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<journal-id journal-id-type="publisher-id">Front. Energy Res.</journal-id>
<journal-title>Frontiers in Energy Research</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Energy Res.</abbrev-journal-title>
<issn pub-type="epub">2296-598X</issn>
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<publisher-name>Frontiers Media S.A.</publisher-name>
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<article-id pub-id-type="publisher-id">1366465</article-id>
<article-id pub-id-type="doi">10.3389/fenrg.2024.1366465</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Energy Research</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>A data-driven ensemble technique for the detection of false data injection attacks in the smart grid framework</article-title>
<alt-title alt-title-type="left-running-head">Gupta et al.</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/fenrg.2024.1366465">10.3389/fenrg.2024.1366465</ext-link>
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<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Gupta</surname>
<given-names>Tania</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
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<name>
<surname>Bhatia</surname>
<given-names>Richa</given-names>
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<xref ref-type="aff" rid="aff2">
<sup>2</sup>
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<name>
<surname>Sharma</surname>
<given-names>Sachin</given-names>
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<sup>3</sup>
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<name>
<surname>Reddy</surname>
<given-names>Ch. Rami</given-names>
</name>
<xref ref-type="aff" rid="aff4">
<sup>4</sup>
</xref>
<xref ref-type="aff" rid="aff5">
<sup>5</sup>
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<contrib contrib-type="author">
<name>
<surname>AboRas</surname>
<given-names>Kareem M.</given-names>
</name>
<xref ref-type="aff" rid="aff6">
<sup>6</sup>
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<name>
<surname>Mobarak</surname>
<given-names>Wael</given-names>
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<xref ref-type="aff" rid="aff7">
<sup>7</sup>
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<aff id="aff1">
<sup>1</sup>
<institution>Department of Electronics and Communication</institution>, <institution>NSUT East Campus (Affiliated to GGSIPU)</institution>, <addr-line>Delhi</addr-line>, <country>India</country>
</aff>
<aff id="aff2">
<sup>2</sup>
<institution>Netaji Subhash University of Technology</institution>, <addr-line>Delhi</addr-line>, <country>India</country>
</aff>
<aff id="aff3">
<sup>3</sup>
<institution>Electrical Engineering Department</institution>, <institution>Graphic Era Deemed to be University</institution>, <addr-line>Dehradun</addr-line>, <country>India</country>
</aff>
<aff id="aff4">
<sup>4</sup>
<institution>Department of Electrical and Electronics Engineering</institution>, <institution>Joginpally B. R. Engineering College</institution>, <addr-line>Hyderabad</addr-line>, <country>India</country>
</aff>
<aff id="aff5">
<sup>5</sup>
<institution>Applied Science Research Center</institution>, <institution>Applied Science Private University</institution>, <addr-line>Amman</addr-line>, <country>Jordan</country>
</aff>
<aff id="aff6">
<sup>6</sup>
<institution>Department of Electrical Power and Machines</institution>, <institution>Faculty of Engineering</institution>, <institution>Alexandria University</institution>, <addr-line>Alexandria</addr-line>, <country>Egypt</country>
</aff>
<aff id="aff7">
<sup>7</sup>
<institution>College of Engineering</institution>, <institution>University of Business and Technology</institution>, <addr-line>Jeddah</addr-line>, <country>Saudi Arabia</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1789076/overview">Flah Aymen</ext-link>, &#xc9;cole Nationale d&#x27;Ing&#xe9;nieurs de Gab&#xe8;s, Tunisia</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1304113/overview">Uchenna Diala</ext-link>, University of Derby, United Kingdom</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1773557/overview">Nikolaos Manousakis</ext-link>, University of West Attica, Greece</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Tania Gupta, <email>taniagupta2409@gmail.com</email>; Ch. Rami Reddy, <email>crreddy229@gmail.com</email>; Wael Mobarak, <email>w.fawzy@ubt.edu.sa</email>
</corresp>
</author-notes>
<pub-date pub-type="epub">
<day>26</day>
<month>04</month>
<year>2024</year>
</pub-date>
<pub-date pub-type="collection">
<year>2024</year>
</pub-date>
<volume>12</volume>
<elocation-id>1366465</elocation-id>
<history>
<date date-type="received">
<day>06</day>
<month>01</month>
<year>2024</year>
</date>
<date date-type="accepted">
<day>01</day>
<month>04</month>
<year>2024</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2024 Gupta, Bhatia, Sharma, Reddy, AboRas and Mobarak.</copyright-statement>
<copyright-year>2024</copyright-year>
<copyright-holder>Gupta, Bhatia, Sharma, Reddy, AboRas and Mobarak</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<p>The major component of the smart grid (SG) is the advanced metering infrastructure (AMI), which monitors and controls the existing power system and provides interactive services for invoicing and electricity usage management with the utility. Including a cyber-layer in the metering system allows two-way communication but creates a new opportunity for energy theft, resulting in significant monetary loss. This article proposes an approach to detecting abnormal consumption patterns using energy metering data based on the ensemble technique AdaBoost, a boosting algorithm. Different statistical and descriptive features are retrieved from metering data samples, which account for extreme conditions. The model is trained for malicious and non-malicious data for five different attack scenarios, which are analyzed on the Irish Social Science Data Archive (ISSDA) smart meter dataset. In contrast to prior supervised techniques, it works well even with unbalanced data. The efficacy of the proposed theft detection method has been evaluated by comparing the accuracy, precision, recall, and F1 score with the other well-known approaches in the literature.</p>
</abstract>
<kwd-group>
<kwd>advanced metering infrastructure</kwd>
<kwd>cyber security</kwd>
<kwd>false data injection attacks</kwd>
<kwd>feature extraction</kwd>
<kwd>machine learning</kwd>
<kwd>smart meter</kwd>
</kwd-group>
<custom-meta-wrap>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Smart Grids</meta-value>
</custom-meta>
</custom-meta-wrap>
</article-meta>
</front>
<body>
<sec id="s1">
<title>1 Introduction</title>
<p>The notable characteristics of the smart grid (SG) that increase the effectiveness of the current power system are indeed the two-way power and information exchange. Energy theft has been a severe challenge in the traditional power grid worldwide. Almost all utilities worldwide suffer significant financial losses due to energy theft, primarily in developing countries (<xref ref-type="bibr" rid="B25">Keping et al., 2015</xref>). Based on the most recent published research by Northeast Group, LLC, stealing energy costs the world $89.3 billion/year, among which the world&#x2019;s top 50 emerging-market countries lose $58.7 billion/year (<xref ref-type="bibr" rid="B44">Xia et al., 2019</xref>). In contrast to the old grid, which manually collects customer billing information monthly, the new SG measures consumer energy consumption minute by minute for each device installed at user premises (<xref ref-type="bibr" rid="B15">Gupta and Bhatia, 2020</xref>). This aids the utility in managing loads, providing user billing information, and managing energy utilization (<xref ref-type="bibr" rid="B48">Yu et al., 2021</xref>). By providing monitoring capabilities through numerous sensors and accurate readings, the SG claims to lower the risk of energy theft by giving the power utility billing data and price information at a higher frequency, i.e., on an hour-to-hour basis (<xref ref-type="bibr" rid="B49">Zhang et al., 2017</xref>).</p>
<p>However, since the SG relies more extensively on information and communication technologies, there are more potential cyber-attack threats, which reduce the SG&#x2019;s reliability and result in significant operational and monetary losses (<xref ref-type="bibr" rid="B5">Attia et al., 2018</xref>; <xref ref-type="bibr" rid="B21">Jiang and Li, 2022</xref>). There are two entities of electricity losses: technical losses (TLs) and non-technical losses (NTLs) in the SG system. TLs are power losses incurred during electricity generation and transmission. The NTL category includes energy theft, and it states that the most common causes of NTLs include conventional methods such as meter reading bypassing, communication network failures, meter spoofing, and tampering with meter readings using a magnet (<xref ref-type="bibr" rid="B26">Kong et al., 2023</xref>). However, due to the introduction of an intelligent digital metering system and the inclusion of an internet layer in the metering system, there are several new entrance points for energy theft in addition to the conventional methods (<xref ref-type="bibr" rid="B39">Sun et al., 2018</xref>; <xref ref-type="bibr" rid="B53">Zhang et al., 2021</xref>). As a result, it draws the attention of researchers to the SG&#x2019;s cyber security (<xref ref-type="bibr" rid="B19">Jain et al., 2022</xref>). Mechanical meters in the old grid can only be adjusted physically. In contrast, advanced metering infrastructure (AMI) metering data open the door for both physical and remote adjustments (<xref ref-type="bibr" rid="B38">Song et al., 2022</xref>). Energy theft attacks against the SG could be initiated by malicious users who manipulate their smart meters to claim lower consumption readings to cut their energy bills illegally (<xref ref-type="bibr" rid="B28">Lin et al., 2022</xref>). Thus, the need to locate that malicious user and secure the system is of utmost importance (<xref ref-type="bibr" rid="B34">Mrabet et al., 2018</xref>; <xref ref-type="bibr" rid="B36">Pengwah et al., 2023</xref>).</p>
<p>Historically, to discover irregular energy usage, technicians must examine consumer monthly consumption data collected over an extended period, and after that, they must physically visit each resident community to confirm the condition and connection of each meter (<xref ref-type="bibr" rid="B11">Cheng et al., 2017</xref>; <xref ref-type="bibr" rid="B50">Zhang et al., 2023a</xref>). Due to research into machine learning (ML) techniques, power utilities now have a new opportunity to identify unusual electricity usage patterns from a variety of energy data (<xref ref-type="bibr" rid="B51">Zhang et al., 2023b</xref>; <xref ref-type="bibr" rid="B40">Tan et al., 2023</xref>). By identifying anomalous patterns, these techniques can reduce the workload for system operators and increase detection accuracy (<xref ref-type="bibr" rid="B13">Guarda et al., 2023</xref>). As per previously available architecture, systems for detecting energy theft are classified into three groups: state-based or power-based, game-based, and artificial intelligence (AI)-based approaches (<xref ref-type="bibr" rid="B24">Jokar et al., 2016</xref>), as depicted in <xref ref-type="fig" rid="F1">Figure 1</xref>. In a state-based approach, specific instruments or metering devices were designed to combat energy theft. For instance, a hardware-based method was proposed for identifying fake users (<xref ref-type="bibr" rid="B30">Liu X. et al., 2023</xref>; <xref ref-type="bibr" rid="B41">Wang et al., 2024</xref>). Various sorts of sensors and radio frequency identification tags are used in this system to identify the malicious user. Additional metrics such as power, voltage, and current are used in the distribution network to detect electricity theft (<xref ref-type="bibr" rid="B42">Wang et al., 2021</xref>; <xref ref-type="bibr" rid="B52">Zhang et al., 2024</xref>). Despite being costly to install and operate, this system has a good detection rate. Extra devices entail additional expenditures, and such device types are challenging to deploy within the current distribution system (<xref ref-type="bibr" rid="B45">Xiao et al., 2013</xref>; <xref ref-type="bibr" rid="B17">Henriques et al., 2014</xref>). The game theory-based approach assumes a game between the service supplier and fraudulent users. This strategy was based on sound assumptions. Actual user consumption data are derived from the game equilibrium. This has been theoretically calculated (<xref ref-type="bibr" rid="B4">Amin et al., 2012</xref>). However, it must still be solved to articulate the utility function of all stakeholders, including attackers, authorities, suppliers, and alternative solutions (<xref ref-type="bibr" rid="B3">Amin, 2015</xref>; <xref ref-type="bibr" rid="B43">Wang et al., 2023</xref>).</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Existing energy theft detection approaches.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g001.tif"/>
</fig>
<p>The third group includes AI-based methods: AMI uses ML algorithms to assess customers&#x2019; metering data and energy usage patterns to identify those who may be committing electricity theft (<xref ref-type="bibr" rid="B16">Gupta et al., 2022</xref>; <xref ref-type="bibr" rid="B29">Liu D. et al., 2023</xref>). In this, there are primarily two sorts of schemes: classification-based and clustering-based. Classification approaches often involve analyzing users&#x2019; past electricity consumption data with labels to identify odd trends and detect suspected electricity theft behaviors. It needs a dataset with labels (<xref ref-type="bibr" rid="B20">Jiang et al., 2021</xref>; <xref ref-type="bibr" rid="B10">Chen Y. et al., 2022</xref>). The metering data are utilized for training the classifier, which then identifies dishonest users. In contrast, clustering approaches rely on the information without labels; i.e., by studying the relationship between users, outliers are identified (<xref ref-type="bibr" rid="B23">Jokar et al., 2013</xref>; <xref ref-type="bibr" rid="B47">Yang et al., 2016</xref>; <xref ref-type="bibr" rid="B37">Sharma et al., 2023</xref>). Consumers often follow the same pattern under normal circumstances; hence, deviations from this pattern may indicate the presence of fraud. The classifier is trained using various ML techniques using a metering dataset available widely for research purposes and further used to detect unusual patterns, such as malicious users (<xref ref-type="bibr" rid="B9">Chen B. et al., 2022</xref>; <xref ref-type="bibr" rid="B32">Ma and Hu, 2022</xref>). The classifier&#x2019;s primary flaw was its poor detection rate and high rate of false positives. Smart meter historical data are the foundation for the clustering models, subject to significant dataset fluctuations that provide a broad range of normal data and low detection rates (<xref ref-type="bibr" rid="B14">Guo and Hu, 2023</xref>; <xref ref-type="bibr" rid="B54">Zheng et al., 2023</xref>). This makes it very likely that the malicious data that the adversaries introduce will go undetected (<xref ref-type="bibr" rid="B27">Li and Li, 2023</xref>; <xref ref-type="bibr" rid="B33">Mo and Yang, 2023</xref>). Therefore, there is a requirement for a detection method that overcomes the abovementioned restrictions.</p>
<p>Unbalanced or abnormal data are one of the alarming issues with the current classifier. Real-time samples of normal data are easily available, but fetching theft samples is difficult. On the other hand, theft samples are rare or non-existent for a customer. In addition, algorithms based on classification are susceptible to attacks on the data values, and accepting faulty consumption values by adversaries can contaminate the dataset (<xref ref-type="bibr" rid="B47">Yang et al., 2016</xref>). If this factor is not considered properly, it results in a higher false-positive rate. According to what the author has revealed, a false positive will prove expensive because when a malicious user is recognized, a significant amount of procedure is required from the utility. In-person inspection is one of the steps that must be completed before an attack can be considered valid for final verification. Therefore, it is essential to create an adequate model of energy theft detection to overcome these limitations.</p>
<p>This research article introduces a robust energy theft detection system leveraging smart metering data using the AdaBoost ensemble method. The proposed approach addresses the evasion techniques observed in existing classification-based theft detection systems. A comprehensive threat model is presented, accounting for various false data injection (FDI) attack scenarios. The system acknowledges non-malicious elements influencing consumption patterns, including occupant changes, weather variations, and appliance modifications. By incorporating these factors, our method achieves a superior detection rate compared to other available schemes. Experimental assessments were conducted across diverse FDI attack scenarios, benchmarking against state-of-the-art methods such as SVM, LR, KNN, NB, and RFC. The comparative analysis encompassed various performance metrics, demonstrating the effectiveness of our proposed system in enhancing energy theft detection accuracy and resilience against deceptive strategies.</p>
<p>The remainder of this paper is structured as follows: in <xref ref-type="sec" rid="s2">Section 2</xref>, the relevant work on FDI threats is discussed. The system model of an SG monitoring system is discussed in <xref ref-type="sec" rid="s3">Section 3</xref>. The suggested attack detection mechanism is described in <xref ref-type="sec" rid="s4">Section 4</xref>. The performance of the suggested approach is examined and compared to other available methods in <xref ref-type="sec" rid="s5">Section 5</xref>. This paper concludes with <xref ref-type="sec" rid="s6">Section 6</xref>.</p>
</sec>
<sec id="s2">
<title>2 Related work</title>
<p>This section discusses the studies conducted on the SG&#x2019;s security. We are using smart meter consumption data to identify unethical users. In conventional power networks, analyzing consumer load profiles for indications of energy theft has drawn the interest of experts in the past (<xref ref-type="bibr" rid="B8">Cao et al., 2020</xref>; <xref ref-type="bibr" rid="B46">Yang et al., 2023</xref>). The majority of recently published works in the literature are devoted to the detection of fraud. AMI daily smart meter readings were used to estimate the consumption pattern of clients using support vector machines (SVMs). The classifier was trained with normal data and thieved sample data from the past. The load profiles of the smart meter malicious user were proposed in a classification-based energy theft detection system. The identifier was educated using historical data from theft and normal sample populations. New samples were categorized based on criteria and SVM outcomes. In a multiclass study, SVM was trained to distinguish between regular and malicious load profile samples. Creating a synthetic dataset addresses and resolves the issue of uneven training datasets (<xref ref-type="bibr" rid="B24">Jokar et al., 2016</xref>; <xref ref-type="bibr" rid="B1">Ahmad et al., 2018</xref>). It is among the most recent models for detecting power theft (<xref ref-type="bibr" rid="B31">Lyu et al., 2024</xref>). It creates a hyperplane to divide the various classes. The XGBoost-supervised technique was proposed to detect the non-malicious user (<xref ref-type="bibr" rid="B7">Buzau et al., 2019</xref>). The method based on this ML approach analyzes customer behavior patterns from past kWh consumption data and identifies anomalous activity. A back propagation neural network was constructed and used to analyze SG energy theft (<xref ref-type="bibr" rid="B12">Depuru et al., 2011</xref>). The SVM parameters were estimated via a neural network model to reduce the training time of the classifier. Additionally, a data encoding technique was suggested to increase the classifier&#x2019;s effectiveness and speed. However, their system only works to identify energy theft attempts that provide zero-use reports. The metering data are encoded into binary values and transformed at one process phase. As a result, various attack types cannot be detected using the suggested categorization approach. To assess SG power theft, a broad and deep convolutional neural network model was created (<xref ref-type="bibr" rid="B55">Zheng et al., 2018</xref>). To investigate the attack path for false data injections against AC-based state estimation in power systems, we presented a new semidefinite programming-based convexification framework that detects globally optimal stealth attacks (<xref ref-type="bibr" rid="B22">Jin et al., 2019</xref>). In <xref ref-type="bibr" rid="B2">Alexopoulos et al. (2020</xref>), in the case of zero-injection buses, FDI attacks against a PMU linear state estimator based on Cartesian formulation were investigated with the presumption that the attacker would probably attempt to tamper with as few measurements as feasible. A novel hybrid attack (<xref ref-type="bibr" rid="B35">Pei et al., 2020</xref>) offered a low-cost attack mechanism that attackers could simply use to target buses with limited connectivity based on state estimation. To achieve observability for the entire system, this algorithm deployed extra-phase measurement units based on a greedy approach after prioritizing the protection of the most susceptible buses in the first phase. The new energy data sample is categorized using the K-means technique based on the similarity measure. It is one of the simplest methods available (<xref ref-type="bibr" rid="B6">Aziz et al., 2020</xref>).</p>
<p>However, many AI-based approaches need more precision for specific reasons. Due to the difficulties in obtaining labeling datasets of electrical thefts, i.e., proper preset thresholds and some external knowledge, the application of classification algorithms is restricted. It makes it harder to achieve in real-time situations, compromising detection accuracy. Unsupervised clustering cannot detect tampered load profiles with standard forms, resulting in low detection precision. Neural networks, for instance, are susceptible to overfitting since they learn the training examples exceptionally well but fail to generalize to new samples. Consequently, an effective system for detecting energy theft is essential to overcome these restrictions. Intending to develop a solution with low computing costs, better accuracy, and fewer false detections, we use the <italic>AdaBoost</italic> method to detect a stealthy attack on smart meter readings in this study. By creating a synthetic attack dataset and assuming that stealing trends are foreseeable, we can solve the issue of unbalanced data. The use of the <italic>AdaBoost</italic> algorithm is motivated by the reasons listed below.<list list-type="simple">
<list-item>
<p>1. Compared to most learning algorithms, the AdaBoost algorithm is less prone to overfitting and corrects misclassifications generated by poor classifiers. The classifiers based on this model have positive performance for anomaly detection problems.</p>
</list-item>
<list-item>
<p>2. Finding relationships between features in large datasets is challenging due to the various feature types. By integrating the weak learners for statistical attributes and descriptive attributes into a strong classifier, the links between these two different types of attributes are managed organically, regardless of any forced conversions between statistical and descriptive features of the dataset.</p>
</list-item>
<list-item>
<p>3. The <italic>AdaBoost</italic> technique is extremely quick when using straightforward, weak classifiers. Considering all the points listed above, we select the ensemble technique. In the proposed design, we put much effort into creating a reliable system that can be installed in the control center and use the data from the smart meter to detect suspicious energy readings and demand data that have been tampered with.</p>
</list-item>
</list>
</p>
<p>The proposed algorithm is created for various FDI attack scenarios to lessen the chance of the power system experiencing financial loss. The suggested approach was created to effectively anticipate various cyberattacks.</p>
</sec>
<sec id="s3">
<title>3 System model</title>
<p>This section discusses the AMI network and attack models used in this article.</p>
<sec id="s3-1">
<title>3.1 Network model</title>
<p>One crucial component of the SG is AMI, which is a network of information and communication, smart meters, and meter data management systems. The home area network (HAN), neighborhood area network (NAN), wide area network (WAN), and utility systems make up the majority of the three significant components that make up the AMI&#x2019;s communication network, as illustrated in <xref ref-type="fig" rid="F2">Figure 2</xref>. Smart meters connected to houses via the HAN are the basis of the AMI. These meters collect current and voltage usage data in real-time and send it across the NAN to a data concentrator. These data are used by the utility for forecasting, demand response (DR), and power billing. WAN links the data concentrator and control center, and smart meters facilitate smooth energy generation and consumption balance. This integration allows for efficient defect detection, real-time customer research, and improved smart grid tracking. Overall, smart meters in AMI improve energy management, billing accuracy, and grid responsiveness.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Network architecture of the advanced metering infrastructure.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g002.tif"/>
</fig>
</sec>
<sec id="s3-2">
<title>3.2 Attack model</title>
<p>The attacker&#x2019;s approach to attempting an attack is proposed here. The control center gathers information to analyze the consumption patterns of consumers and detect faults. An attacker uses this fine-grained consumption reading and can send false information to utilities to reduce their bill illegally. The primary goal of a consumer stealing electricity is to obtain the expended energy for less money than it is worth. Illegally reporting false bills creates a financial loss to the utility and a disturbance to energy management. A list of the many possible energy attacks against the AMI systems is illustrated in <xref ref-type="table" rid="T1">Table 1</xref>.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Summary of energy theft attack targeted at AMI systems.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Attack type</th>
<th align="center">Description</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td rowspan="3" align="center">Physical attack</td>
<td align="left">&#x2022; Tampering meter readings illegally</td>
</tr>
<tr>
<td align="left">o By-passing meter readings using a magnet</td>
</tr>
<tr>
<td align="left">o Fake metering</td>
</tr>
<tr>
<td rowspan="3" align="center">Cyber attack</td>
<td align="left">Eavesdropping on confidential information</td>
</tr>
<tr>
<td align="left">Gaining privileged system access</td>
</tr>
<tr>
<td align="left">Tampering with energy meter storage</td>
</tr>
<tr>
<td rowspan="3" align="center">Data attack</td>
<td align="left">Targeting the metering values</td>
</tr>
<tr>
<td align="left">Purposely changing consumption to zero</td>
</tr>
<tr>
<td align="left">Revealing the private information of the user</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>In the proposed threat model, fraudulent data have been introduced into the system at the consumer&#x2019;s location primarily for financial advantage. The paper analyzes the two classes of FDI attacks, as listed in <xref ref-type="table" rid="T2">Tables 2</xref>, <xref ref-type="table" rid="T3">3</xref>, where <inline-formula id="inf1">
<mml:math id="m1">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the user&#x2019;s actual energy consumption throughout the time interval <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents fraudulent energy consumption data collected using the smart meter.</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Mathematical expression of partial reduction-based FDI attack class.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Type</th>
<th align="center">Definition</th>
<th align="center">Attack class</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">Attack 1 (A<sub>1</sub>)</td>
<td align="center">
<inline-formula id="inf4">
<mml:math id="m4">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">Partial reduction of consumption</td>
</tr>
<tr>
<td align="center">Attack 2 (A<sub>2</sub>)</td>
<td align="center">
<inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf6">
<mml:math id="m6">
<mml:mrow>
<mml:mn>0.1</mml:mn>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>0.9</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">Partial reduction of consumption</td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap id="T3" position="float">
<label>TABLE 3</label>
<caption>
<p>Mathematical expression of price control-based FDI attack class.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Type</th>
<th align="center">Definition</th>
<th align="center">Attack class</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">Attack 3 (A<sub>3</sub>)</td>
<td align="center">
<inline-formula id="inf7">
<mml:math id="m7">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mn>24</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="normal">t</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">Price based</td>
</tr>
<tr>
<td align="center">Attack 4 (A<sub>4</sub>)</td>
<td align="center">
<inline-formula id="inf8">
<mml:math id="m8">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf9">
<mml:math id="m9">
<mml:mrow>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the mean of previous day <inline-formula id="inf10">
<mml:math id="m10">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> values, <inline-formula id="inf11">
<mml:math id="m11">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; random (0.1,0.9)</td>
<td align="center">Price based</td>
</tr>
<tr>
<td align="center">Attack 5 (A<sub>5</sub>)</td>
<td align="center">
<inline-formula id="inf12">
<mml:math id="m12">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf13">
<mml:math id="m13">
<mml:mrow>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the mean of previous</td>
<td align="center">Price based</td>
</tr>
</tbody>
</table>
</table-wrap>
<sec id="s3-2-1">
<title>3.2.1 Partial reduction-based FDI attack class</title>
<p>The primary goal of the user in this kind of FDI attack is to lower the amount of energy used to benefit financially. The attacker can inject the reduced consumption as compared to the actual value for that purpose. The mathematical representation of partial reduction-based FDI attacks is listed in <xref ref-type="table" rid="T2">Table 2</xref>. The target of each attack is to decrease energy usage by the factor <inline-formula id="inf14">
<mml:math id="m14">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. The objective of the first attack A<sub>1</sub> is to reduce <inline-formula id="inf15">
<mml:math id="m15">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> by a flat reduction ratio <inline-formula id="inf16">
<mml:math id="m16">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf17">
<mml:math id="m17">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is a fixed number from random (0.1, 0.9). In contrast, the objective of attack A<sub>2</sub> is to dynamically reduce consumption by the factor <inline-formula id="inf18">
<mml:math id="m18">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf19">
<mml:math id="m19">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> varies from <inline-formula id="inf20">
<mml:math id="m20">
<mml:mrow>
<mml:mn>0.1</mml:mn>
<mml:mo>&#x3c;</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>0.9</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
</sec>
<sec id="s3-2-2">
<title>3.2.2 Price control-based FDI attack class</title>
<p>In this, the attacker aims to cause financial loss to the utility by changing the energy meter data so that total power consumption will not change but attack the effect financially. These attacks happen when the DR is used, and the price varies throughout the day. The mathematical expressions for the price control attacks A<sub>3</sub>, A<sub>4</sub>, and A<sub>5</sub> are listed in <xref ref-type="table" rid="T3">Table 3</xref>. Reversing the day&#x2019;s consumption order is done in A<sub>3</sub>. In A<sub>4</sub>, the malicious reading of energy consumption <inline-formula id="inf21">
<mml:math id="m21">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is equal to the mean of power readings <inline-formula id="inf22">
<mml:math id="m22">
<mml:mrow>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> of the previous day multiplied by a fixed random value <inline-formula id="inf23">
<mml:math id="m23">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. A<sub>5</sub> multiplies each meter reading with a random value ranging from 0.1 to 0.9 with <inline-formula id="inf24">
<mml:math id="m24">
<mml:mrow>
<mml:msub>
<mml:mover accent="true">
<mml:mi>e</mml:mi>
<mml:mo>&#xaf;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
<p>
<xref ref-type="fig" rid="F3">Figure 3</xref> is the graphical representation of FDI attacks (A<sub>1</sub>&#x2013;A<sub>5</sub>) and no attack scenario for 1&#xa0;day. Consumption includes all five types of attacks and consumption by the user without attack with respect to the time of 1&#xa0;day, i.e., 24&#xa0;hr.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>An illustration of malicious and non-malicious user&#x2019;s consumption patterns for a day for different attack scenarios.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g003.tif"/>
</fig>
<p>
<statement content-type="algorithm" id="Algorithm_1">
<label>Algorithm 1</label>
<p>Algorithm of the proposed theft detection system, TDS.<list list-type="simple">
<list-item>
<p>
<bold>Input:</bold>&#x2003;Energy consumption data of N&#xa0;days, with each day having i energy measurement time slots; <inline-formula id="inf25">
<mml:math id="m25">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
<mml:mn>1</mml:mn>
</mml:msubsup>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mo>&#x2026;</mml:mo>
</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
<mml:mi>i</mml:mi>
</mml:msubsup>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x2026;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>N</mml:mi>
<mml:mn>1</mml:mn>
</mml:msubsup>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x22ef;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>N</mml:mi>
<mml:mi>i</mml:mi>
</mml:msubsup>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<p>
<bold>Output:</bold>&#x2003;The measurement value <inline-formula id="inf26">
<mml:math id="m26">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> for that selected day p and time slot c denoted as <inline-formula id="inf27">
<mml:math id="m27">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> either belongs to the faulty (false) class or non-faulty (true) class</p>
</list-item>
<list-item>
<label>1</label>
<p>&#x2003;<bold>for</bold> <inline-formula id="inf28">
<mml:math id="m28">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>t</mml:mi>
<mml:mi>o</mml:mi>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> <bold>do</bold>
</p>
</list-item>
<list-item>
<label>2</label>
<p>&#x2003;&#x2003;&#x2003;<bold>If</bold> <inline-formula id="inf29">
<mml:math id="m29">
<mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="bold-italic">e</mml:mi>
<mml:mi mathvariant="bold-italic">p</mml:mi>
<mml:mi mathvariant="bold-italic">c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> is missed, <bold>then</bold>
</p>
</list-item>
<list-item>
<label>3</label>
<p>&#x2003;&#x2003;&#x2003;<inline-formula id="inf30">
<mml:math id="m30">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mi mathvariant="italic">max</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> (filled with the most repeated value)</p>
</list-item>
<list-item>
<label>4</label>
<p>&#x2003;&#x2003;<bold>End</bold>
</p>
</list-item>
<list-item>
<label>5</label>
<p>&#x2003;<bold>End</bold>
</p>
</list-item>
<list-item>
<label>6</label>
<p>&#x2003;p &#x2190; p &#x2b; 1</p>
</list-item>
<list-item>
<label>7</label>
<p>&#x2003;&#x2003;Generation of synthetic attack pattern <inline-formula id="inf31">
<mml:math id="m31">
<mml:mrow>
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mi>b</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> for different attack scenarios</p>
</list-item>
<list-item>
<label>8</label>
<p>&#x2003;<bold>for</bold> b &#x3d; 1 to 5 <bold>do</bold>
</p>
</list-item>
<list-item>
<label>9</label>
<p>&#x2003;&#x2003;&#x2003;Generation of a synthetic attack pattern for each value of b</p>
</list-item>
<list-item>
<label>10</label>
<p>&#x2003;&#x2003;&#x2003;Merging <inline-formula id="inf32">
<mml:math id="m32">
<mml:mrow>
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mi>b</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> with <inline-formula id="inf33">
<mml:math id="m33">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> and generating a combined dataset.</p>
</list-item>
<list-item>
<label>11</label>
<p>&#x2003;&#x2003;Select the combined meter measurement value <inline-formula id="inf34">
<mml:math id="m34">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>q</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> of the latest q days as a training set</p>
</list-item>
<list-item>
<label>12</label>
<p>&#x2003;&#x2003;&#x2003;Adaptive Boost Ensemble Method (training dataset)</p>
</list-item>
<list-item>
<label>13</label>
<p>&#x2003;&#x2003;&#x2003;Given: <inline-formula id="inf35">
<mml:math id="m35">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf36">
<mml:math id="m36">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>&#x2208;</mml:mo>
<mml:mi mathvariant="script">E</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>&#x2208;</mml:mo>
<mml:mo>{</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mrow>
<mml:mfenced open="" close="}" separators="|">
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<label>14</label>
<p>&#x2003;//Initialization <inline-formula id="inf37">
<mml:math id="m37">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>r</mml:mi>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<label>15</label>
<p>&#x2003;&#x2003;&#x2003;<bold>for</bold> (<inline-formula id="inf38">
<mml:math id="m38">
<mml:mrow>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>3</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mi>T</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) classifiers <bold>do</bold>
</p>
</list-item>
<list-item>
<label>16</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;Train weak learners with the <inline-formula id="inf39">
<mml:math id="m39">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> distribution</p>
</list-item>
<list-item>
<label>17</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;get weak hypothesis <inline-formula id="inf40">
<mml:math id="m40">
<mml:mrow>
<mml:msub>
<mml:mi>h</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; <inline-formula id="inf41">
<mml:math id="m41">
<mml:mrow>
<mml:mi mathvariant="fraktur">R</mml:mi>
<mml:mo>&#x2192;</mml:mo>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="|">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>l</mml:mi>
<mml:mi>s</mml:mi>
<mml:mi>e</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>u</mml:mi>
<mml:mi>e</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<label>18</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;Aim: Select <inline-formula id="inf42">
<mml:math id="m42">
<mml:mrow>
<mml:msub>
<mml:mi>h</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> with low weight error:</p>
</list-item>
<list-item>
<label>19</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;<inline-formula id="inf43">
<mml:math id="m43">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:msub>
<mml:mi>r</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>h</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2260;</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<label>20</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;Choose <inline-formula id="inf44">
<mml:math id="m44">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi mathvariant="italic">ln</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</list-item>
<list-item>
<label>21</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;Update, <bold>for</bold> l &#x3d; 1 to <inline-formula id="inf45">
<mml:math id="m45">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> <bold>do</bold>
</p>
</list-item>
<list-item>
<label>22</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;&#x2003;<inline-formula id="inf46">
<mml:math id="m46">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi mathvariant="italic">exp</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>h</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>, where</p>
</list-item>
<list-item>
<label>
</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;&#x2003;&#x2003;<inline-formula id="inf47">
<mml:math id="m47">
<mml:mrow>
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the normalization factor</p>
</list-item>
<list-item>
<label>23</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;<bold>end</bold>
</p>
</list-item>
<list-item>
<label>24</label>
<p>&#x2003;&#x2003;&#x2003;<bold>End</bold>
</p>
</list-item>
<list-item>
<label>25</label>
<p>&#x2003;&#x2003;&#x2003;<bold>for</bold> <inline-formula id="inf48">
<mml:math id="m48">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
<mml:mi>&#x3f5;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> testing dataset to <bold>do</bold>
</p>
</list-item>
<list-item>
<label>26</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;<bold>if</bold> <inline-formula id="inf49">
<mml:math id="m49">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
<mml:mi>&#x3f5;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> faulty user, <bold>then</bold>
</p>
</list-item>
<list-item>
<label>27</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;&#x2003;false class (malicious consumer)</p>
</list-item>
<list-item>
<label>28</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;<bold>else</bold>
</p>
</list-item>
<list-item>
<label>29</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;&#x2003;<inline-formula id="inf50">
<mml:math id="m50">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> true class (non-malicious consumer)</p>
</list-item>
<list-item>
<label>30</label>
<p>&#x2003;&#x2003;&#x2003;&#x2003;<bold>end</bold>
</p>
</list-item>
<list-item>
<label>31</label>
<p>&#x2003;&#x2003;&#x2003;<bold>End</bold>
</p>
</list-item>
<list-item>
<label>32</label>
<p>&#x2003;<bold>End</bold>
</p>
</list-item>
</list>
</p>
</statement>
</p>
</sec>
</sec>
</sec>
<sec id="s4">
<title>4 Proposed energy theft detection model under varying attack scenarios</title>
<p>Our proposed model framework comprises four modules: proposed ensemble modeling technique, data preprocessing, training phase, and testing phase. The first subsection of this section presents a description of the proposed methodology, an ensemble modeling-based <italic>AdaBoost</italic> technique. Then, the subsequent parts cover the remaining three modules of the theft detection approach, which we use to foil attempts on the integrity of our energy meter data. The steps of our proposed approach&#x2019;s framework for detecting electricity theft are presented in <xref ref-type="statement" rid="Algorithm_1">Algorithm 1</xref>.</p>
<sec id="s4-1">
<title>4.1 Proposed ensemble modeling technique</title>
<p>AdaBoost is a supervised ML-based boosting algorithm to help classification models perform better. AdaBoost sequentially creates several learning models. The first model is created by conventionally fitting the classifier to the given dataset. The second model is then created by training a second instance of the classifier using the same set of data, with an emphasis on the samples that the previous model incorrectly identified. The third model then uses the weak classifiers from the prior model to train the classifier. By integrating misfit samples of the classifier into a robust classifier or merging weak learners&#x2019; decision trees from learning models, very accurate predictions may be made to improve the final predictive performance of the system.</p>
<p>Let <inline-formula id="inf51">
<mml:math id="m51">
<mml:mrow>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>3</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mi>T</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represent the collection of generated weak learners of classifiers. Here, the training dataset is represented as <inline-formula id="inf52">
<mml:math id="m52">
<mml:mrow>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula>, where <italic>p</italic> represents days taken in the training set, with each day having <italic>i</italic> measurement time slots. <inline-formula id="inf53">
<mml:math id="m53">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <italic>w</italic>here <inline-formula id="inf54">
<mml:math id="m54">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>&#x2208;</mml:mo>
<mml:mi mathvariant="script">E</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
<mml:mo>&#x2208;</mml:mo>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="|">
<mml:mrow>
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<mml:mn>1</mml:mn>
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<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the training set containing <italic>l</italic> samples, where all e inputs are an element of total set <inline-formula id="inf55">
<mml:math id="m55">
<mml:mrow>
<mml:mi mathvariant="script">E</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and outputs are an element of <italic>a</italic> set comprising only two values, &#x2212;1 (false class, i.e., malicious user) and &#x2b;1 (true class, i.e., non-malicious user). <italic>X</italic> is the weight of the samples, and <italic>i</italic> is the <italic>i</italic>th training sample. <inline-formula id="inf56">
<mml:math id="m56">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>r</mml:mi>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>; initialize all the weights of your samples to 1 divided by the number of training samples <italic>l</italic>. In <inline-formula id="inf57">
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<mml:mrow>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
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<mml:msub>
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<mml:mi>i</mml:mi>
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</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf58">
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<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:msub>
<mml:mi>r</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the probability, <inline-formula id="inf59">
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<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the minimum misclassification error for the model, and <inline-formula id="inf60">
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<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the weight of the classifier. Assume that <inline-formula id="inf61">
<mml:math id="m61">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>l</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are the weights assigned to dataset samples to show the importance of the data points, where <italic>l</italic> is the <inline-formula id="inf63">
<mml:math id="m63">
<mml:mrow>
<mml:msup>
<mml:mi>l</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> training sample. Some of the key points of the AdaBoost-based algorithm for attack detection are summarized below.<list list-type="simple">
<list-item>
<label>
</label>
<p>&#x2022; Set weights <inline-formula id="inf64">
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<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> <italic>for</italic> <inline-formula id="inf65">
<mml:math id="m65">
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, satisfying Eq. <xref ref-type="disp-formula" rid="e1">(1)</xref>.</p>
</list-item>
</list>
<disp-formula id="e1">
<mml:math id="m66">
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>c</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(1)</label>
</disp-formula>
<list list-type="simple">
<list-item>
<label>
</label>
<p>&#x2022; Consider Eq. <xref ref-type="disp-formula" rid="e1">(1)</xref> condition for <inline-formula id="inf66">
<mml:math id="m67">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
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<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
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<mml:mi>T</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> classifiers.</p>
</list-item>
<list-item>
<label>
</label>
<p>&#x2022; Update the weights according to Eq. <xref ref-type="disp-formula" rid="e2">(2)</xref> for <inline-formula id="inf67">
<mml:math id="m68">
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mi>l</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
</list-item>
</list>
<disp-formula id="e2">
<mml:math id="m69">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi mathvariant="italic">exp</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>h</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mfrac>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2)</label>
</disp-formula>where <inline-formula id="inf68">
<mml:math id="m70">
<mml:mrow>
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is a normalization factor and <inline-formula id="inf69">
<mml:math id="m71">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the weight of the classifier.<list list-type="simple">
<list-item>
<label>
</label>
<p>&#x2022; Choose the generated weak classifier that minimizes the sum of weighted classification errors.</p>
</list-item>
<list-item>
<label>
</label>
<p>&#x2022; The classifier&#x2019;s weight is adjusted after each iteration to make it focus on sample points that are challenging to categorize correctly. After an iteration, this is accomplished by updating misclassified sample points with higher weights. In the following iteration, our learning system would pay more attention to these sample points by assigning them increased weights. In contrast, classifiers would assign less weight to the well-categorized sample points and give less attention in the next iteration. The final prediction is then calculated by summing the weighted predictions from all classifiers.</p>
</list-item>
<list-item>
<label>
</label>
<p>&#x2022; It has been demonstrated that using the AdaBoost method, if the misclassification rates of the weak classifiers are less than 50%, then the weighted classification error rate of the strong classifier will converge to zero as the number of iterations increases, i.e., when</p>
</list-item>
</list>
<disp-formula id="e3">
<mml:math id="m72">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>&#x221e;</mml:mi>
<mml:mo>,</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mo>&#x200d;</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mi>H</mml:mi>
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<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2260;</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>
<list list-type="simple">
<list-item>
<label>
</label>
<p>&#x2022; The basis of Eq. <xref ref-type="disp-formula" rid="e3">(3)</xref> is that misclassification rates for the weak classifiers are less than 50%.</p>
</list-item>
<list-item>
<label>
</label>
<p>&#x2022; By merging the decision trees for the descriptive and statistical aspects of the smart meter into a robust classifier, the linkages between these features are naturally handled. This is the primary reason why our <italic>AdaBoost-based</italic> algorithm achieves good attack detection results.</p>
</list-item>
</list>
</p>
<p>The decision trees reduce the total of the incorrect classification outputs for true (faulty) and false (non-faulty) samples. The misclassification rates for the selected weak learners are guaranteed to be lower than fifty percent, assuring the algorithm&#x2019;s convergence.</p>
</sec>
<sec id="s4-2">
<title>4.2 Data pre-processing</title>
<p>The first step toward training the detection model is data pre-processing, which includes cleaning the raw data, filling in the missing values, and removing extreme values. Our power theft detection model uses energy consumption measurements from a real smart meter dataset of 5,000 customers for training and evaluation purposes (<xref ref-type="bibr" rid="B18">ISSDA, 2020</xref>). This dataset comprises energy consumption readings from residential and business users from 2009 to 2010, spanning 533 days. To enhance the financial analysis for a statewide deployment, the main purpose of this study is to assess the impact of user power to find energy theft. Six data sample files containing records of 533 days in each file made up the raw dataset. Each file has three columns: the smart meter identifier, the encoded date and time, and the amount of energy used in kWh. Every document includes 533 days&#x2019; worth of metering information for every client, captured every half hour, i.e., each user&#x2019;s daily consumption data presented by 48 vector components. All of the consumers&#x2019; consumption is included in the raw data collection. To prepare the data for our experiment, we divided the raw files by meter ID into many consumption datasets.</p>
<p>Assume <inline-formula id="inf70">
<mml:math id="m73">
<mml:mrow>
<mml:mi>e</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>3</mml:mn>
</mml:msub>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>.</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mn>48</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> as the customer&#x2019;s energy consumption in a day, which is recorded in kWh to the data concentrator unit of AMI for each 30&#xa0;min. The whole dataset is represented as <inline-formula id="inf71">
<mml:math id="m74">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
<mml:mn>1</mml:mn>
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</mml:mtd>
<mml:mtd>
<mml:mrow>
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</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mn>1</mml:mn>
<mml:mi>i</mml:mi>
</mml:msubsup>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x2026;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>N</mml:mi>
<mml:mn>1</mml:mn>
</mml:msubsup>
</mml:mtd>
<mml:mtd>
<mml:mo>&#x22ef;</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:msubsup>
<mml:mi>e</mml:mi>
<mml:mi>N</mml:mi>
<mml:mi>i</mml:mi>
</mml:msubsup>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf72">
<mml:math id="m75">
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the total consumption days, with each day having <inline-formula id="inf73">
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<mml:mrow>
<mml:mi>i</mml:mi>
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</inline-formula> measuring slots. We use the attack scenarios in <xref ref-type="table" rid="T2">Tables 2</xref> and <xref ref-type="table" rid="T3">3</xref> to create attack samples.</p>
<p>Missing values are those in which the smart meter cannot record the meter readings for reasons such as an error in transmission, a component break, and a bad connection. When missing values are incorrectly handled, a biased ML model is created, producing unreliable results. The most repeated value imputation method is used to fill in the missing value in the proposed method. The mathematical representation is as follows:<disp-formula id="e4">
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<p>where mode <inline-formula id="inf74">
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<sec id="s4-2-1">
<title>4.2.1 Feature extraction</title>
<p>In the second phase of the cleaning process crucial for time-series classification, extreme values are eliminated from the raw dataset. This step is pivotal for accurate classification results. Effective feature extraction is vital for enhanced accuracy and interpretability. The dynamic nature of an individual user&#x2019;s daily consumption pattern necessitates stable features reflecting daily and weekly load patterns. To achieve this, descriptive and statistical features, detailed in <xref ref-type="table" rid="T4">Table 4</xref>, are extracted monthly for each time slot reading across the entire period. Extreme values, indicative of unusual activities such as vacations or changes in appliances or residences, are removed to ensure data integrity. This refined dataset forms the basis for robust time-series classification.</p>
<table-wrap id="T4" position="float">
<label>TABLE 4</label>
<caption>
<p>List of features extracted.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Descriptive feature</th>
<th align="left">Statistical feature</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Maximum of weekend</td>
<td align="left">Mean of weekend</td>
</tr>
<tr>
<td align="left">Minimum of weekend</td>
<td align="left">Mean of week</td>
</tr>
<tr>
<td align="left">Total of week</td>
<td align="left">Auto-correlation</td>
</tr>
<tr>
<td align="left">Maximum of week</td>
<td align="left">Median of week</td>
</tr>
<tr>
<td align="left">Total of weekday</td>
<td align="left">Range</td>
</tr>
<tr>
<td align="left">Maximum of weekdays</td>
<td align="left">Entropy</td>
</tr>
<tr>
<td align="left">Total of weekend</td>
<td align="left">Quartile 25</td>
</tr>
<tr>
<td align="left">Minimum of week</td>
<td align="left">Standard deviation</td>
</tr>
<tr>
<td align="left">Minimum of weekdays</td>
<td align="left">Quartile deviation</td>
</tr>
<tr>
<td align="left">Minimum of weekend</td>
<td align="left">Coefficient of quartile deviation</td>
</tr>
<tr>
<td align="left"/>
<td align="left">Quartile 75</td>
</tr>
<tr>
<td align="left"/>
<td align="left">Variance of week</td>
</tr>
<tr>
<td align="left"/>
<td align="left">Interquartile range</td>
</tr>
<tr>
<td align="left"/>
<td align="left">Mean of weekday</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
<sec id="s4-3">
<title>4.3 Training phase</title>
<p>The next module is used to train a model with the energy meter data readings for the detection of energy theft. For that purpose, we need both benign data and malicious data; otherwise, the classifier will face the problem of data balancing and make the efficiency of the theft detection system low. As the malicious data are not available and it is difficult to gather faulty readings, we propose synthetically generating the malicious dataset for different types of five FDI attack scenarios to address this issue. The attacks in <xref ref-type="table" rid="T2">Table 2</xref> are based upon the partial reduction of A<sub>1</sub> and A<sub>2</sub> attacks, and in <xref ref-type="table" rid="T3">Table 3</xref>, price-based attacks A<sub>3</sub>, A<sub>4</sub>, and A<sub>5</sub>. Energy theft aims to record less usage than the user uses or shift high usage to low-tariff times. Therefore, it is easy to produce malicious samples using benign samples. The suggested ensemble approach is used to detect intruders using meter reading data once the data have been properly formatted. We randomly choose 50% of the data for each user to create five synthetic attack patterns. After generating attack patterns <inline-formula id="inf78">
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</inline-formula>, the non-malicious values are mixed, and the combined dataset is generated. For training the model, we select the historical data (i.e., measurement values) from the most recent m&#xa0;days from the combined dataset. As a result, we have 70% of the data for model training and 30% for model testing. The flow diagram of the proposed algorithm is shown in <xref ref-type="fig" rid="F4">Figure 4</xref>. The model is also trained for existing AI approaches as per the survey, including support vector machine, logistics regression (LR), K- nearest neighbor (KNN), na&#xef;ve Bayes (NB), and random forest classifier (RFC) to demonstrate the efficacy of the suggested strategy.</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Framework of the proposed algorithm.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g004.tif"/>
</fig>
</sec>
<sec id="s4-4">
<title>4.4 Testing phase</title>
<p>Following the training set, pre-processing and format conversion are performed on each new smart meter reading. Determining whether data are genuine or false, i.e., if testing data belong to the non-malicious or malicious type, enables us to make detection decisions for false data. After introducing a synthetic attack, the <italic>AdaBoost</italic> ensemble technique is applied to a fresh meter reading to assess whether it belongs to the faulty or non-faulty class. The newly created sample is uploaded to the genuine dataset, and the appropriate attack patterns are created and added to the attack dataset. When the fresh sample presented to the classifier identified an assault, the smart meter&#x2019;s suspicious behavior was notified. After that, more data samples of the same meter ID were tested, and suspicious behavior was reported <inline-formula id="inf80">
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</inline-formula> times, indicating energy theft was discovered. Once energy theft is identified, the required measures, such as an on-site examination, are taken. Repetition is essential so that, for any change (change in an appliance, vacation, or seasonal change), a non-malicious user is not reported, owing to the cost of the on-site inspection. Priority inspection is assigned to a certain region based on the number of customers determined to be defective. If the theft is confirmed, the specific consumer values are included in the attack data values; otherwise, they remain in the authentic dataset. To show the effectiveness of our system, we have implemented experiments as mentioned with other well-known ML techniques and compared them with the proposed method for each attack type, as discussed in detail in <xref ref-type="sec" rid="s5">Section 5</xref> of this article.</p>
</sec>
</sec>
<sec id="s5">
<title>5 Results and discussions</title>
<p>To verify the efficacy of the proposed approach, various supervised algorithms are applied to the data sample described in <xref ref-type="sec" rid="s4">Section 4</xref>. The performance of our scheme was assessed using the metrics accuracy (<inline-formula id="inf81">
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<p>The model performance is better when <inline-formula id="inf92">
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</inline-formula>. Experimental analysis for these five types of attacks is discussed in subsequent subsections.</p>
<sec id="s5-1">
<title>5.1 Experiment no. 1</title>
<p>In the first experiment, different existing AI techniques are applied to the smart meter data for A<sub>1</sub>. In our experiment, we assume the <inline-formula id="inf96">
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</mml:mrow>
</mml:math>
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</inline-formula>, <inline-formula id="inf99">
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<mml:mi>R</mml:mi>
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</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf100">
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<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are evaluated for each method and compared with the proposed method, as shown in <xref ref-type="table" rid="T5">Table 5</xref>. For the proposed method, <inline-formula id="inf101">
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<mml:mrow>
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is 85%, whereas for the SVM, it is 83%; for LR, it is 83%; and KNN it is 79%; additionally, for NB, it is 72%; and for RFC, it is 84%. <xref ref-type="fig" rid="F5">Figure 5</xref> shows the region of convergence (ROC) curve of all the models mentioned on which the experiment is conducted. ROC is the graph between <inline-formula id="inf102">
<mml:math id="m108">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>p</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf103">
<mml:math id="m109">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>p</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, representing the performance measurement for the classifier.</p>
<table-wrap id="T5" position="float">
<label>TABLE 5</label>
<caption>
<p>Performance parameter comparison for A<sub>1</sub>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Technique</th>
<th colspan="4" align="center">Performance comparison</th>
</tr>
<tr>
<th align="center">
<italic>A</italic>
<sub>
<italic>c</italic>
</sub>
</th>
<th align="center">
<italic>P</italic>
<sub>
<italic>r</italic>
</sub>
</th>
<th align="center">
<italic>R</italic>
<sub>
<italic>e</italic>
</sub>
</th>
<th align="center">
<italic>F</italic>
<sub>
<italic>s</italic>
</sub>
</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">SVM</td>
<td align="center">0.838</td>
<td align="center">0.656</td>
<td align="center">0.836</td>
<td align="center">0.735</td>
</tr>
<tr>
<td align="center">LR</td>
<td align="center">0.834</td>
<td align="center">0.671</td>
<td align="center">0.753</td>
<td align="center">0.710</td>
</tr>
<tr>
<td align="center">KNN</td>
<td align="center">0.793</td>
<td align="center">0.598</td>
<td align="center">0.712</td>
<td align="center">0.650</td>
</tr>
<tr>
<td align="center">NB</td>
<td align="center">0.727</td>
<td align="center">0.496</td>
<td align="center">0.932</td>
<td align="center">0.648</td>
</tr>
<tr>
<td align="center">RFC</td>
<td align="center">0.841</td>
<td align="center">0.692</td>
<td align="center">0.740</td>
<td align="center">0.715</td>
</tr>
<tr>
<td align="center">Our method</td>
<td align="center">
<bold>0.852</bold>
</td>
<td align="center">
<bold>0.694</bold>
</td>
<td align="center">
<bold>0.808</bold>
</td>
<td align="center">
<bold>0.747</bold>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Comparison of the ROC curve of the proposed method with existing methods for A<sub>1</sub>.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g005.tif"/>
</fig>
</sec>
<sec id="s5-2">
<title>5.2 Experiment no. 2</title>
<p>This experiment is conducted for A<sub>2</sub> belonging to the partial reduction FDI attack class, in which a synthetic attack pattern is generated using the definition mentioned in <xref ref-type="table" rid="T2">Table 2</xref> and merged with the normal smart meter data. In this, the malicious value <inline-formula id="inf104">
<mml:math id="m110">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is generated by multiplying the real-time energy consumption value <inline-formula id="inf105">
<mml:math id="m111">
<mml:mrow>
<mml:msub>
<mml:mi>e</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> of the user with the <inline-formula id="inf106">
<mml:math id="m112">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> factor, whose value is in the dynamic range from <inline-formula id="inf107">
<mml:math id="m113">
<mml:mrow>
<mml:mn>0.1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> to <inline-formula id="inf108">
<mml:math id="m114">
<mml:mrow>
<mml:mn>0.9</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. In the combined dataset, we take the ratio of 70:30% for genuine and non-genuine data. The different evaluation metrics are listed in <xref ref-type="table" rid="T6">Table 6</xref> for our method and other compared techniques. For the proposed system, <inline-formula id="inf109">
<mml:math id="m115">
<mml:mrow>
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>is shown as 84%, whereas for the SVM, it is 67%; for LR, it is 70%; for the RFC algorithm, it is 80%; and for KNN, it is 66%. The ROC curve for the differences is compared in <xref ref-type="fig" rid="F6">Figure 6</xref>. As per the result obtained, our method outperforms attack 2 compared to other methods.</p>
<table-wrap id="T6" position="float">
<label>TABLE 6</label>
<caption>
<p>Performance parameter comparison for A<sub>2</sub>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Technique</th>
<th colspan="4" align="center">Performance comparison</th>
</tr>
<tr>
<th align="center">
<italic>A</italic>
<sub>
<italic>c</italic>
</sub>
</th>
<th align="center">
<italic>P</italic>
<sub>
<italic>r</italic>
</sub>
</th>
<th align="center">
<italic>R</italic>
<sub>
<italic>e</italic>
</sub>
</th>
<th align="center">
<italic>F</italic>
<sub>
<italic>s</italic>
</sub>
</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">SVM</td>
<td align="center">0.670</td>
<td align="center">0.656</td>
<td align="center">0.836</td>
<td align="center">0.735</td>
</tr>
<tr>
<td align="center">LR</td>
<td align="center">0.700</td>
<td align="center">0.682</td>
<td align="center">0.169</td>
<td align="center">0.270</td>
</tr>
<tr>
<td align="center">KNN</td>
<td align="center">0.663</td>
<td align="center">0.484</td>
<td align="center">0.337</td>
<td align="center">0.397</td>
</tr>
<tr>
<td align="center">NB</td>
<td align="center">0.585</td>
<td align="center">0.439</td>
<td align="center">0.921</td>
<td align="center">0.594</td>
</tr>
<tr>
<td align="center">RFC</td>
<td align="center">0.800</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
</tr>
<tr>
<td align="center">Our method</td>
<td align="center">
<bold>0.848</bold>
</td>
<td align="center">
<bold>0.740</bold>
</td>
<td align="center">
<bold>0.831</bold>
</td>
<td align="center">
<bold>0.783</bold>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Comparison of the ROC curve of the proposed method with existing methods for A<sub>2</sub>.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g006.tif"/>
</fig>
</sec>
<sec id="s5-3">
<title>5.3 Experiment no. 3</title>
<p>The price control-based FDI attack A<sub>3</sub> was the focus of experiment 3, in which the altered meter reading <inline-formula id="inf110">
<mml:math id="m116">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the reverse of the day&#x2019;s readings. This assault on the loading mechanism involves changing the price of energy at various times of the day while keeping the overall amount of electricity used constant and reporting used to occur at low-tariff times. The experimental results of this attack by applying our proposed method are listed in <xref ref-type="table" rid="T7">Table 7</xref>. A comparison of different performance metrics shows that our proposed system achieves an accuracy of 83%, outperforming other methods such as SVM (71%), LR (69%), RFC algorithm (67%), and KNN (63%). The ROC curve for the different methods is compared in <xref ref-type="fig" rid="F7">Figure 7</xref>.</p>
<table-wrap id="T7" position="float">
<label>TABLE 7</label>
<caption>
<p>Performance parameter comparison for A<sub>3</sub>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Technique</th>
<th colspan="4" align="center">Performance comparison</th>
</tr>
<tr>
<th align="center">
<italic>A</italic>
<sub>
<italic>c</italic>
</sub>
</th>
<th align="center">
<italic>P</italic>
<sub>
<italic>r</italic>
</sub>
</th>
<th align="center">
<italic>R</italic>
<sub>
<italic>e</italic>
</sub>
</th>
<th align="center">
<italic>F</italic>
<sub>
<italic>s</italic>
</sub>
</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">SVM</td>
<td align="center">0.717</td>
<td align="center">0.656</td>
<td align="center">0.836</td>
<td align="center">0.735</td>
</tr>
<tr>
<td align="center">LR</td>
<td align="center">0.694</td>
<td align="center">0.682</td>
<td align="center">0.169</td>
<td align="center">0.270</td>
</tr>
<tr>
<td align="center">KNN</td>
<td align="center">0.637</td>
<td align="center">0.437</td>
<td align="center">0.972</td>
<td align="center">0.603</td>
</tr>
<tr>
<td align="center">NB</td>
<td align="center">0.334</td>
<td align="center">0.294</td>
<td align="center">0.966</td>
<td align="center">0.451</td>
</tr>
<tr>
<td align="center">RFC</td>
<td align="center">0.670</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
</tr>
<tr>
<td align="center">Our method</td>
<td align="center">
<bold>0.839</bold>
</td>
<td align="center">
<bold>0.736</bold>
</td>
<td align="center">
<bold>0.675</bold>
</td>
<td align="center">
<bold>0.704</bold>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F7" position="float">
<label>FIGURE 7</label>
<caption>
<p>Comparison of the ROC curve of the proposed method with existing methods for A<sub>3</sub>.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g007.tif"/>
</fig>
</sec>
<sec id="s5-4">
<title>5.4 Experiment no. 4</title>
<p>Experiment 4 is conducted for A<sub>4</sub> of the price control FDI class, similar to A<sub>3</sub>, where fraudulent customers attempt with the same motive. A faulty meter malicious reading is generated by multiplying the mean of the whole day consumption by the random number <inline-formula id="inf111">
<mml:math id="m117">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, taken as <inline-formula id="inf112">
<mml:math id="m118">
<mml:mrow>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. Experimental results are listed in <xref ref-type="table" rid="T8">Table 8</xref> by applying our theft detection method, and compared with other methods described above, the comparison graph is depicted in <xref ref-type="fig" rid="F8">Figure 8</xref>. The detection accuracy of our system is 90%, which is higher than other techniques.</p>
<table-wrap id="T8" position="float">
<label>TABLE 8</label>
<caption>
<p>Performance parameter comparison for A<sub>4</sub>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Technique</th>
<th colspan="4" align="center">Performance comparison</th>
</tr>
<tr>
<th align="center">
<italic>A</italic>
<sub>
<italic>c</italic>
</sub>
</th>
<th align="center">
<italic>P</italic>
<sub>
<italic>r</italic>
</sub>
</th>
<th align="center">
<italic>R</italic>
<sub>
<italic>e</italic>
</sub>
</th>
<th align="center">
<italic>F</italic>
<sub>
<italic>s</italic>
</sub>
</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">SVM</td>
<td align="center">0.817</td>
<td align="center">0.672</td>
<td align="center">0.773</td>
<td align="center">0.719</td>
</tr>
<tr>
<td align="center">LR</td>
<td align="center">0.799</td>
<td align="center">0.718</td>
<td align="center">0.550</td>
<td align="center">0.623</td>
</tr>
<tr>
<td align="center">KNN</td>
<td align="center">0.740</td>
<td align="center">0.538</td>
<td align="center">0.740</td>
<td align="center">0.694</td>
</tr>
<tr>
<td align="center">NB</td>
<td align="center">0.507</td>
<td align="center">0.378</td>
<td align="center">0.976</td>
<td align="center">0.545</td>
</tr>
<tr>
<td align="center">RFC</td>
<td align="center">0.692</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
</tr>
<tr>
<td align="center">Our method</td>
<td align="center">
<bold>0.905</bold>
</td>
<td align="center">
<bold>0.799</bold>
</td>
<td align="center">
<bold>0.917</bold>
</td>
<td align="center">
<bold>0.854</bold>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F8" position="float">
<label>FIGURE 8</label>
<caption>
<p>Comparison of the ROC curve of the proposed method with existing methods for A<sub>4</sub>.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g008.tif"/>
</fig>
</sec>
<sec id="s5-5">
<title>5.5 Experiment no. 5</title>
<p>A synthetic attack pattern was generated for A<sub>5</sub> in this test. Malicious reading is obtained by multiplying the real meter reading value with a random value <inline-formula id="inf113">
<mml:math id="m119">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> varying from 0.1 to 0.9. Experiment results in comparison are listed in <xref ref-type="table" rid="T9">Table 9</xref> by applying our ensemble boosting technique and compared with other existing methods. The detection accuracy comparison graph of our technique with others is shown in <xref ref-type="fig" rid="F9">Figure 9</xref>. <inline-formula id="inf114">
<mml:math id="m120">
<mml:mrow>
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> of our system is 92% higher than that of other existing techniques. To validate the effectiveness of the proposed approach on unbalanced data, the area under the ROC curve (AUC) has been accessed, showing a comparison of different attack scenarios (A<sub>1</sub>&#x2013;A<sub>5</sub>) in <xref ref-type="fig" rid="F10">Figure 10</xref>. The AUC is determined by plotting the receiver characteristics curve, which depicts the relationship between the false-positive and true-positive rates. It serves as a comprehensive measure of classification performance. By leveraging these established metrics, we ensure a thorough demonstration of the proposed scheme&#x2019;s robustness and suitability for handling unbalanced datasets in classification scenarios.</p>
<table-wrap id="T9" position="float">
<label>TABLE 9</label>
<caption>
<p>Performance parameter comparison for A<sub>5</sub>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Technique</th>
<th colspan="4" align="center">Performance comparison</th>
</tr>
<tr>
<th align="center">
<italic>A</italic>
<sub>
<italic>c</italic>
</sub>
</th>
<th align="center">
<italic>P</italic>
<sub>
<italic>r</italic>
</sub>
</th>
<th align="center">
<italic>R</italic>
<sub>
<italic>e</italic>
</sub>
</th>
<th align="center">
<italic>F</italic>
<sub>
<italic>s</italic>
</sub>
</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">SVM</td>
<td align="center">0.710</td>
<td align="center">0.656</td>
<td align="center">0.836</td>
<td align="center">0.735</td>
</tr>
<tr>
<td align="center">LR</td>
<td align="center">0.681</td>
<td align="center">0.682</td>
<td align="center">0.169</td>
<td align="center">0.270</td>
</tr>
<tr>
<td align="center">KNN</td>
<td align="center">0.665</td>
<td align="center">0.463</td>
<td align="center">0.978</td>
<td align="center">0.629</td>
</tr>
<tr>
<td align="center">NB</td>
<td align="center">0.404</td>
<td align="center">0.324</td>
<td align="center">0.977</td>
<td align="center">0.487</td>
</tr>
<tr>
<td align="center">RFC</td>
<td align="center">0.70</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
<td align="center">0.697</td>
</tr>
<tr>
<td align="center">Our method</td>
<td align="center">
<bold>0.923</bold>
</td>
<td align="center">
<bold>0.817</bold>
</td>
<td align="center">
<bold>0.945</bold>
</td>
<td align="center">
<bold>0.877</bold>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F9" position="float">
<label>FIGURE 9</label>
<caption>
<p>Comparison of the ROC curve of the proposed method with existing methods for A<sub>5</sub>.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g009.tif"/>
</fig>
<fig id="F10" position="float">
<label>FIGURE 10</label>
<caption>
<p>Comparison of the AUC score of different attack scenarios (A<sub>1</sub>&#x2013;A<sub>5</sub>) for the proposed method with other state-of-the-art methods.</p>
</caption>
<graphic xlink:href="fenrg-12-1366465-g010.tif"/>
</fig>
</sec>
</sec>
<sec id="s6">
<title>6 Conclusion</title>
<p>In this study, we provide an ensemble <italic>AdaBoost</italic> approach for depicting the relationship between false-positive and true-positive rates. It serves as a comprehensive measure of classification performance. By leveraging these established metrics, we ensure a thorough demonstration of the proposed scheme&#x2019;s robustness and suitability for handling unbalanced datasets in classification scenarios or identifying fraudulent users of the SG framework. Numerous models are combined sequentially using the ensemble approach to enhance the ultimate prediction performance. The approach involves providing high weightage to the misclassified user&#x2019;s data samples and iterating again to give better predictions while reducing the false positive rate <inline-formula id="inf115">
<mml:math id="m121">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>p</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. The whole algorithm used in this article does not require a predetermined threshold or any external knowledge. Different statistical and descriptive features are extracted to consider the extreme conditions in data samples, as incorrect identification leads to expensive on-site inspections. The experiment&#x2019;s results demonstrate that the algorithm can more effectively identify faulty data in the AMI through a mix of theoretical analysis and performance simulation, achieving higher detection accuracy than current methods. Similar tests on well-known data analysis algorithms such as SVM, LR, KNN, NB, and RFC were undertaken for performance evaluation. Moreover, the proposed method exhibits a higher detection accuracy of 85.2%&#x2013;92.3% for attacks 1&#x2013;5 than that of other state-of-the-art methods, surpassing well-known data analysis algorithms like SVM, LR, KNN, NB, and RFC. The recommended solution uses extensive experimentation on a real-world dataset of 5,000 customers and provides good performance even with a low sample rate, protecting users&#x2019; privacy.</p>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s7">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.</p>
</sec>
<sec id="s8">
<title>Author contributions</title>
<p>TG: conceptualization, data curation, methodology resources, software, writing&#x2013;original draft, and writing&#x2013;review and editing. RB: data curation, methodology, supervision, validation, project administration, and writing&#x2013;review and editing. SS: conceptualization, data curation, investigation, methodology, project administration, software, supervision, and writing&#x2013;original draft. CR: investigation, methodology, writing&#x2013;original draft, and writing&#x2013;review and editing. KA: conceptualization, investigation, software, and writing&#x2013;review and editing. WM: data curation, methodology, and writing&#x2013;review and editing.</p>
</sec>
<sec sec-type="funding-information" id="s9">
<title>Funding</title>
<p>The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.</p>
</sec>
<sec sec-type="COI-statement" id="s10">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s11">
<title>Publisher&#x2019;s note</title>
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