of the network does not change.

In this paper, the current generator model is adopted. To calculate the DFVs at every bus of the grid in the presence of multiple DGs, the superposition principle is applied. Figure 1 shows the flow chart of the proposed method, which includes the iterative procedure. For every type of fault, the main steps of the flow chart are as follows:

-

Step 1. Three-phase Load-Flow is performed to know the pre-fault voltages at all the buses of the grid;

-

Step 2. The short-circuit impedance matrix of the network is computed;

-

Step 3. Three-phase FPM is applied by computing and calculating all DFV of the network without DGs;

-

Step 4. For the assigned P* and for the values of DFV at the DG supply terminal obtained at Step 3, the current value injected by the DG modeled by a current generator is obtained;

-

Step 5. The voltage contribution coming from the DG is calculated by multiplying the injected current value of Step 4 and the short-circuit impedance matrix of the network. The DFV matrix of Step 3 is updated.

Steps 4 and 5 are repeated iteratively until convergence is achieved. Convergence is determined by the error between the injected real power P from the DG unit and the variation in the DFV matrices in two consecutive iterations. When these errors fall below a predefined threshold, the iterative process terminates. In case of multiple DGs, the risk of divergence due to the increase in variation in the DFV matrices in consecutive steps is highly reduced by the limitation imposed on the maximum output current of the inverter.

2.3. Advanced Indices as Tools for Ascertaining Unbalanced Voltage Dips and Swells

The indices used to classify and quantify the severity of unbalanced voltages primarily concern unaffected phase voltages, meaning those not influenced by temporary events like voltage dips and swells. The basic formulation of the unbalance index relates to IEC (International Electrotechnical Commission). Once the phasors of the positive V₁ and negative V₂ sequence components are defined, the well-known Voltage Unbalance Factor (VUF) is obtainable by

$$\mathrm{V}\mathrm{U}\mathrm{F}={\displaystyle \frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}}.100$$

(6)

Further indices are available in the specific technical and scientific literature [26], like the Line Voltage Unbalance Ratio (LVUR), defined by NEMA (National Electrical Manufacturers Association); the Unbalance Factor, defined by CIGRE (Conseil International des Grands Réseaux Électriques); and the Phase Voltage Unbalance Ratio 1 and 2 (PVUR1, PVUR2), defined by IEEE (Institute of Elctrical and Electronic Engineers).

The original formulation of the aforementioned indices cannot be directly applied when one or more phase voltages are subject to temporary dips or swells. When the unbalanced factors are applied to sagged voltages, typically, the denominator approaches zero, driving the values toward infinity. The values saturate at very high levels, making it impossible to adequately differentiate between sites for comparative purposes. To address this limitation, new indices have been proposed in [24] that are capable of quantifying unbalanced phase voltages even when affected by dips or swells. This includes the voltages observed during symmetrical or unsymmetrical faults in unbalanced systems, modeled and computed as shown in the previous Sections.

Referring to a bus k, it was proposed to use the pre-fault voltage positive sequence at the denominator of the VUF (#); at the numerator, the positive sequence of the DFV of the same bus is used. This leads to the definition of the ${\mathrm{M}}_{\mathrm{D},\mathrm{k}}$ index (7):

$${\mathrm{M}\mathrm{D}}_{\mathrm{k}}={\displaystyle \frac{{\left|{\overline{\mathrm{V}}}_{\mathrm{d},\mathrm{k}}\right|}_{\mathrm{d}\mathrm{f}}}{{\left|{\overline{\mathrm{V}}}_{\mathrm{d},\mathrm{k}}\right|}_{\mathrm{p}\mathrm{f}}}}$$

(7)

By placing the DFV negative sequence at the numerator once, and the DFV zero sequence too, ${\mathrm{M}}_{\mathrm{I},\mathrm{k}}$ and ${\mathrm{M}}_{\mathrm{O},\mathrm{k}}$ indices are

$${\mathrm{M}\mathrm{I}}_{,\mathrm{k}}={\displaystyle \frac{{\left|{\overline{\mathrm{V}}}_{\mathrm{i},\mathrm{k}}\right|}_{\mathrm{d}\mathrm{f}}}{{\left|{\overline{\mathrm{V}}}_{\mathrm{d},\mathrm{k}}\right|}_{\mathrm{p}\mathrm{f}}}}$$

(8)

$${\mathrm{M}\mathrm{O}}_{\mathrm{k}}={\displaystyle \frac{{\left|{\overline{\mathrm{V}}}_{\mathrm{o},\mathrm{k}}\right|}_{\mathrm{d}\mathrm{f}}}{{\left|{\overline{\mathrm{V}}}_{\mathrm{d},\mathrm{k}}\right|}_{\mathrm{p}\mathrm{f}}}}$$

(9)

The definitions (7)–(9) require the measurement of the phase voltages both in the pre-fault condition and during the fault. Applying the method proposed in this paper, the pre-fault phase voltages are the direct outcomes of Step 1 of the procedure depicted in Figure 1; the phase voltages during the fault are the outcomes of the FPM.

2.4. Monte Carlo Simulations Method

In this paper, the Monte Carlo simulation (MCS) probabilistic method has been employed to accurately assess the probability of the impact of single-phase distributed generators (DGs) on phase voltage unbalance under short-circuit conditions, while also considering the stochastic nature of faults. Figure 2 shows the flow chart of the MCS method applied in this paper.

• The MCS method is well known in the literature and can be summarized in the following steps:
• Initially, the probability density functions (pdfs) of the independent variables are assumed to be known, and the computational algorithm is initialized with these given values;
• Next, the analytical calculation of the dependent variables of interest is performed;
• A new iteration begins. The number of iterations is chosen to ensure an accurate determination of the pdf of the relevant variables.

For the presented case study, the Monte Carlo simulation (MCS) method utilizes Gaussian probability density functions (PDFs) to model both the power injected by each distributed generator (DG) and the power demanded by loads within the residential subnetwork. This standard initial assumption in probabilistic power system analyses is primarily supported by the Central Limit Theorem, which suggests a normal distribution for aggregated independent random variables, and the Gaussian’s maximum entropy property, which implies minimal implicit assumptions regarding the underlying data.

3. Numerical Applications

This Section presents the numerical results of 5000 Monte Carlo iterations performed on the test network [25], fixing bus R7 as the fault node and R4 as the observation bus, regardless of the short-circuit types. The latter are Line-to-Ground fault (LG) and Three-Phase-to-Ground fault (LLLG). Section 3.1 describes the LV Distribution Network where the short-circuit analysis is performed. Section 3.2 presents short-circuit analysis results.

3.1. Distribution Test Network

Figure 3 shows the reference test network [25] studied in this paper, with the indication of the installed DGs and of the fault locations.

Table 1. Location, power size, and phase of the installed DGs.

Bus	Mean Value of Injected Power	Injecting on Phase (1-2-3)
R8	15 kWp	1
R11	15 kWp	1
R15	15 kWp	1
R16	15 kWp	1
R17	15 kWp	1
R18	15 kWp	1

presents the buses where the DGs are installed, along with the mean value of the injected power; the standard deviation (Std Deviation) is assumed to be 10% for all busbars. The network load demand is set to its default value from the test network [25], with a standard deviation of 10%.

3.2. Short-Circuit Analysis Results

New indices based on symmetric components have been proposed in [24] to characterize voltage asymmetries in fault conditions of unbalanced distribution networks. Results have shown that the occurrence of voltage dips and swells can always be ascertained by approximate thresholds found in this study. By observing the values of the proposed indices, it is possible to discern voltage dips from swells and to estimate the intensity of the occurred event.

The first findings in [24] motivated us to conduct a more comprehensive analysis on the probabilistic frame. Several tests were performed in the next step, and the following cases are discussed:

• Case A: Faults on bus R7 and observing bus R4;
• Case B: Faults on bus R13 and observing bus R13; faults on bus R17 and observing bus R7.

Case A

Figure 4 shows the DFVs of the bus R4 for a Line to the Ground (LG) on the first phase of bus R7.

Figure 4a shows the pdf of the first phase voltage, V1, which is also the one that is faulted. It is significantly different from the other phase voltages. Figure 4b for phase 2 and Figure 4c for phase (3) show that they have approximately similar values under frequent conditions of overvoltages. The bimodality of the pdf V1 is caused by the presence of the DGs (which are injecting real power only in phase 1) and by the considered variations in the load demand, which also causes more imbalances in the grid, both in pre- and during-fault conditions.

Figure 5 shows the DFVs on bus R4 for a three-phase-to-ground fault (LLLG).

As can be noted, balanced faults in a slightly unbalanced electrical system can cause different values of the phase voltages: Figure 5, indeed, shows undervoltage conditions for phase 1 (Figure 5a), and similar values between phases V2 (Figure 5b) and V3 (Figure 5c); the highest values are of V2 and V3, also for the support given by the DGs during faults.

The following figures show the pdfs of the proposed indices, MD, MI, and MO, at bus R4. Figure 6 shows the pdfs for LG faults, while Figure 7 shows the pdfs for LLLG faults. The indices are normalized to per unit (p.u.) values based on the nominal voltage, spanning a range of [0, 1] p.u.

Of almost 2.5 million LG fault scenarios simulated on bus R7, a voltage swell was observed in at least one phase of the observation bus R4. An MD (Figure 7a) value approximating 1 p.u., concurrently with MI (Figure 7b) and MO (Figure 7c) values close to zero, denotes symmetrical operation, even in the presence of fault conditions. To discriminate voltage swells, empirical thresholds can be established. We verified that an MD value not exceeding 0.85 (MD ≤ 0.85) and an MO value exceeding 0.40 (MO ≥ 0.40) were effective in 90.22% of instances. The cases not classified by both proposed indices represent situations where only one of the two thresholds was satisfied. The following Figure 8a,b shows the distribution of the amplitude of the voltage swells on the x-axis and the number of cases on the y-axis classified and not classified by MD and MO, respectively.

From Figure 8a,b, it is of paramount importance to note that for the proposed indices, the higher the value of voltage swells observed on a bus, the higher the correlation to the empirical thresholds found (

Table 2. Mean and standard deviations of some variables of interest.

Event/Variable	Mean [p.u.]	Std Deviation [p.u.]
Voltage swell values classified by MD—MO	1.166	0.032
Voltage swell values not classified by MD—MO	1.143	0.017

).

In other words, the proposed indices, MD and MO, demonstrate a strong potential and high probability in discerning which buses in the electrical system, in the occurrence of an LG fault at a certain bus, exhibit significant voltage swells compared to those that exhibit lower levels. The following tables sum up the DFVs and the proposed index values in terms of mean and standard deviation at the bus of observation R4, while LG faults occur on bus R7 (

Table 3. (a). LG1 at bus R7—observed bus: R4. (b). LG2 at bus R7—observed bus: R4. (c). LG3 at bus R7—observed bus: R4.

(a)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	0.4381	0.0128	MD	0.8777	0.0027
V2	1.0513	0.0039	MI	0.1368	0.0034
V3	1.0771	0.0038	MO	0.2894	0.0072
(b)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	1.1165	0.0051	MD	0.8811	0.0007
V2	0.4197	0.0007	MI	0.1188	0.0010
V3	1.0581	0.0021	MO	0.3356	0.0028
(c)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices [p.u.]	Mean [p.u.]	Std Deviation [p.u.]
V1	1.0691	0.0044	MD	0.8633	0.0005
V2	1.0343	0.0025	MI	0.1379	0.0019
V3	0.4260	0.0004	MO	0.2936	0.0013

) in different phases (LG1, LG2, LG3), respectively. In all three cases, the indices MD, MO, and MI quantify the unsymmetrical dips and swells; in particular, when the fault occurs in LG2 at the bus R7, the highest value of MO indicates the presence of the highest swell on V1 (value equal to 1.1165).

Case B

The following cases are reported:

-

LG faults occur on bus R13, observing R13 (

Table 4. (a). LG1 at bus R13—observed bus: R13. (b). LG2 at bus R13—observed bus: R13. (c). LG3 at bus R13—observed bus: R13.

(a)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	0.0570	0.0045	MD	0.7934	0.0009
V2	1.2105	0.0026	MI	0.2288	0.0010
V3	1.1213	0.0031	MO	0.5465	0.0020
(b)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	1.1905	0.0070	MD	0.7977	0.0011
V2	0.0074	0.0006	MI	0.2002	0.0016
V3	1.2174	0.0028	MO	0.6054	0.0028
(c)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	1.2253	0.0051	MD	0.7751	0.0009
V2	1.0841	0.0032	MI	0.2239	0.0022
V3	0.0067	0.0005	MO	0.5584	0.0014

);

-

LG faults occur on bus R7, observing R17, provided with a DG (

Table 5. (a). LG1 at bus R7—observed bus: R17. (b). LG2 at bus R7—observed bus: R17. (c). LG3 at bus R7—observed bus: R17.

(a)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	0.1237	0.0191	MD	0.8034	0.0047
V2	1.1411	0.0087	MI	0.2297	0.0063
V3	1.1917	0.0090	MO	0.5537	0.0152
(b)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	1.2707	0.0107	MD	0.8080	0.0015
V2	0.0141	0.0016	MI	0.1932	0.0014
V3	1.1590	0.0042	MO	0.6289	0.0051
(c)
DFVs	Mean [p.u.]	Std Deviation [p.u.]	Proposed Indices	Mean [p.u.]	Std Deviation [p.u.]
V1	1.1669	0.0083	MD	0.7749	0.0013
V2	1.1038	0.0051	MI	0.2168	0.0030
V3	0.0204	0.0016	MO	0.5474	0.0025

).

Table 4 and Table 5 summarize the statistics of the DFVs and of the proposed indices in terms of mean and standard deviation. Table 4 and Table 5 demonstrate the effectiveness of the indices MD and MO to discriminate the voltage swells.

In all cases when voltage swells occur, the MD < 0.85 and MO > 0.40. However, Table 3 illustrates a singularity (MO < 0.4 and MD > 0.85), resulting in an unexpected voltage swell of V1.

4. Conclusions

This paper provides a comprehensive analysis of unbalanced conditions in a low-voltage distribution network, covering both pre-fault and post-fault operations. The network was modeled in phase coordinates, and a three-phase load flow was executed to obtain pre-fault phase voltages at each bus. Distributed generation units were integrated using a current source model, applying an iterative procedure for power injection from each inverter during-fault conditions. We extended the analysis into a probabilistic perspective for short-circuit events, employing the Monte Carlo simulation method.

This paper’s main innovative aspect is the proposal of novel MD, MI, and MO indices for quantifying the severity of unbalanced voltage dips and swells. Beyond this, the analysis adopts a data-driven approach, efficiently utilizing the extensive data from Monte Carlo simulations, which are wholly based on the system’s model. A key benefit of our methodology is the use of three-phase load flow, which will allow for the segregation of unbalances attributed to the network’s inherent form from those originating from asymmetrical faults. A key advantage of our approach is that the Monte Carlo simulation’s computational efficiency and precision remain unaffected by the type of probability density function used. This will allow for straightforward incorporation of non-Gaussian probability density functions, diverse phase combinations, multiphase distributed generation, and seasonal variations.

The main outcomes are as follows:

• For three-phase-to-ground faults, the MD index alone proves sufficient for classifying severe voltage unbalanced dips;
• For line-to-ground faults, the simultaneous behaviors of the MD and MO indices are critical. Specifically, an MD value below 85% in conjunction with an MO value exceeding approximately 40% adequately ascertains the presence of voltage swells.

Further studies are underway:

• To effectively differentiate between the impacts of network imbalance and asymmetrical faults, perform correlation analyses between the proposed indices and various system operating conditions; this will be facilitated by utilizing three-phase flow models of the pre-fault conditions;
• To evaluate and potentially confirm the heuristic thresholds identified in this work;
• To correlate extreme indicator values of the proposed indices with the specific input conditions that generated them;
• To involve computing the proposed indices in three-phase electrical networks that exhibit significant load and structural imbalances.

Eventually, different inverter capabilities and fault ride-through strategies will be tested in future work to assess any influence they have on the obtained outcomes.