A Novel Methodology to Estimate Probability Density Function of Voltage Sag Duration and Failure Rates on Power Distribution Systems

Voltage sags and power interruptions are important power quality problems that affect sensitive customers, mainly because they cause annual massive economical losses to the industrial sector as a result of unexpected production process disruptions. In this sense, to propose corrective and preventive measures and improve the power quality of the distribution systems, stochastic methodologies have been proposed in the literature to estimate annual voltage sags and power interruptions. However, these methodologies, generally, use typical cumulative distribution functions of voltage sag duration (PSgD), which may not reflect the real estate of the network under study. To solve this constraint, this paper proposes a novel methodology to estimate a proper PSgD considering information of the distribution network (i.e., topology and coordination schemes of the protection system) and the stochastic behaviors of short-circuits that can affect the distribution system. Moreover, the proposed methodology allows estimating permanent failure rates and average repair time considering known or expected values of reliability indicators. The results show that this proposed methodology is capable to adapt from an initial PSgD curve to another one with fidelity, in order to achieve real values of expected annual power interruptions.


INDEX TERMS
Annual average hours that isolated sections are without service. R L/N Ratio of annual LDI to the total annual events among voltage sags, SDI and LDI for the whole grid. VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ R S/L (s) Ratio of P SDI (s) to P LDI (s) for a s. N sag/yr Annual voltage sags for the whole grid. N SDI/yr Annual SDI for the whole grid. N LDI/yr Annual LDI for the whole grid.

I. INTRODUCTION
Voltage sags and power interruptions are common electrical disturbances that affect the power systems. Their disruptive effects on sensitive equipment and production processes represent significant financial losses to commercial and industrial customers [1].
Magnitude and duration are the main features to characterize voltage sags. To assess voltage magnitude, probabilistic methods have been proposed based on analytical formulation [2] or Monte Carlo simulations (MCS) [3]. For analytical formulation, mathematic expressions are proposed to calculate the remaining voltage due to faults at buses and along the lines [4]. However, these expressions assume some simplifications, which may lead to less accurate results. Probabilistic methods based on MCS allow inclusion of uncertainties of each variable that takes part of the short circuit faults, so the results are closer to the real, but they need more computational efforts. In turn, voltage sag duration (d sag ) depends on the fault clearance process and mostly on the protection schemes performance. The prediction of d sag is treated in the literature by using fixed values or probability distribution functions (e.g., normal [5], lognormal, Weibull [6], etc.) to represent the fault clearance time.
Power interruptions can be classified as short and long durations [7]. Short duration interruptions (SDIs) are caused by reclosers, which are activated to eliminate momentary faults and avoid unnecessary permanent power interruptions. Long duration interruptions (LDIs) are caused by permanent faults, and their durations are linked to the performance of the power system operation to restore the normal energy supply to faulty section. In this sense, indicator as System Average Interruption Frequency Index (SAIFI) and System Average Interruption Duration Index (SAIDI) are commonly used to measure distribution system reliability regarding electrical faults.
In the literature [3], [4], [5], [6], [7], [8], [9], d sag , permanent failures rates (λ LDI ) and annual average hours in which the isolated sections run out of service (t Is ) are important parameters used by researchers to assess reliability indicators, such as SAIFI and SAIDI. Then, socioeconomic parameters, such as energy not supplied (ENS) and financial losses due to process trips (FL PT ), are used by electric planners to estimate the distribution network performance. In particular, the parameters d sag , λ LDI and t Is are usually obtained from statistical analysis of short circuits that occur throughout the distribution network. However, studies reported in [10] and [11] pointed out difficulties to apply this statistical approach. In general, it is very challenging to have enough raw data to get reliable, meaningful and conclusive results. In effect, the failures in components (lines, transformers, etc.) that may cause short circuits are dependent on different factors, such as weather conditions, component type, size and geographical location. Thus, a component may not fail according to the history of similar components.
In this paper, to eliminate such difficulties, a novel methodology is proposed to estimate and adapt a cumulative distribution function (CDF) of voltage sag duration on distribution networks. Moreover, an iterative methodology is proposed to estimate λ LDI and t Is using known values of SAIFI and SAIDI as input parameters.
It must be emphasized that the proposed procedure allows obtaining reliability indicators and socioeconomic parameters closer to reality. Thus, the main contributions of this paper are summarized as follows: -A CDF of d sag definition that considers the stochastic behaviors of short-circuits and the operating characteristics of the protection system in distribution networks. This definition allows distribution planners to propose suitable preventive or corrective actions to improve the power quality in the grid. -An estimation of proper values of λ LDI and t Is to obtain accurate reliability indicators for expansion planning studies. Accurate failure rates allow distribution utilities better planning and operation of their systems. The rest of the paper is organized as follows. Section II describes an overview of voltage sags and power interruptions. Section III presents the proposed methodology to estimate a cumulative distribution function of voltage sag duration. Section IV introduces the proposed methodology for the estimation of annual voltage sag, interruptions and reliability indicators. Section V shows the numerical results obtained on the 34-busbars distribution system. Finally, section VI presents the conclusions.

II. OVERVIEW OF VOLTAGE SAGS AND INTERRUPTIONS IN POWER DISTRIBUTION SYSTEM
A voltage sag (the term dip or simply sag are also used by the power quality community) is a decrease between 0.1 and 0.9 pu in the root-mean-square (RMS) value of an AC voltage (at the power frequency) for a duration from 0.5 cycles to 1 min (some particularities can be applied by some countries, e. g. the Brazilian regulation sets from 1 cycle to 3 min) [12], [13]. Voltage sags that last less than half a cycle are considered transients, mainly because these events cannot be characterized effectively by a change in the RMS value of the voltage at the fundamental frequency. Voltage sags that last longer than 1 min are classified as long-duration variations, and these events generally are not associated with electrical faults [14]. Also, in [12], interruptions are defined as undervoltages (decreased voltage) more than 0.1 p.u. of the supply voltage for a period not exceeding 1 min. In case of supply voltage goes to zero for a period in excess of 1 min, the term used is sustained interruptions.
Voltage sag occurs due to the short duration increase of the current magnitude elsewhere in the power systemtransformer energization [15], starting of large motors or heavy load as causes of voltage sags [16]. However, in these cases, the disruptive effects on customers can be mitigated by installing modern equipment, such as voltage regulators, FACTS, etc. [17]. These actions can be sponsored by electrical utilities or the affected customers themselves. Furthermore, system faults are also the main causes of voltage sags [18], but these types of events can happen either close to or far from the customers, leading to relevant consequences to utilities and end users, mainly because faults are associated with random behaviors that are difficult to predict.
Electrical faults (or short circuits) can cause voltage sags and/or interruptions (temporary -SDI or permanent -LDI) to some customers depending on the fault location and the duration of the fault in the power system. This process is illustrated in Fig. 1(a), where a distribution system with two feeders is affected by an electrical fault at feeder B. Also, Fig. 1(b) shows different waveforms for two customers A and B during an electrical fault. When a fault affects feeder B, the circuit breaker CB2 opens to clear the short-circuit current after d sag seconds (or milliseconds) the fault occurred. In order to reestablish the continuity of the service and avoid transitory fault, reclosing can occur several times. After that, a permanent power interruption is setted for feeder B. The values of d SDI and d LDI refine the durations of SDI and LDI.

A. VOLTAGE SAGS AND INTERRUPTION DURATION
Voltage sag duration is defined as the time interval between the point on wave of sag initiation and its ending. Definitions and discussions about the identification of this time interval can be found in [19] and [20].
The protection system existing on the distribution system plays an important role in the identification of the voltage sag and SDI durations. From Fig. 1, customer A is facing a sag in the interval [T sag , T SDI ]. After T SDI , customer A is facing similar sags up to T LDI . On the other hand, customer B is facing SDI in the interval [T SDI , T LDI ] due to reclosing attempts of CB2. From Fig. 1 Voltage sag duration, or d sag, is limited according to the type of protective device that acts to clear the fault, e.g., fuses, reclosers, etc. Moreover, coordination schemes (as fuse-blowing and fuse-saving) used by utilities define the voltage sag and SDI durations. For fuse-blowing scheme, a fuse is blown when any fault occurring on its protected zone, whether the fault is permanent or temporary [21]. Using this scheme, the frequency and duration of permanent faults are generally increased. For the fuse-saving scheme, a recloser is activated before a fuse blows to save fuses, i.e., the upstream recloser isolates the fault even before the fuse can blow, so the service can be restored for temporary faults. Values of voltage sag duration (d sag , see Fig. 1(b)) can be obtained from time-current characteristic (TCC) curves of the protective devices. For that, the short-circuit current is a piece of important information to select the protective device that is been activated to isolate the faulted sections [22].
Automatic circuit breakers and reclosers are the main cause of SDI. Generally, these devices are designed to reclose two or three times in order to give chances to clear persistent transient faults [23]. Also, it is common to call the first reclose (d 1F , see Fig. 1(b)) as an instantaneous reclose, which may result in closure in 12 to 30 cycles (i.e., 0.2 s to 0.5 s for 60 Hz system) for line reclosers and substation breakers or 2 s or 5 s when distributed generation is installed in the grid. After the instantaneous reclose, time-delayed operations are executed in uniform intervals (d 1D and d 2D , see Fig. 1(b)). Typical values are in 1 to 2 s for line reclosers or d 1D ≈ 15 s and d 2D ≈ 30 s for substation breakers [24].
In case of sustained interruptions, the duration of LDI (d LDI , see Fig. 1(b)) is related to the performance of electrical utility to deal with the permanent interruption. Switches with manual or automated operation may be used to transfer the power supply of some customers located in healthy section to neighboring feeders to reduce reliability indices. In case of faulted section, maintenance crews are necessary to repair the fault and restore the power supply.

III. PROPOSED METHODOLOGY FOR ESTIMATING CUMULATIVE DISTRIBUTION FUNCTION OF VOLTAGE SAG DURATION
Electrical faults (short circuits) can happen anytime and anywhere on the network. Their impacts on the grid and on the VOLUME 11, 2023 customers are analyzed using data collected from measurement campaigns and through methodologies that reflect the real behavior of each variable that takes part of a short circuit.
To characterize a short-circuit event, it is necessary to know its magnitude and duration. Thus, this paper proposes a stochastic methodology based on Monte Carlo simulation (MCS) to predict possible magnitudes of short-circuit currents that can affect the electrical network and the cumulative distribution function (CDF) of the voltage sag duration, i.e., P SgD , which reflect the probabilistic effect of duration for each simulated short-circuit event. Fig. 2 shows the detailed flow chart to carry out the estimation of the P SgD . The procedure is executed in three steps and is summarized below: -First, N sc values of short circuit currents (I sc ) are obtained by considering the variables (i.e., type, position and impedance) that conform to each short circuit s simulated [3]. Then, these variables are represented by probability distribution functions (PDF) obtained from historical data (for existing networks) [25], or expected values (for nonexistent networks, e.g., during distribution expansion planning studies). -Second, N sc values of d sag are obtained using each I sc and the time-current characteristic curve of the protected device which is used to eliminate each I sc (i.e., faultclearing time).
-Third, N s values of d sag are stored to execute a statistical analysis to obtain P SgD (t). For example, from Fig. 2, it is possible to estimate that P SgD (0.04 s) = 0.3, and P SgD (0.06 s) = 0.75, i.e., there is the probability of 0.75 that d sag is up to 0.06 s. It is relevant to remark that, although duration can be included as a variable within MCS, this is an inefficient procedure as it increases the number of short-circuit simulated. In this sense, P SgD brings information about the probability of voltage sag for a defined duration based on the fault-clearing time, whose complement of this value is the probability of power interruption (momentary and/or permanent). However, P SgD is a curve that needs to be known in order to provide the appropriate values of sag and interruption probabilities. A generic P SgD curve may not reflect the real state of the network with respect to short circuits. In fact, the estimated annual voltage sags, SDI and LDI that affect the network (i.e., N sag/yr , N SDI/yr and N LDI/yr , respectively) may results in different values from the real data. To solve this problem, we propose, at first, the application of the parameter R L/N , which represents the ratio of annual LDI to the total annual events (among voltage sags, SDI and LDI) that really affect the whole grid, and, at second, a practical iterative procedure to adapt P SgD (t) in order to achieve the real network conditions R L/N . This procedure allows P SgD (t) to stretch and compress depending on R L/N (more details are shown in the appendix). Although, P SgD (t) can be modified on the vertical and horizontal axes, only horizontal stretches and compressions are relevant for estimating cumulative distribution function of voltage sag duration, mainly because P SgD (t) must be less than 100% for any value of t.
The step-by-step of the proposed procedure is shown in Fig. 3 and explained as follow: • Step 1: Iteration n = 0. To define a R L/N . This value is generally between 5 and 9. To define an error%.
• Step 2: To elaborate N sc short-circuit conditions using MCS. If s = 1, . . . , N sc , each s will produce a I sc (s) and a d sag (s).
• Step 3: To elaborate P SgD (t (n) ). Later, n = n+1 and to go to step 4. Otherwise, go to step 7 • Step 7: To stop. During step 2, the simulated fault locations consider the failure rate of each line. Moreover, a uniform distribution can be used for the entire line to define the fault point. For the type of fault to be selected, single-phase, two-phase, two-phaseto-ground, and three-phase faults can use typical occurrence rates. For the fault impedance (Z f ), a uniform distribution can be used between zero and the maximum value (Z fmax ), Z f is generally considered purely resistive for all types of faults [26].
To elaborate P  LDI/yr in step 4 is introduced in section IV.
The step 6 allows P SgD to adapt its shape depending on fs (n) , i.e., P SgD can be stretched or compressed on the X axis (d sag axis) to approximate the values of R (n) L/N to R L/N .

IV. PROSED METHODOLOGY FOR ESTIMATING ANNUAL VOLTAGE SAGS, INTERRUPTION, AND RELIABILITY INDICATORS
Such as proposed in [3] and [18], transients, voltage sags, SDI and LDI can be considered independent electrical From (1), P trans (s), P sag (s), P SDI (s) and P LDI (s) can be obtained by taking into account the time interval of each event. Depending on historical data, each interval can be linked with a respective PDF. For example, as shown in Fig. 4, T sc (s), T sag (s), T SDI (s) and T LDI (s) represent the beginnings of short circuit, voltage sag, SDI and LDI, respectively for s. The probability of each phenomenon (P f ) is calculated as the integral of PDF of d sag or PDF dsag (t) between a time interval [t 1 , t 2 ], i.e., the difference between P SgD (T 2 ) and P SgD (T 1 ) as expressed by (2).
It is important to mention that after T SDI (s), momentary and permanent interruptions may happen. Thus, P SDI (s) + P LDI (s) = 1 − P SgD (T SDI (s)). The duration of SDI and LDI will depend on the performance of the protection system to restore the normal energy supply. Taking the abovementioned into account, in this paper, the follow issues are considered: • P trans (s) can also be evaluated from P SgD for T sag (s) equal to a half cycle (or a cycle as in Brazilian regulation [13]). Thus: P trans (s) = P SgD (T sag (s)) • P sag (s) can also be obtained from P SgD for the interval between T sag (s) and T SDI (s). Thus, we can write: P sag (s) = P SgD (T SDI (s)) − P SgD (T sag (s)) (3) • From (1), P SDI (s) + P LDI (s) = 1 − P trans (s) − P sag (s). Based on the above consideration, we have: P SDI (s) + P LDI (s) = 1 − P SgD (T SDI (s)).
• For each s, the relation between P SDI (s) and P LDI (s) can be considered constant, i.e.: R S/L (s) = P SDI (s)/P LDI (s). It is noticeable that, like R S/L (s) is between 5 and 8, according to historical data [27], or zero when there are no reclosing attempts VOLUME 11, 2023 for momentary faults, the P SDI (s) and P LDI (s) can be evaluated by (4). P LDI (s) = 1 − P SgD (T SDI (s)) 1 + R S/L (s) P SDI (s) = R S/L (s) · 1 − P SgD (T SDI (s)) 1 + R S/L (s) (4) • The probability of s to affect the power grid is evaluated as 1/N s , where N s is number of short circuits simulated. Thus, the probability of a LDI to affect the power grid P N LDI(s) due to s can be obtained from (5).
We can now estimate the annual events among transients, voltage sags, SDI and LDI that can affect the power grid (N tot/yr ), according to (6).
where λ LDI represents the permanent failure rate per km per year of the feeder and L grid represents the total length of the feeder in km (trunk and all branch lines). Thus, the annual voltage sags, SDI and LDI that affect the whole power grid can be estimated by (7)

A. FINANCIAL LOSSES IN COSTUMERS DUE TO VOLTAGE SAGS, SDI AND LDI
Such as proposed in [3] and [28], annual process trips for each customer c due to voltage sags (N sag p/yr(c) ), SDI (N SDI p/yr(c) ) and LDI (N LDI p/yr(c) ) and the financial losses due to process trips for the whole distribution network (FL PT ) can be estimated by considering four relevant parameters: -The uncertainty of production process trip (Un PT (s,c)) of c for each s; -The probabilities of each phenomenon (among sag, SDI and LDI) to affect the point of common coupling b (i.e., local where c is connected to the utility); -The activity factors for c (F a (c)); and -The unitary costs with respect to voltage sags, SDI and LDI for the customer c, i.e., C sag (c), C SDI (c) and C LDI (c). The values of Un PT (s,c) depend on, mainly, the type of equipment or production process and the magnitude and duration of voltage sag. More information about Un PT (s,c) for different equipment and industrial processes can be found in [3] and [28]. The probabilities of sag, SDI and LDI can be obtained by (3) and (4). Values of F a (c), C sag (c), C SDI (c) and C LDI (c) can be obtained from the literature (adapted values from [29] and [30] are shown in TABLE 4). N s · F a(c) · N tot/yr (12) Thus, FL PT can be estimated by (13).

B. ASSESSMENT OF RELIABILITY INDICATORS
The reliability indicators SAIFI and SAIDI can be evaluated considering each simulated event s obtained by MCS. The evaluation of annual SAIFI and SAIDI can be made separately depending on the position of each customer with respect to s in the distribution network, i.e., customers located in the isolated section (IS). SAIFI and SAIDI are calculated by (15) (17) where, N T is total of customers in the distribution network; N Is c(s) is number of customers located in IS; t Is (s) is the average repair time of isolated sections; f Is (s) is factor of IS to take into account local agency regulations to evaluate SAIFI and SAIDI, e.g., for the Brazilian regulation, SAIFI and SAIDI only are counted for events longer than 3 min [13]; P N LDI(s) can be obtained by (5). These factors can be obtained by (16).
In case of annual ENS, it can be calculated by (17), where i is indices of customers located in IS during a simulated event s. P kW(i,s) is active power of each customer i during a simulated event s.
From (5), R S/L mainly impacts the probability of a permanent interruption affecting the network. In other words, the increment of R S/L means that there is an increment in the number of momentary interruptions in relation to the permanent ones. Changes in the protection system (for example, adjustments in the reclosing sequences of reclosers) can lead to this effect. This increment mainly reduces, which in turn leads to the reduction of SAIFI, SAIDI and END (this effect can be seen in (14) - (17)), mostly because the network faces less permanent interruptions. R S/L values can be obtained from known historical data of each energy utility, which are constantly audited by regulators. In addition, with the inclusion of intelligent monitoring and data collection systems in the distribution networks, based on the concepts of smart grid, the identification and quantification of the number of momentary and permanent interruptions (and consequently R S/L ), within a time interval, are becoming more and more efficient, accurate and reliable.

C. ESTIMATING PERMANENT FAILURE RATE AND AVERAGE REPAIR TIME
Depending on λ LDI and t Is selected to calculate SAIFI yr and SAIDI yr , the values may not reflect the real network behavior for a specific year. Therefore, we propose a practical iterative procedure to estimate suitable λ LDI and t Is values, which is achieved as follows: The iterative process shown in Fig. 5 allows to estimate proper values of λ LDI and t Is to achieve known or expected annual values of SAIFI and SAIDI. Thus, economic indicators such as ENS and FL PT can be obtained with better accuracy.

V. NUMERICAL APPLICATIONS
The proposed methodology is applied in a feeder based on the IEEE 34 test case [31]. This feeder is an unbalanced, long (approx. 90 km), radial distribution system. This network is composed by a main three-phase (3F) trunk, with singlephase (1F) laterals.

A. MODIFIED IEEE 34 TEST FEEDER
The single line diagram of the test feeder is shown in Fig. 6, which includes new components related to the protection system. All typical components, including transformers, capacitors, loads and lines stay unchanged from the original IEEE 34 bus test case. The relevant changes are listed here: • The 3-phase short-circuit fault level is 676 MVA, with a positive sequence X/R ratio of 4.33.
• The 1-phase short-circuit fault level is 467 MVA, with zero sequences X/R ratio of 4.02. VOLUME 11, 2023   • The load allocated in 890 busbars (original IEEE 34 test case [31]) is aggregated to 832 busbars in order to analyze only one voltage level of 24.9 kV.
• The position of voltage regulator VR1 is changed to 812-814 line from the original 814-850 line. The taps for VR1 are fixed in: phase A -tap 5, phase B and C -tap 0.
• The taps for voltage regulator VR2 are fixed in: phase A -tap 14, phase B and C -tap 11.
• The base rating of the transformer is changed to 5 MVA, and the percent impedance is 7%, with an X/R ratio of 12.
• One recloser and seven fuses are included in the truck and laterals . TABLE 1 and TABLE 2 show the fuses curves used for each lateral and the recloser settings of R1, respectively. Moreover, the coordination curves among fuses and recloser can be seen in Fig. 7.
• Residential, commercial and industrial customers are allocated in each busbar as shown in TABLE 3.
• Values of activity factor by customer type and unitary costs due to voltage sag, SDI and LDI as shown in TABLE 4.     Fig. 8(a). It is possible to observe that there is a higher frequency (approx. 48% of total events) between 0.04 s to 0.055 s. This concentration is caused by triggering the fast curve of recloser R1 (see Fig. 7) to respect the fuse-saving protection scheme.
Four CDF are used to represent the behavior of P SgD (see Fig. 8(b)), which are: empirical, normal-1 (µ = 0.04098 s, and σ = 0.0129 s), Weibull (α = 0.01291 s, β = 0.0322 s,  and γ = 1.9487), and normal-2 (µ = 0.08 s, and σ = 0.01 s), where µ and σ are the mean and the standard deviation, respectively, and α, β, and γ are position, scale, and shape parameters for Weibull, respectively. The parameters for the first three curves are obtained after statistical analysis using the histogram from Fig. 8(a). Thus, these curves present similar behaviors with some changes at certain intervals of d sag . In turn, the parameters for the fourth CDF are randomly selected to test the ability of the proposed methodology to adapt the form of the CDF to defined values of R S/L (s) and R L/N . It is important to highlight that normal-2 presents lower P sag values than the previous three CDF for the same d sag value. TABLE 6 shows annual values of voltage sags, interruptions, reliability indicators, ENS and F L for each one of the four CDFs. For these evaluations, it is expected N LDI/yr = 9.1 annual permanent interruptions for the whole distribution network (considering L grid = 91 km).
Such as shown in TABLE 6, on the one hand, the application of empirical, normal-1 and Weibull CDFs for P SgD leads to higher values of N sag/yr (46.1, 45.7, and 48, respectively) than N SDI/yr and N LDI/yr . However, on the other hand, the application of normal-2 leads to a N sag/yr much smaller than N SDI/yr and N LDI/yr . The explanation is that in the range of 0.04 s to 0.055 s (interval with the higher frequency of events, see Fig. 8(a)), normal-2 presents lower P sag values. Thus, the application of normal-2 could lead to inaccurate estimations of annual indicators.
Reliability indicators (SAIFI and SAIDI) and ENS also are shown in TABLE 6. They present standard deviations below 1% considering empirical, normal-1 and Weibull CDFs. Thus, the applications of these three curves lead to similar values of SAIFI, SAIDI and ENS. However, SAIFI, SAIDI and ENS are lower than normal-2. The main reason for this is that normal-2 lead to more SDI than the other three curves, i.e., SDI events lasting less than 3 minutes are not considered in SAIFI and SAIDI. With respect to annual financial losses, these values are: £1188k for empirical, £1195k for normal-1, £1203k for Weibull, and £830k for normal-2. In fact, the smallest value is obtained for normal-2, mainly because of the reduced number of annual voltage sags.

C. CONVERGENCE PROCESS TO ADAPT PROBABILITY DISTRIBUTION FUNCTION OF VOLTAGE SAG DURATION
The proposed methodology allows CDFs to adapt its form to achieve defined R L/N values. Thus, the four CDFs introduced above are adjusted using the proposed methodology to achieve the same expected value of R S/L (s) = 5 and R L/N = 1/8 = 0.125. Fig. 9 shows the shape of each curve modified by the proposed methodology. The modification of P SgD curves aims to adjust the initial values of R L/N (see TABLE 6) to the expected value of R L/N = 0.125. For empirical, normal-1 and Weibull, due to their R L/N values are less than 0.125, their initial P SgD curves must be stretched. In turn, for normal-2, with initial R L/N value of 0.180, its initial P SgD curve must be compressed. From Fig. 9, up to 0.046 s normal-2 presents lower values of P sag than the other three curves. This characteristic changes after 0.048 s when normal-2 presents higher values of P sag than the other three curves. This new shape allows normal-2 to obtain annual indicators to the expected R L/N = 0.125. Thus, there is 95% of probability than voltage sag duration is up to: 0.073 s for empirical, 0.075 s for normal-1, 0.088 s for Weibull, and 0.058 s for normal-2.
The annual values of the seven indicators for each one of the four CDFs after the applications of the proposed methodology are shown in TABLE 7. The results show that even different CDFs, the indicators converge in similar values (σ < 1.5%), for that it is necessary to execute between 7 and 9 iterations. Considering normal-2, before the adjustment procedure SAIFI, SAIDI and END represent 5.6, 15.4 h and 26855 kWh, respectively. After the adjustment procedure, these values go to 6.1, 16.8 h and 29496 kWh, i.e., increments  of 9.3%, 9.3% and 9.8 %, respectively. These variations lead to an even greater variation in financial losses from £830k to £1036k (increment of 23.5 %).

D. TUNNING RELIABILITY INDICATORS
The proposed methodology is applied to different values of SAIFI and SAIDI extracted from [32], which correspond to the average annual values in Brazil. Although the voltage level of the IEEE 34 network is different from that used in Brazilian electrical networks, SAIFI and SAIDI values can be used as references to evaluate the adjustment in λ LDI and t Is . However, other reliability indicators can also be used without complying with the proposed methodology. Moreover, to simplify the model, during the simulations, the natural growth of demand and the number of consumers is not considered. For all cases, errorSAIFI% = errorSAIDI% = 0.5 %. TABLE 8 shows estimated values of λLDI and tIs applying the proposed methodology for the IEEE 34 feeder. This table introduces twelve combination values of annual SAIFI and SAIDI from 2010 and 2021. The trend shows important reductions in both reliability indices in the last 12 years (47 % to SAIFI and 36 % to SAIDI). With respect to the estimated parameters, there are a reduction of 47 % to λ LDI and an increment of 21 % to t Is . Thus, the proposed methodology allows evaluating the most suitable λ LDI and t Is values that lead to real values of SAIFI and SAIDI.

VI. CONCLUSION
This paper introduces a novel methodology to estimate P SgD and λ LDI on distribution systems. These variables allow to calculate the data with greater precision socioeconomic indicators such as ENS and FL PT .
The estimation of annual voltage sags and interruptions, ENS, FL PT , and reliability indicators enables the utility to identify regions that need investments, in order to improve the energy quality delivered to end consumers. It is important to mention that, at first, variables such as P SgD and λ LDI are generally obtained from statistical treatments of historical data from many years of monitoring, and, at second, they need to be tuned to represent the current state of the distribution network. Thus, the proposed methodology performs the tuning process considering as reference R L/N , which can be known by the utility for specific years or expected for network expansion planning studies.
Moreover, the proposed methodology may help network operators during the analysis of simulated scenarios with different R S/L , R L/N or P SgD . The comparison process can identify suitable reinforcement in the power system (for example, changes in time-current curves of protection systems, new substations, etc.) that can be executed in the short or long term to obtain better reliability indicators.
Finally, the proposed methodology is an important tool for operation and expansion planning studies, mainly because the probabilistic variables can complement known variables coming from real-time measurements. Thus, the treatment of probabilistic and known variables allows obtaining an overview closer to reality, without the need for rigorous and exhaustive system monitoring, which improves the quality of results, and can be applied to existing and future distribution networks.

A. STRETCHING AND COMPRESSING GRAPHS
If we multiply a function f (x) by a coefficient a, we achieve a function g (x) = af (x) whose graph will be stretched or compressed vertically with respect to the graph of f (x). Moreover, if we multiply a function's input x of f (x) by a positive coefficient b, we get a function h (x) = f (bx) whose graph is stretched or compressed horizontally. It is important to mention that, for all cases, the shape of the graph is not affected. The TABLE 9 shows the graph of g(x) and h(x) depending on the values of coefficients a and b.