Definition of Regulatory Targets for Electricity Default Rate in Brazil: Proposition of a Fuzzy Inference-Based Model

Abstract

The current electricity default rates in continental countries, such as Brazil, pose risks to the economic stability and investment capabilities of distribution utilities. This situation results in higher electricity tariffs for regular customers. From a regulatory perspective, the key issue regarding this challenge is devising incentive mechanisms that reward distribution utilities for their operational and investment choices, aiming to mitigate or decrease electricity non-payment rates and avoid tariff increases for regular customers. Despite adhering to the principles of incentive regulation, the Brazilian Electricity Regulatory Agency (ANEEL) uses a methodological approach to define regulatory targets for electricity defaults tied to econometric models developed to determine targets to combat electricity non-technical losses (NTLs). This methodology has been widely criticized by electricity distribution utilities and academics because it includes many ad hoc steps and fails to consider the components that capture the specificities and heterogeneity of distribution utilities. This study proposes a fuzzy inference-based model for defining regulatory default targets built independently of the current methodological approach adopted by ANEEL and aligned with the principles of incentive regulation. An empirical study focusing on the residential class of electricity consumption demonstrated that it is possible to adopt a specific methodology for determining regulatory default targets and that the fuzzy inference approach can meet the necessary premises to ensure that the principles of incentive regulation and the establishment of regulatory targets are consistent with the reality of each electricity distribution utility.

1. Introduction

The increasing demand for electricity, coupled with intricate socioeconomic circumstances in continental nations such as Brazil, creates economic and regulatory hurdles that directly impact the sustainability of electricity distribution systems [1,2,3,4,5,6,7,8,9,10,11]. Specifically, the existing levels of electricity non-payment in these nations pose risks to economic stability, restrict the investment potential of distribution utilities, and lead to higher electricity tariffs for ordinary consumers [5,6,7,8,9,10,11]. From a regulatory viewpoint, the fundamental issue concerning this matter revolves around creating incentive mechanisms that incentivize distribution utilities to make operational and investment choices aimed at mitigating or lowering electricity non-payment rates.

Incentive-based regulatory approaches are widely used in the electricity sector in most countries. Incentive-based regulations typically involve setting performance targets for regulated companies and providing financial rewards or penalties based on their performance relative to these targets. This regulatory approach encourages cost-effective investments, promotes innovation, and improves the overall efficiency in the sector. Incentive regulation can be divided into three modalities: earnings sharing, yardstick competition, and price caps [12,13,14,15,16].

The regulatory methodology adopted by the Brazilian Electricity Regulatory Agency (ANEEL) follows the principles of incentive regulation, using yardstick competition mechanisms and seeking fair comparisons among electricity distribution utilities to create incentive mechanisms that ensure cost reduction, innovation, and efficiency [17,18]. In 2015, ANEEL developed various econometric model specifications for panel data regressions considering socioeconomic and market variables related to all distribution utilities in the country. The goal was to create an index that reflects the challenge of addressing non-technical losses (NTLs) based on the unique characteristics of each distribution area [19]. More recently, the Agency perceived that it would be necessary to review this methodology owing to a series of problems pointed out by agents of the Brazilian electricity sector and academic experts (for example, [20,21]). In 2020, the Agency revised its regulatory methodology, which can be found in Technical Note No. 46/2020 [22].

Concerning the issue of electricity defaults in Brazil, there has been a notable increase in default rates over the past decade. The twelve-month average default rate rose from 2.4% in 2016 to 4.7% in 2022, nearly doubling during this period [11]. Despite adhering to the principles of incentive regulation, ANEEL uses a methodology to define regulatory targets to limit or reduce electricity default rates tied to econometric models developed to establish targets to combat NTLs. Conceiving an alternative methodological approach independent of the current ANEEL methodology is of fundamental importance for distribution utilities because they may lose revenue owing to tight benchmarks. It also affects consumers, who may be burdened by their tariffs because of distribution utilities’ ineffective management of default levels.

In addition to the problem of methodological dependence of econometric models used by ANEEL to establish targets to combat NTLs, other criticisms include (i) the complexity of the methodology, expressed by numerous ad hoc steps, and the low explanatory ability of the adopted econometric models; (ii) calculation of the Social Complexity Index (SCI) itself, which does not consider the heterogeneity or specificity of each distribution utility; (iii) the non-use of the time variable explicitly; and (iv) the lack of treatment of outliers [23]. Moreover, the SCI is one of the pillars of defining regulatory targets to limit or reduce electricity default rates. The calculation involves estimating correlation coefficients between the percentage of NTLs in the low-voltage distribution market and a series of socioeconomic variables from concession areas. This is done using a panel data model with random effects. ANEEL’s assumption of using the same social complexity ranking is based on the principle that both NTLs and unrecoverable revenues are affected by the socioeconomic variables of concession areas [23].

Aiming to identify conceptual and empirical studies addressing the central issues of this study, namely, electricity defaults, incentive-based regulatory approaches, and fuzzy inference-based models, a literature review of reference documents and scientific articles published between 2000 and 2024 yielded only a limited number of documents, revealing research gaps associated with the interplay between these three issues [5,6,7,8,9,10,11,23,24,25].

The first research gap refers to the lack of studies in the field of electricity defaults from an incentive-based regulatory perspective [5,6,7,8,9,10,11]. Although considerable effort has been devoted to developing different models to assist ANEEL in better defining regulatory targets for combating electricity NTLs in Brazil [23,24,25], no prior study has specifically examined the establishment of regulatory targets aimed at reducing or limiting electricity bill defaults independently of the current methodological approach used by ANEEL for regulating non-technical losses. This is the second research gap explored in this study by developing a new application of fuzzy inference systems (FIS) in the context of incentive regulation, specifically, its modality yardstick competition.

In this context, this study addresses these research gaps by investigating the following research questions:

• RQ1: What are the main challenges faced by ANEEL in defining regulatory targets to limit or reduce electricity default rates for electricity distribution utilities in Brazil?
• RQ2: What are the essential premises that must be considered when developing a conceptual model to define regulatory targets to limit or reduce electricity default rates that meet the fundamental requirements of incentive-based regulation?
• RQ3: To what extent can an alternative model based on fuzzy inference methodology better assist ANEEL in defining regulatory targets for limiting or reducing electricity default rates in Brazil?
• RQ4: Is it feasible to demonstrate the applicability of the proposed model by focusing on the residential class of electricity consumption in Brazil?

From a regulation-oriented perspective, this study proposes a fuzzy inference-based model to establish regulatory targets to limit or reduce electricity default rates in Brazil, irrespective of the current methodological approach employed by ANEEL for setting regulatory targets to address NTLs in distribution utilities.

An empirical study focusing on the residential class of electricity consumption demonstrated that it is possible to adopt a specific model to determine regulatory targets to limit or reduce electricity default rates and that the fuzzy inference methodology can meet the essential premises to ensure that the principles of incentive regulation and the establishment of regulatory targets are consistent with the reality of each electricity distribution utility. Notwithstanding the empirical demonstration of its applicability, by focusing on the residential class of electricity consumption in Brazil, the proposed model is applicable to all classes of electricity consumption.

This paper is organized into seven sections. The second section provides a literature review covering the central issues of this study, namely, electricity defaults, incentive-based regulatory approaches, and fuzzy inference-based models. Section 3 briefly presents the research design and methodology. Section 4 introduces a fuzzy inference-based model for defining regulatory targets to limit or reduce electricity default rates in Brazil independently of the model adopted by ANEEL to combat NTLs and aligned with incentive regulation principles. Section 5 describes the results of an empirical study aimed at demonstrating the applicability of the proposed model, focusing on the residential class of electricity consumption in Brazil. In this study, a fuzzy inference system (FIS) was used to estimate the default rates of 59 Brazilian electricity distribution utilities. The results of the empirical study provide strong support for the formulated hypotheses. Section 6 discusses the methodological differentials of the proposed model compared with those currently adopted by ANEEL and highlights the study’s contributions to the research fields of electricity default from a regulatory perspective and fuzzy inference systems (FIS), as it addresses a new application for this approach, namely, the application of FIS in the context of incentive regulation. Finally, Section 7 presents concluding remarks and policy implications for those interested in advancing the knowledge of regulations to limit or reduce electricity default rates at the national level.

2. Literature Review

A literature review covering the central issues of this study, namely, electricity defaults, incentive-based regulatory approaches, and fuzzy inference-based models, focused on reference documents and scientific articles published between 2000 and 2024. The findings from this review reveal research gaps associated with the interplay between these three issues, as pointed out in the introduction.

2.1. Electricity Defaults

One of the initial findings in the exploratory phase of this study was that very few previous studies on electricity defaults were published in the period covered by the literature review (2000–2024), and none of them were written from an incentive-based regulatory perspective [5,6,7,8,9,10,11].

Araújo [5] conducted a statistical analysis of electricity NTLs and default rates in Brazilian distribution utilities and proposed a regression model to characterize electricity defaults. This model was able to explain over 50% of the variation in electricity default rates observed by distribution utilities using the average tariff of the distribution utility, intensity of indigence, and urbanization rate as explanatory variables.

Zhow et al. [6] focused on analyzing key variables affecting charge arrears, using logistic regression to establish a model for recognizing the possibility of arrears. By predicting the default probability, electricity enterprises can shift from the passive occurrence to the active prevention of arrears.

Szabo and Ujhelyi [7] explored the effectiveness of a water education campaign to increase payments for public utilities in a low-income periurban area in South Africa. Overall, the study’s findings underscore the significance of targeted information campaigns in influencing consumer behavior and addressing nonpayment challenges in the water sector. By emphasizing the role of consumer perceptions, reciprocity, and cost considerations, the research demonstrated that while increasing consumer information is important, interventions should also focus on building trust, fostering positive perceptions, and ensuring cost-effective strategies to promote sustainable payment behavior and efficient use of public utilities. They conclude that policymakers in developing countries can benefit from exploring a range of interventions beyond traditional enforcement methods to address non-payment issues. Strategies focusing on building relationships, providing information, and fostering reciprocity between service providers and consumers can contribute to sustainable solutions.

Khana et al. [8] found that pervasive non-payment also imposes a financial burden on utilities across India. Low revenue collection rates make it difficult for utilities to maintain and upgrade infrastructure, which could further worsen the service quality. This study emphasizes the importance of balancing affordability and service quality in the electricity sector, as well as the need for regulatory interventions to improve reliability for all customer segments.

Fowlie et al. [9] analyzed default effects and follow-on behavior in an electricity pricing program in Sacramento (USA), yielding several key findings. The results indicated a strong default effect, with a significant majority of customers choosing to remain on time-varying pricing programs when it was the default option, compared to a much lower opt-in rate. Interestingly, customers who were expected to benefit from the program without changing their behavior were not more likely to enroll, suggesting that factors beyond cost–benefit calculations influenced their decisions. The study also highlighted that more engaged customers were more likely to respond to pricing changes, indicating a correlation between attentiveness and behavioral adjustments in response to dynamic pricing. In conclusion, this study demonstrated that default settings play a crucial role in shaping consumer decisions in the context of electricity-pricing programs. The findings suggest that consumers may be nudged onto certain pricing plans because of inattention or other factors, but once in the program, they can adapt and even prefer new pricing regimes.

Murwirapachena et al. [10] investigated non-payment culture’s impact on the financial performance of South African municipalities, specifically focusing on the electricity provision function. This study aims to fill a gap in the existing literature by quantifying the impact of nonpayment for municipal services on the financial performance of municipalities in South Africa using comprehensive municipal data and robust statistical models. Additionally, this study seeks to model non-payment as a determinant of multinational financial performance, with bad debts written off by the electricity department of each municipality serving as a proxy for non-payment.

Moita et al. [11] analyzed the factors affecting default rates among residential electricity consumers in Brazil using billing data. The key factors influencing default rates include electricity tariffs; enforcement actions, such as power cuts for non-payment, which play a significant role in combatting defaults; conjunctural economic shocks to consumers, which can also influence electricity bill defaults; and the income level of households, a crucial factor in determining their ability to pay electricity bills on time. They conclude that by understanding and addressing these key factors, policymakers and electricity distribution utilities can work towards reducing default rates and ensuring electricity affordability for residential consumers in Brazil.

Collectively, these studies underscore the multifaceted nature of the electricity default problem and disclose several unresolved regulatory issues, particularly the definition of regulatory targets addressed to limit or reduce electricity default rates at the national level.

2.2. Incentive-Based Regulatory Approaches

Drawing on recognized references in the field of incentive regulation [12,13,14,15,16], incentive-based regulations generally entail establishing performance targets for registered firms and offering financial incentives or penalties based on their performance compared with these targets. This regulatory approach stimulates cost-effective investments, fosters innovation, and enhances overall sectoral efficiency. Incentive regulation can be categorized into three main types: revenue sharing, yardstick competition, and price caps. In this study, we concentrate on yardstick competition because ANEEL uses this regulatory modality.

Schleifer [16] introduced the concept of yardstick competition as a regulatory mechanism to incentivize cost reductions in regulated monopolies. By comparing a firm’s costs to those of similar firms, regulators can infer the attainable cost level of the firm and encourage it to operate more efficiently. This approach aims to eliminate the dependence of a firm’s price on its chosen cost level, ultimately leading to improved performance and cost efficiency.

In the electricity distribution sector, yardstick competition can be a valuable regulatory tool for promoting efficiency and cost reduction. Regulators can use cost data from comparable electricity distribution firms to set prices for individual firms, creating a competitive environment in which firms are incentivized to minimize costs [17,18].

A key advantage of yardstick competition in the electricity distribution sector is its ability to promote competition and efficiency without requiring detailed knowledge of firms’ cost structures. Regulators can rely on cost comparisons across similar firms to determine price levels, thus making the regulatory process more straightforward and transparent [17,18].

By using yardstick competition to define regulatory default targets, regulators can enhance transparency and accountability in the regulatory process. The use of sectoral benchmarks and cost comparisons makes it easier for regulators to objectively assess the performance of distribution utilities and to hold them accountable for meeting established targets. This transparency can help build trust among regulators, utilities, and consumers, leading to more effective regulation and oversight of the sector.

Furthermore, yardstick competition can improve resource allocation and investment decisions in the electricity distribution sector. By incentivizing firms to operate more efficiently and reduce costs, regulators can ensure that resources are allocated effectively and that investments are made in a way that maximizes social welfare. This can result in a more competitive and dynamic industry landscape, with firms striving to outperform their peers and deliver value to their consumers.

2.3. Fuzzy Inference-Based Models

Fuzzy inference-based models [26,27,28] were reviewed in this study, particularly the seminal work of Lotfi Zadeh on fuzzy algorithms for complex systems and decisions [27,28]. The decision to develop a fuzzy inference model was based on the robustness, precision, and stability of this methodological approach, combined with the simplicity of implementation and proven effectiveness in various areas of application, such as for designing and modelling solar energy systems [29] and the development of biomedical devices [30].

A fuzzy inference system (FIS) uses fuzzy set theory to map inputs (features in the case of fuzzy classification) to outputs (classes in the case of fuzzy classification) [26,27,28]. There are two main types of fuzzy inference systems: (i) Mamdani-type [31] and (ii) Sugeno-type [32].

In the Mamdani-type inference, the output membership functions are fuzzy sets. After the aggregation process, each output variable is represented by a fuzzy set that requires defuzzification. Alternatively, a singleton output membership function can be used, which simplifies the defuzzification process by representing the output as a single spike instead of a distributed fuzzy set. This approach enhances efficiency by reducing the computational complexity of finding the centroid of a two-dimensional function [31].

In contrast, the Sugeno fuzzy inference method is similar to the Mamdani method, but with linear or constant output membership functions [32]. Unlike the Mamdani method, the Sugeno FIS does not employ an output membership function. Instead, the output is a crisp number calculated by multiplying each input by a constant and then summing the results.

This study employed the Mamdani-type FIS method [31], which comprises the following steps: (i) fuzzification of input variables, (ii) fuzzy inference, and (iii) defuzzification (Figure 1).

In the first step, called the fuzzification of the input variables (crisp), linguistic variables are created and associated with them. Thus, a given variable can be classified into different degrees of membership in a linguistic variable or a fuzzy group (good, bad, high, or low). Thus, the fuzzifier transforms an absolute numerical variable (crisp) into a linguistic one.

In the second step, different associations are linked through a series of linguistic sentences established by rules. In a simplified manner, it can be said that such rules describe the relationships between the input variables (antecedents) and output variables (consequents). Each input value (the antecedent of a rule) activates one or more fuzzy groups with a certain membership function. A rule can be determined by an expert or obtained directly through the analysis of a database, such as one concerning the actual default rate. Thus, an ‘if/then’ rule can be defined as follows: if the price is ‘high’ and quantity is ‘low’, then the risk is ‘high’.

The third and final steps are characterized by transforming linguistic variables into output variables (crisp). This step, known as defuzzification, generates an output (numerical value) that can be evaluated. Some metrics can be used in defuzzification, with the most common being the calculation of the center of gravity (centroid), average of the maximum, maximum, minimum of the maximum, and bisector.

3. Research Design and Methodology

Based on the procedural model outlined by Martins et al. [33], this study was structured into three phases and six stages. The research questions were written as descriptive questions to provide a clear foundation for the development of a set of hypotheses. Thus, the following hypotheses were formulated as more formal predictions of research outcomes.

Hypothesis 1 (H1). 

The fuzzy inference-based model can better assist ANEEL in defining regulatory default targets in Brazil, as it will be developed independently of the NTL methodology adopted by the Agency.

Hypothesis 2 (H2). 

The input variables under categories ‘vulnerability of the concession area’ and ‘size of the distribution utility’ will significantly influence the determination of regulatory default targets in the fuzzy inference-based model, with variations in these two categories of variables leading to distinct target outcomes for electricity default rates.

Hypothesis 3 (H3). 

The fuzzy inference approach is effective in determining regulatory default targets in the electricity sector, which is consistent with the reality of each electricity distribution utility.

Hypothesis 4 (H4). 

The fuzzy inference-based model provides regulatory default targets that are comparable to or more accurate than those established by ANEEL to limit or reduce default rates in the electricity sector.

Thus, the research questions and hypotheses clarified the main purpose and specific objectives of the study, which in turn dictated the research design, direction, and outcome, as summarized in

Table 1. Research design.

Phase	Stage	Research Questions [Sections]
Motivation (Why?)	Problem definition and the rationale for the research	Why should we propose a fuzzy inference-based model to assist ANEEL in defining regulatory targets to limit or reduce electricity default rates in Brazil? (Section 1).
Conceptualisation and development (What and how?)	2. State of research on the central issues and identification of research gaps and unsolved problems	What is the state of research on regulatory models and instruments to limit or reduce electricity default rates in distribution utilities, with special attention paid to those works aligned with incentive-based regulation? (Section 2). What are the main limitations of the current methodologial approach adopted by ANEEL to define regulatory targets for limiting or reducing electricity default rates? (Section 1 and Section 2).
	3. Definition of the research design and methodology	From a regulatory perspective, how can a fuzzy inference-based model be developed and validated in the context of Brazil’s electricity distribution sector to limit or reduce electricity default rates? [Section 3].
	4. Development of a fuzzy inference-based model to define regulatory targets for default rates to be accomplished by Brazilian electricity distribution utilities	What are the essential premises that must be considered when developing a conceptual model to define regulatory targets for limiting or reducing electricity default rates that meet the fundamental requirements of incentive-based regulation? (Section 4). To what extent can a fuzzy inference-based model better assist ANEEL in defining regulatory targets for limiting or reducing electricity default rates in Brazil? (Section 4).
Validation (How to demonstrate the applicability of the proposed technique?)	5. Demonstration of the applicability of the proposed model by focusing on the residential class of electricity consumption in Brazil	Is it feasible to demonstrate the applicability of the proposed model by focusing on the residential electricity consumption class in Brazil? (Section 5). Can the empirical results evidentiate the benefits of using a fuzzy inference-based model to establish regulatory targets to limit or reduce electricity default rates? (Section 5 and Section 6).
	6. Discussion of the research results and policy implications	What are the differentials of the proposed methodological approach over the current econometric models used by ANEEL to define regulatory targets for limiting or reducing electricity default rates in Brazil? (Section 5 and Section 6).

.

The stages in each phase are outlined in Table 1 as follows: (i) defining the research problem and rationale (first stage); (ii) literature review on the central research issues, identifying research gaps and unresolved questions (second stage); (iii) developing the research methodology (third stage); (iv) creating a fuzzy inference-based model to establish regulatory targets for limiting or reducing default rates in Brazilian electricity distribution utilities (fourth stage); (v) applying the model to the residential electricity consumption class to better assist ANEEL in setting regulatory targets for reducing default rates (fifth stage); and (vi) discussing the research findings and policy implications, highlighting the differences between the proposed methodological approach and the current models used by the Brazilian Agency (sixth stage).

The first two stages involved a literature review of scientific articles and reference documents published between 2000 and 2024 to identify conceptual and empirical works related to the central theme of this study. Systematic searches were conducted in major scientific databases (e.g., Scopus and Web of Science), as detailed in Appendix A. These searches were supplemented with subsequent searches using Google Scholar. Subsequently, the bibliographic review was deepened by analyzing the references cited in the selected articles (backward search) and identifying the most relevant previous works, including grey literature.

Considering the criticisms of the methodological approach adopted by ANEEL for defining regulatory targets to limit or reduce electricity default rates, a special attempt was made to (i) identify the main models developed within the context of incentive regulation applicable to the problem in focus and (ii) gather methodological insights to establish the essential premises to be considered when developing a conceptual model to define regulatory targets for electricity default rates that meet the fundamental requirements of incentive regulation.

The third stage involves outlining the research methodology, which includes (i) conducting a literature review of scientific articles and documents published between 2000 and 2024 related to central research issues, with a particular focus on previous studies developed within the framework of incentive-based regulation; (ii) defining the essential premises to be considered during the modeling phase; (ii) developing a fuzzy inference model to assist ANEEL in establishing regulatory targets for limiting or reducing electricity defaults in distribution utilities; and (iii) demonstrating the applicability of the proposed model by focusing on the residential electricity consumption class in Brazil.

The choice of an alternative model to that adopted by ANEEL to define regulatory targets for limiting or reducing electricity default rates should be aligned with incentive regulation principles. Furthermore, the heterogeneity of the concession areas of electricity distribution should consider the heterogeneity of the concession areas of distribution utilities in a country with as many inequalities as Brazil has. In this sense, an intelligent system that feeds into actual information and is capable of modelling scenarios based on uncertainty can be a viable alternative, as demonstrated in Section 4.

The exploratory phase of modeling began with experimental tests of different models reported in the literature, such as the dynamic clustering model [34], other clustering models [35], models based on factor homogenization [36,37], and panel data regressions [38,39]. However, none of these approaches has proven to be suitable for addressing the research gaps posed in Section 1. Subsequently, the fuzzy inference-based models reviewed in Section 2 were analyzed [26,27,28,31,32], particularly the seminal work of Lotfi Zadeh on fuzzy algorithms for complex systems and decisions [27,28]. The decision to develop a fuzzy inference model was based on the robustness, precision, and stability of this methodological approach combined with the simplicity of implementation and proven effectiveness in various areas of application, such as designing and modelling solar energy systems [29] and developing biomedical devices [30].

The application of a fuzzy inference system, as shown in Figure 1, to establish regulatory targets for limiting or reducing electricity default rates at the national level is the fourth stage of research design. Before proceeding to this application, a set of essential premises was established for developing the intended model. Based on these premises and following the Mamdani-type FIS method [31], an algorithm was developed in the MATLAB^® software package version R2022a (MathWorks, Natick, MA, USA) [40] to create an FIS capable of modeling the default rates of 59 electricity distribution utilities in Brazil. The essential premises for modeling and a detailed description of the fuzzy inference-based model are presented in Section 3.

Finally, the applicability of the proposed model is demonstrated by focusing on the residential class of electricity consumption in Brazil. The empirical results presented in Section 5 provide strong support for the formulated hypotheses. They are discussed in Section 6, highlighting the differences between the proposed model and the methodological approach currently used by ANEEL, as well as previous studies addressing the problem of electricity default rates in other countries.

4. Proposition of a Fuzzy Inference-Based Model to Establish Regulatory Targets for Electricity Default

This section introduces a fuzzy inference-based model to assist ANEEL in establishing regulatory targets to limit or reduce electricity default rates in Brazil. Figure 2 shows a general view of the proposed model.

4.1. Phase 1: Definition and Classification of Input Variables

Based on the literature review presented in Section 2, before proceeding to the definition and classification of input variables, a set of essential premises was established to develop an alternative model to better assist ANEEL in setting regulatory targets for limiting or reducing electricity default rates, aligned with the principles of incentive regulation. They are:

• Maintain the principles of incentive regulation, which can be considered a gain from the evolution of methodologies adopted by ANEEL;
• Capture the heterogeneity of the concession areas of distribution utilities;
• Provide equal treatment to all distribution utilities;
• Robust and proven effective methodology;
• Use the default rates’ database of distribution utilities to extract learning and observed patterns;
• Be easy to implement.

Once the essential premises are defined, the next step is a deep documentary analysis of the current ANEEL methodology to define FIS input variables.

The next step is to group these variables into two categories: ‘vulnerability of the concession area’ and ‘size of the distribution utility’, as described below.

• Vulnerability of the concession area: Consists of variables indicating the socioeconomic conditions of each distribution utility concession area.
• Size of the distribution utility: Consists of variables associated with the size of distribution utilities, such as kilometers of the transmission line network of a given electricity distribution utility.

4.2. Phase 2: Database Preparation and Inputs for Fuzzy Inference

This phase begins with the preparation of the database, where the input variables are normalized to obtain the average of all inputs for the use of a single value in each of the categories (size of distribution utilities and vulnerability of concession areas). Normalization is performed by subtracting the average of the values (of each variable) for each distribution utility, then dividing by the standard deviation of each variable calculated for each utility.

Next, five fuzzy groups are created to classify each input and output variable. The groups are named ‘very low’ (VL), ‘low’ (L), ‘moderate’ (M), ‘high’ (H), and ‘very high’ (VH). The groups should be represented by triangular membership functions and, for the last group, a trapezoidal function. Each membership function is defined by values relative to the three vertices of a triangle and four vertices of a trapezoid.

The limits of the fuzzy groups can be obtained using the following steps:

• Step 1: Determination of the range of each variable in the model, obtaining equal class intervals for the fuzzy groups, represented by N1, N2, N3, N4, N5, N6;
• Step 2: Once the ranges are defined, the vertices of the membership functions can be determined as follows:
  • VL fuzzy group—Triangular with vertices at: N1; N1; N2;
  • L fuzzy group—Triangular with vertices at: N1; N2; N3;
  • M fuzzy group—Triangular with vertices at: N2; N3; N4;
  • H fuzzy group—Triangular with vertices at: N3; N4; N5;
  • VH fuzzy group—Trapezoidal with vertices at: N4; N5; N6; N6.

Figure 3 shows the fuzzy groups created to classify each input and output variable. The fuzzy groups are represented by triangular membership functions, and the trapezoidal membership function is used for the last group. Each membership function is defined by values relative to the three vertices of a triangle and four vertices of a trapezoid.

Subsequently, the rules are defined using a database analysis. The variables are classified into five groups using the class intervals mentioned earlier. The procedure adopted was to fix a pair of input variables (‘vulnerability of the concession area’ and ‘size of the distribution utility’) and analyze the output variable (‘default rate’). A pattern of evidence was used to form this rule.

It should be noted that the rules were established considering the principles of incentive regulation. Therefore, when conflicting patterns were obtained in the process, the rule definition was made according to the pattern presented by the company(ies) with the best performance. In other words, once the two input variables ‘vulnerability’ and ‘size’ were fixed at values within a fuzzy group (high, for example), if different “default rates” (output variable) were presented with equal probability among the analysis of each of the distribution utilities, the rule definition would be based on the most efficient pattern.

4.3. Phase 3: Fuzzy Inference, Defuzzification, and Definition of Regulatory Targets

During this phase, an algorithm for fuzzy inference in the MATLAB^® software package was developed. The Mamdani-type FIS method [36] was used for the construction of the fuzzy rule base by employing the concept of a minimum (∧, linguistic connector ‘and’) and maximum (∨, linguistic connector ‘or’) so that output conditions could be inferred. In this study, considering the principles of incentive regulation, the minimum was used to activate the rules and form the output. The adoption of the minimum, in this case, makes the targets more restrictive, and thus closer to the results of the most efficient distribution utilities, aligning with the concept of yardstick competition.

The defuzzification method was chosen by testing all the major methods described in the literature [41] and available in the MATLAB^® software package [40]. Thus, a routine in the MATLAB^® software package was written to systematically calculate the default rates (with each of the defuzzification methods described earlier) and the root mean squared errors (RMSE) between (i) default estimatives using the fuzzy inference system and (ii) default rates measureded by the distribution utilities. This choice was based on the defuzzification method with the lowest RMSE.

After choosing the defuzzification method, the regulatory targets for limiting or reducing the electricity default rates estimated by this fuzzy inference approach should be compared with the targets determined by ANEEL and the actual default rates of the distribution utilities, which are discussed in detail in the next section.

5. Demonstration of the Applicability of the Proposed Model Focusing on the Residential Class of Electricity Consumption in Brazil

This section presents the results of an empirical study that aimed to demonstrate the applicability of the proposed model, focusing on the residential class of electricity consumption in Brazil. In this study, a fuzzy inference system (FIS) was used to estimate the default rates of 59 Brazilian electricity distribution utilities.

The central questions of this empirical study are as follows:

• Can we effectively demonstrate the applicability of the proposed model by focusing on the residential electricity consumption class in Brazil?
• Can the empirical results evidence the benefits of using a fuzzy inference-based model to establish regulatory targets to limit or reduce electricity default rates?

According to the Statistical Yearbook of 2023 (2022 Baseline Year), published by the Energy Research Office in Brazil, the residential class represents 45% of captive consumption, being the largest class in terms of consumption and consumers, where the representation percentage is 86% of the total consumers in the Brazilian captive market [42]. Therefore, a large part of distribution utilities’ billing default rates comes from this consumption class. This segment is undoubtedly strongly impacted by the socioeconomic conditions in which it operates, as well as by the size of the distribution utilities, because the larger the area to supervise, the lower their effectiveness.

5.1. Phase 1: Definition and Classification of Input Variables

Aligned to essential premises for developing an alternative model to define regulatory targets for electricity default rates in Brazil, a deep documentary analysis of the current ANEEL methodological approach was conducted to survey which input variables should be used. Thus, as a result of the efforts in Phase 1, tests were conducted with various variables and, ultimately, the selected input variables were classified into two categories: (i) vulnerability of the concession area and (ii) size of the distribution utility. The previous study published by Araujo [5] was the basis for the choice and classification of the input variables, as shown in

Table 2. Definition, classification of input variables and information sources.

Categories of Input Variables	Input Variables	Definition	Sources
Vulnerability of the concession area	Pob	Percentage of people with per capita income less than half the minimum wage	Brazilian Institute of Geography and Statistics (IBGE) and Institute of Applied Economic Research (IPEA)
	Vio	Deaths due to assault	Department of Informatics of the Unified Health System (DATASUS/Ministry of Health)
	Sub2	Percentage of people in subnormal households	Brazilian Institute of Geography and Statistics (IBGE)
	Gini	Gini Index	Brazilian Institute of Geography and Statistics (IBGE)
	Inadpf	Percentage of default rate among individuals in the credit sector	Central Bank of Brazil (BACEN)
	Rendtotal	Average income from formal and informal work	Brazilian Institute of Geography and Statistics (IBGE)
	Ocupform	Percentage of individuals aged 16 and over, formally employed during the reference week	Brazilian Institute of Geography and Statistics (IBGE)
	RGA	Percentage of households with piped water supply in at least one room (estimates based on growth rates from similar tables of PNAD/IBGE at the state level).	National Household Sample Survey (PNAD/IBGE)
	ADMD	Percentage of households with more than three residents per bedroom (high density of residents per household). Estimates based on growth rates from similar tables of PNAD/IBGE at the state level.	National Household Sample Survey (PNAD/IBGE)
	EEM	Percentage of individuals over 25 years old who dropped out of high school (estimates based on growth rates from similar tables of PNAD/IBGE at the state level).	National Household Sample Survey (PNAD/IBGE)
Size of the distribution utility	Market (GWh)	Billed consumption of electricity distribution utilities	Market Information Monitoring System for Economic Regulation (SAMP/ANEEL).
	Number of consumers	Total number of consumers of electricity distribution utilities	Market Information Monitoring System for Economic Regulation (SAMP/ANEEL).
	Transmission network length	Kilometers of transmission line network of electricity distribution utilities	Market Information Monitoring System for Economic Regulation (SAMP/ANEEL).

.

5.2. Phase 2: Database Preparation and Inputs for Fuzzy Inference

The value of input variables chosen to define ‘vulnerability of the concession area’ and ‘size of the distribution utility’ were normalized so that they could be compared on a single basis, subsequently obtaining their averages in each case. This process was conducted by calculating the mean and standard deviation of the different distribution utilities analyzed for each of the variables described in Table 2. Once these mean and standard deviation values were calculated, each variable (relative to a given distribution utility) was subtracted from its mean and the result was divided by the respective standard deviation. By repeating this process for all input variables (Table 2), a new set of normalized variables was obtained.

Finally, the ‘vulnerability of the concession area’ was calculated by averaging all normalized input variables recognized as representing this category. Similarly, ‘size of the distribution utility‘ was calculated. Thus, in Phase 2, fuzzy groups were obtained for the two categories of input variables (‘vulnerability of the concesssion area’ and ‘size of the distribution utility’) and the output variable (‘electricity default rate’), which can be visualized in Figure 4.

Another extremely important step in the specification of the model is the creation of fuzzy rules, where the adopted procedure is to fix a pair of inputs (‘vulnerability’ and ‘size’) and analyze the output variable (‘electricity default rate’).

A highlighted pattern is used to form the rule. However, it was observed that there was not always unanimity in the indication of the default rate for the same pair of inputs. In some cases, only one distribution utility was left as the representative of a rule. Thus, weights were assigned to the rules, with 1 assigned to those with well-defined patterns (at least two distribution utilities with the same pattern) and 0.6 for the other rules (one single distribution utility representing a rule and patterns of unclear repetition).

By way of illustration, in two cases with very high vulnerability and small distribution utilities—one with a very low default rate and the other with a low default rate—the rule is defined more restrictively (very low default), and a lower weight is assigned to the rule. Thus, less weight (less strength) is assigned to rules that have less clarity in their formation. It is important to mention that the intensity of the weight was defined through successive tests with different weight values until the value that provided the best response was reached. The shape and range of the membership functions only indicate the degree to which each rule is activated to generate an appropriate response.

Figure 5 presents the rules defined for the default rate of the residential consumption class. These rules can be read by observing the crossing of the rows (fuzzy groups of the first category of input variables—‘vulnerability of the concession area’) with the columns (fuzzy groups of the second category—‘size of the distribution utility’), resulting in an indication for the output variable (‘electricity default rate’).

From Figure 5, it can be inferred that a distribution utility in a concession area with a very high vulnerability index (VH) and small size (L) has a moderate default rate (M), and its fuzzy rule can be read as follows: If the vulnerability of the concession area is very high (VH) and the size of the distribution utility is small (L), then its default rate is moderate (M). However, the formation of this rule did not present a well-defined pattern, so a weight of 0.6 was assigned to this rule (hatched cell).

5.3. Phase 3: Fuzzy Inference, Defuzzification, and Definition of Regulatory Targets

Regarding the defuzzification methods, the choice was made by testing the main methods described in the literature [39] and available in the MATLAB^® software package [38] (i.e., centroid, bisector, mean of maximum (MoM), smallest of maximum (SoM), and largest of maximum (LoM)) and comparing the root mean squared errors (RMSE) of the fuzzy inference system estimates with those default rates declared by the distribution utilities. As can be seen in Figure 6, the centroid was the best method for defining regulatory targets for the residential consumption class.

Figure 7 shows a comparative analysis between the default rates declared by the distribution utilities (in the year base), the ANEEL regulatory targets, and default targets defined by the fuzzy inference system (FIS). It consists of a macro analysis in which the comparison is made through the ranges of the limits established for the fuzzy rule formulation. Therefore, it should be noted that the defined fuzzy rules were able to capture well the actual default rate (‘Actual’) declared by the distribution utilities, along with the input variables of ‘size of the distribution utility’ and ‘vulnerability of the concession area’ categories.

When the actual default rate is very low, this level is captured by both the ANEEL methodology and the modeled fuzzy inference system. However, at higher levels of actual default rates, the targets fixed by the fuzzy inference system proved to be more conservative than the ANEEL methodology ones, considering that it is based on the behavior of distribution utilities of the same ‘size’ and ‘vulnerability’ that present lower default rates, explicitly identifying those distribution utilities that are at a higher level than expected in the same context of similar utilities.

Figure 8 presents a graphic example of the application of the 15 fuzzy rules implemented on the two categories of input variables (‘vulnerability’ and ‘size’) and their consequences given by the default rate variable. The graphs in the first column show the amplitudes of the vulnerability variables on the horizontal axis. Similarly, the graphs in the second column represent the amplitude of the size variable on the horizontal axis, and the graphs in the third column represent the amplitude of the default rate variable. The red line indicates when the rules are activated to have the default rate in the output; the graph in the last line represents the output, considering the centroid as a metric.

Figure 8 shows that for a certain input value (represented by the red line), only some fuzzy rules are activated, namely, rules 4, 5, 7, and 8. Other fuzzy rules, such as 2, 3, 6, 9, 11, 12, and 15, do not generate an output variable (‘default rate’) because they do not present simultaneous activation of the two input variables. It is interesting to note that the rules that generate an output are always limited by the smallest activation between the inputs. This occurs because in the formulation of the fuzzy rules, the ‘AND’ connector was used, which represents the intersection or minimum.

In this example, a distribution utility with a low size and low vulnerability generates output responses corresponding to very low and low default rates. The union of all fuzzy output groups generates a fuzzy result, which when subjected to a defuzzification process, produces a numerical result. The results of the analysis performed for all the analyzed distribution utilities are presented next.

Figure 9, Figure 10 and Figure 11 show a point-by-point analysis comparing the deviations calculated against the actual default rates declared by the distribution utilities, considering targets determined by the fuzzy inference system and the ANEEL methodology. The results are described in the order of default rate intensity to facilitate visualization. It is possible to verify the greater stability of the FIS model compared to the ANEEL model. Furthermore, companies with low default rates are able to achieve their targets, which makes sense in yardstick competition mechanisms.

The graph in Figure 9 shows that 18 distribution utilities had residential default rates lower than 0.2%. The regulatory targets defined by the modeled fuzzy inference system suggest higher values than those actually achieved, but predominantly below 0.4%. It is noteworthy that both models showed much higher values for distribution utility, represented by the number 5. This distribution utility falls under the rules of moderate vulnerability (M), very low size (VL), and very low default rate (VL). This rule had a weight of 1, meaning that it was established because it had the majority of cases within the same pattern. However, the distribution utility under analysis has the lowest actual default rate in its group, averaging 97% below the level of the other distribution utilities.

The regulatory targets established by ANEEL for distribution utilities 9 and 16 also proved to be much higher than their declared default rates, with deviations of −0.7 and −0.4 percentage points, respectively.

The average deviation from the actual default rates was −0.35 percentage points for the modeled fuzzy inference system and −0.09 percentage points for the ANEEL methodology. Regarding the fuzzy inference system, all distribution utilities obtained targets that were above the actual level. When the same comparison was made regarding the ANEEL methodology, eight distribution utilities achieved targets that were higher than the actual ones.

Figure 10 shows the next 20 distribution utilities with the moderate actual default rates. It can be inferred that in eight cases (40% of the distribution utilities is in this group), the modeled fuzzy inference system suggests a regulatory target much higher than the actual default rates. In other cases, the estimate values were closer to the actual values than the regulatory targets established by ANEEL. In 25% of the distribution utilities in this group, the deviation between the actual default rates and targets defined by the fuzzy inference system was less than 0.09%. The same analysis regarding ANEEL methodology was found in 15% of the distribution utilities. The average deviation from the actual default rates was −0.08% for the targets defined by the fuzzy inference system and +0.08 percentage points for the targets determined by the ANEEL methodology. It is also possible to note the greater stability of the FIS model compared with the ANEEL methodology. Distribution utilities with average default rates can mainly achieve targets with the FIS model and do not have as many deviations from these targets.

Figure 11 compares the distribution utilities with the highest actual default rates. Again, the FIS model can be perceived as more stable compared with the ANEEL methodology. Furthermore, it is more difficult for distribution utilities in this group to meet their targets with the FIS model when compared to ANEEL, which aligns with the theory of yardstick competition.

When it comes to distribution utilities with higher default rates, as can be seen in Figure 11, the application of the modeled fuzzy inference system shows consistency with incentive regulation principles, performing well in suggesting lower regulatory targets for distribution utilities that are above where they should be, considering the pattern of other distribution utilities with similar ‘vulnerability’ and ‘size’ conditions. Thus, the average deviation from the actual value (actual less estimate) was higher than that presented in the other graphs, at 1.07 percentage points for the application of the fuzzy inference system and 0.67 percentage points for the ANEEL methodology.

In 71% of the distribution utilities in this group, the actual declared default rates were above the fuzzy inference estimates. The same analysis using the ANEEL methodology showed a percentage of 67%. Both methodologies showed a 62% deviation in this group of distribution utilities of less than one percentage point.

Based on the results presented in this section, it can be seen that, particularly among the distribution utilities with the lowest percentages of actual declared default rates, the application of the fuzzy inference system lacks greater accuracy. However, 76% of the deviations presented by this application are less than or equal to 0.5 percentage points in absolute values, which is equivalent to the ANEEL methodology, which has the same percentage of 75%.

The results of the empirical study provide strong support for the formulated hypotheses. Specifically, the findings reveal that the fuzzy inference-based model for defining regulatory default targets demonstrates clear alignment with the principles of yardstick competition regulation in the electricity sector. By incorporating these principles into the model design and rule formulation process, the targets generated by the model reflected the desired regulatory outcomes aimed at promoting efficiency and performance improvement among distribution utilities (H1).

The results of the empirical study focusing on the residential class of electricity consumption provided compelling evidence that the fuzzy inference approach effectively determined regulatory default targets that were tailored to the specific characteristics and operational realities of each electricity distribution utility in Brazil. This demonstrated the model’s ability to adapt to diverse contexts and deliver targeted regulatory solutions (H2).

The study findings indicated that the regulatory default targets derived from the fuzzy inference-based model were comparable to or even more accurate than the targets established by ANEEL for limiting or reducing default rates in the electricity sector. This validation underscored the model’s capacity to provide regulatory guidance that aligns with industry standards and regulatory objectives (H3). In fact it represents an improvement over the ANEEL methodology as it is a model that is easy to apply and was developed exclusively for defining regulatory default targets, detached from the methodology for calculating non-technical losses, and easy to apply. It was also observed that the more adherent the variables were to the reality of the concession areas and distribution utilities, the better the results.

Additionally, the study confirmed that variations in the input variables ‘vulnerability of the concession area’ and ‘size of the distribution utility’ significantly influenced the determination of regulatory default targets in the fuzzy inference-based model, validating the fourth hypothesis (H4). The observed impact of these variables on target outcomes highlighted the model’s sensitivity to key factors affecting default rates and its ability to adjust targets based on varying concession area vulnerabilities and distribution utility sizes.

6. Discussion

The results of the empirical study emphasise the efficacy of the proposed fuzzy inference-based model in improving incentive-based regulatory practices to limit or reduce electricity default rates in Brazil. They also highlighted the study’s contributions to the research fields of electricity default regulation and the applications of fuzzy inference systems in the regulation of electricity distribution at the national level. From the incentive-based regulatory perspective, the key issue regarding this challenge is devising incentive mechanisms that reward distribution utilities for their operational and investment choices, aiming to mitigate or decrease electricity non-payment rates and increase electricity tariffs for regular customers.

In Brazil, the definition of regulatory default targets integrates the methodology used by ANEEL to establish regulatory targets to combat non-technical losses (NTLs). Consequently, this methodological approach does not consider some crucial variables in the context of defaults on energy bills, which have been criticized by agents of the Brazilian electricity sector and academic experts [18,19]. Therefore, a major challenge is to find a robust methodology that is simple to implement and capable of capturing the heterogeneity that exists in concession areas while maintaining the incentive for efficiency and innovation.

The results presented in Section 4 were compared with the methodological approach employed by ANEEL. Overall, there were no significant differences between the application of the fuzzy inference system and the ANEEL methodology in terms of deviations. However, among the distribution utilities with the lowest and highest actual declared default rates, the results indicated that the fuzzy inference system exhibits greater accuracy. Another issue that deserves special attention is the distribution of utilities with default rates that are close to zero. Such distribution utilities should be treated differently because they are already very close to the optimal efficiency level. In this case, it is worth evaluating whether they can further reduce their default rates or whether their current level should be fully recognized as an incentive for other distribution companies to pursue such targets.

The fuzzy inference-based model demonstrated superior capabilities in defining regulatory targets for limiting or reducing electricity default rates compared with the existing methodological approach. The proposed model exhibits robustness and precision in addressing the complexity associated with the definition of regulatory targets for electricity default rates. This comparison underscores the potential of the fuzzy inference-based model in enhancing the regulatory framework for electricity defaults in Brazil by providing a more effective and tailored solution.

This study’s contribution to the research field of electricity default regulation is noteworthy, particularly in light of the limited existing literature on this subject [5,6,7,8,9,10,11]. The previous studies reviewed in Section 1 show the many dimensions of the electricity default issue and reveal various unresolved regulatory considerations, particularly methodologies for defining regulatory targets aimed at limiting or decreasing electricity default rates at national or municipal levels.

Zhow et al. [6] established a logistic regression model to predict default probability based on the latest data from the client that the electricity distribution utility could acquire. This model is attractive for forecasting the default effect of a single distribution. However, in this case, it is only interesting if the data have to be seen as panel data, with the variable considered for all distribution utilities.

Murwirapachena et al. [10] employed panel data with random effects to estimate the relationship between municipal financial performance and non-payment for 28 municipalities in South Africa. This method is interesting because it is similar to that used by ANEEL to predict NTL, in which the outputs are input for the model to establish default rates proposed by this agency. As expected, the results confirm that non-payment negatively impacts municipalities’ financial performance; the same happens with utilities in the case of the model used by ANEEL.

Moita et al. [11] demonstrated that an increase (decrease) in tariffs leads to an increase (decrease) in default occurrences. Owing to the higher tariff, consumers need help to reduce their electricity consumption in the short term, increasing defaults. These results are corroborated by the study proposed by Fowlie et al. [9], who found that consumers adjusted their electricity consumption in response to time-varying prices, even if they did not actively select them.

The findings of these two studies [9,11] deserve mention in this discussion, as they show the importance of the incentive regulation model proposed here. Thus, distribution utilities have incentives to reduce their default rates considering the targets imposed by the regulator; only when they reach these targets can they pass through the investment costs to reduce these rates to the energy tariff. If they do not, the tariff tends to be at a lower price level, which may reduce these rates. Therefore, it is necessary to have a tariff at a level at which the default rate can be controlled: the equilibrium point. This balance can be achieved if the targets for reducing default rates are well calibrated. Low revenue collection rates make it difficult for utilities to maintain and upgrade infrastructure, which could further worsen the service quality [8]. The FIS model proposed in this study can contribute to this objective because strategies focusing on building relationships, providing information, and fostering reciprocity between service providers and consumers can contribute to sustainable solutions, as stated by Szabo and Ujhelyi [7].

Compared with the methodological approach adopted by ANEEL and the previous studies reviewed in Section 3, this study offers a novel approach for addressing electricity default rates in Brazil through the proposition of a fuzzy inference-based model. Aligned with the principles of incentive regulation, this model represents a comprehensive method for setting regulatory targets that incentivize distribution utilities to effectively address nonpayment issues.

Aiming to fill a crucial gap in the literature, the proposed model provides a holistic and innovative solution to the challenges posed by electricity defaults in the regulatory context, thereby advancing the understanding and practice of incentive regulatory mechanisms in the electricity sector. As discussed below, no prior research has specifically focused on establishing regulatory targets to limit or reduce electricity default rates independently of the current ANEEL methodological approach for non-technical loss regulation.

Another notable contribution of this study is the application of fuzzy inference systems (FIS) in the context of yardstick competition regulation. By adapting FIS to address the research gaps posed in Section 1, this study demonstrates the versatility and effectiveness of fuzzy logic in regulatory decision-making processes. The innovative use of FIS in this context highlights their adaptability and potential to enhance regulatory frameworks in the energy sector. This study expands the scope of FIS applications and underscores their relevance in addressing real-world regulatory challenges, thereby contributing to the advancement of regulatory practices in the electricity distribution sector.

As previously stated, studies that address the problem of electricity default rates will contribute to modelling and predicting such a phenomenon in the future, or at least explain it. The model proposed here for defining default rates in the electricity market was developed from this perspective. Through the proposition of a fuzzy inference-based model aligned with the principles of incentive regulation, this study offers a novel approach for addressing electricity default rates with the potential to enhance regulatory frameworks in the electricity distribution sector in Brazil.

7. Conclusions

In this paper a fuzzy inference-based model is presented to effectively support the definition of regulatory targets for limiting or reducing electricity default rates at the national level. The definition of regulatory targets for default rates is of utmost importance for electricity distribution utilities, national regulatory organisms, and all consumers because it strongly impacts energy tariffs. A significant challenge is finding a robust methodology that is easy to implement and capable of capturing the heterogeneity existing in distribution concession areas while maintaining incentives for efficiency and innovation.

The main conclusions drawn from this study are as follows:

• The proposed fuzzy inference-based model offers a promising approach for defining regulatory targets to limit or reduce default electricity rates in Brazil. The model demonstrates superior capabilities compared with the current methodology employed by ANEEL;
• This study contributes significantly to the research on electricity defaults from a regulatory perspective by introducing a novel and effective method for addressing non-payment issues in the electricity distribution sector. The model aligns with the principles of incentive regulation and provides a comprehensive solution to incentivize distribution utilities to mitigate their default rates;
• The exploration of fuzzy inference systems (FIS) in the context of incentive regulation, particularly within the yardstick competition modality, showcases the adaptability and effectiveness of fuzzy logic in regulatory decision-making processes. This study expands the application of the FIS and highlights its potential to enhance regulatory frameworks in the electricity distribution sector;
• The empirical results obtained by applying the fuzzy inference-based model to the residential class of electricity consumption in Brazil demonstrated the feasibility and efficacy of the model in setting regulatory targets. The ability of the model to align with the reality of each electricity distribution utility underscores its practical relevance and applicability across different categories of electricity consumption.

Considering the research questions formulated in Section 1, particularly the first and second questions, the main limitation of the current methodological approach adopted by ANEEL for defining regulatory targets for electricity default rates in Brazil is that it is linked to the methodology for combating non-technical losses, which has been criticized by agents of the electricity distribution sector and the academic community. In addition, studies in the literature reveal many facets of the electricity default issue, revealing various unresolved regulatory questions, including methodologies for defining regulatory targets aimed at limiting or decreasing electricity default rates in a country.

The proposed fuzzy inference-based model could capture the heterogeneity of concession areas and be simple to implement, providing a sense of reassurance and confidence. In addition, an independent methodology for defining regulatory default targets benefits all stakeholders involved because fuzzy inference is a simple implementation methodology that is widely used in environments with some degree of uncertainty. This allows the implementation of rules arising from the performance of distribution utilities in different socioeconomic situations and constitutes a powerful tool for defining regulatory default targets.

Regarding the third research question, the results of using a fuzzy inference-based model to obtain regulatory targets for default rates in the residential consumption class could demonstrate that it is possible to persist in this direction, as the proposed methodology is detached from the non-technical losses methodology, maintaining the principles of incentive regulation and capturing the heterogeneity of the various concession areas. The fuzzy inference-based model yielded direct results that were very close to those obtained using the ANEEL methodology.

To address the fourth research question, we established a set of essential premises before defining and classifying input variables. Once the essential premises were defined, the next step was a deep documentary analysis of the current ANEEL methodology to define FIS input variables. The next step was to group these variables into two categories: (i) vulnerability and (ii) size of the distribution utilities. Using only two input categories, that is, the vulnerability and size of the distribution utilities, we could conduct a correct classification of distribution utilities in different concession areas and simplify the implementation of the model while ensuring comparability between distribution utilities and maintaining incentive regulation. We conclude that choice of variables comprising these two inputs is fundamental to the efficiency of the model. This topic could be discussed in future work with the distribution utilities themselves, adding more value to the analysis given their accumulated experience.

With respect to the fifth and sixth research questions, it was feasible to demonstrate the applicability of the proposed model by focusing on the residential electricity consumption class in Brazil. The empirical results demonstrated the benefits of using a fuzzy inference-based model to establish regulatory targets to limit or reduce electricity default rates. It was possible to notice that in terms of deviations, there were no significant differences between the application of the FIS and that of the ANEEL methodology. The fuzzy inference system proposed in this study represents an improvement over the ANEEL methodology, as it is a model that is easy to apply and was developed exclusively for defining regulatory default targets, is detached from the method of calculating non-technical losses, and is easy to use. It was also observed that the more adherent the variables were to the reality of the concession areas and distribution utilities, the better the results.

The findings of this study have significant implications for policymakers in the energy sector, particularly for regulatory authorities, such as ANEEL. The fuzzy inference-based model offers a practical and data-driven approach to setting regulatory targets that align with incentive regulation principles, since the impacts of electricity default directly or indirectly influence the disposable income of the population, especially those with a higher percentage of income committed to paying the energy bill, who may be affected by the portion of unrecoverable revenue transferred to the tariffs of all consumers. Policymakers can leverage this model to incentivize distribution utilities to improve operational efficiency, reduce default rates, and enhance the overall system sustainability. It also serves as a policy to incentivize investments and efficiency in distribution utilities, making it a topic that deserves special attention from ANEEL. Implementing such innovative regulatory tools can lead to more effective decision making and better outcomes for both utilities and consumers.

Although defaults by electricity distribution utilities are related to the non-technical loss (NTL) problem, it is crucial that ANEEL uses a separate methodology for measuring it, distinct from that used for defining NTLs, as discussed before. Thus, the main policy implication of the future adoption of the model proposed in this study is its decoupling from the NTL methodology. This differentiation is significant because some distribution utilities exhibit low default rates despite high rates of non-technical losses and vice versa.

Despite the advancements made in this study, some limitations should be acknowledged. One limitation pertains to the focus on the residential electricity consumption class, which may not fully capture the complexity of default rates across all consumer categories. Additionally, regarding the rule formation process, we decided to establish rules only through the database, which creates gaps owing to possible rules that need to be more representative in the database.

To build upon the findings of this study and further advance in this research field, several avenues for future work can be explored.

First, expanding the application of the fuzzy inference-based model to other electricity consumption classes could provide a more comprehensive understanding of its effectiveness in diverse contexts. Additionally, its robustness can be tested using different input variables. Regarding the definition of rules, it is also possible to consider establishing rules through research by specialists working in distribution utilities themselves.

Additionally, the construction of specific factors for classes linked to the public sector can be evaluated, as the vulnerability of these segments involves different issues. Alternatively, for the entire public segment, one could think of more significant penalties in the event of default to encourage payment in real time, which, coming from the public sector, would be the minimum expected. In this way, lower default rates (very close to zero) can be fully recognized in consumer tariffs without establishing regulatory default targets. In addition, in future work, rules can be created based on the same formation logic as the proposal in this study, with consultation and validation by experts.

Finally, incorporating real-time data and advanced modeling techniques such as machine learning algorithms can enhance a model’s predictive capabilities and responsiveness to dynamic market conditions. Furthermore, conducting longitudinal studies to assess the long-term impact of regulatory target sets using a fuzzy inference-based model can offer valuable insights into its sustainability and efficacy over time.