Description and Analysis of Data Security Based on Differential Privacy in Enterprise Power Systems

Abstract

The pursuit of environmental sustainability, energy conservation, and emissions reduction has become a global focal point. Electricity is the primary source of energy in our daily lives. Through the analysis of smart power systems, we can efficiently and sustainably harness electrical energy. However, electric power system data inherently contain a wealth of sensitive user information. Therefore, our primary concern is protecting these sensitive user data while performing precise and effective analysis. To address this issue, we have innovatively proposed three granular information models based on differential privacy. In consideration of the characteristics of enterprise electricity consumption data and the imperative need for privacy protection, we implement a reasonable modeling process through data processing, information granulation expression, and the optimization analysis of information granularity. Our datasets encompass enterprise electricity consumption data and related attributes from Hitachi Building Technology (Guangzhou) Co., Ltd’s cloud computing center. Simultaneously, we have conducted experiments using publicly available datasets to underscore the model’s versatility. Our experimental results affirm that granular computation can improve the utility of differential privacy in safeguarding data privacy.

1. Introduction

Big data have brought new vitality to the development of the Ubiquitous Power Internet of Things (UPIoT). UPIoT technologies enable real-time monitoring, collection, and analysis of vast amounts of electricity information for more effective electricity distribution management and optimization of power networks. Additionally, UPIoT can be employed to monitor the status of electrical equipment, facilitating the early detection of potential faults or issues. Moreover, UPIoT assists both businesses and consumers in gaining a deeper understanding of energy consumption patterns, thereby enabling measures to reduce energy consumption and costs [1].

However, UPIoT gathers a substantial number of data regarding the electricity consumption habits of individuals or businesses. If these data are not adequately protected, they could potentially expose sensitive information related to their production and daily activities. For businesses, the inadvertent disclosure of sensitive operational activities or manufacturing processes may indirectly reveal trade secrets and information about the operation and capacity of factory equipment. Furthermore, as internet-based enterprises increasingly rely on large data supercomputing platforms such as cloud computing, network security issues also become a significant concern [2].

To effectively analyze electricity consumption data for reducing energy conservation and emissions, as well as to address the issue of privacy leakage in electricity consumption data, various data privacy protection methods have been proposed, such as data anonymization and data encryption [3,4]. In data anonymization methods for data privacy protection, background information has a significant impact on the security of data privacy, particularly in the linkage attacks [5]. Data encryption is influenced by the encryption algorithm, the speed of information transmission, and the computational speed of encryption and decryption, making operations on massive data highly complex. Among them, differential privacy (DP) has emerged as an essential and prevalent privacy model in the field of data security since it is not affected by background knowledge [6]. Differential privacy achieves data security by introducing privacy protection mechanisms, which add too much noise to the data. Therefore, it is overwhelmingly necessary to improve the utility of the perturbed data and help the data express semantics reasonably and accurately.

Motivation: Due to the collection and analysis of fine-grained electricity consumption data, privacy concerns have garnered significant attention from businesses. For businesses, the exposure of electricity consumption data could result in the leakage of sensitive business secrets, such as production activities, energy costs, manufacturing processes, and related information. Under the premise of data security and privacy protection, it is a serious challenge to realize the accurate modeling and analysis of intelligent systems. On the one hand, big data needs to be protected by feasible technologies such as DP. On the other hand, because disturbance noise is added to protect data security, data availability is affected. Therefore, reasonable data modeling should reduce the impact of noise on data semantic expression.

Contribution: This work adopts a novel data-modeling method called granular computing, which enhances the semantic expression capability of perturbed data in privacy computation problems. It achieves accurate prediction and data release in intelligent systems and improves the data-mining ability of machine learning or deep learning models. Granular computing helps intelligent systems understand user needs and achieve the goal of human–machine interaction and promotes computer interaction that is “human-centered”. Considering the key role of information granules in data mining, we first granulate user electricity consumption data to increase data-expression capabilities and enhance semantics and then further study the security issues of information granules. In the process of releasing and analyzing electricity consumption information granules, it is extremely important to protect the security of the information granules.

The remainder of this paper is organized as follows. We provide an introduction to the background of relevant technologies such as differential privacy and Granular Computing, as well as the research progress in data security protection within the context of the power system in Section 2. Section 3 delves into the concepts and implementation methods of differential privacy. Section 4 demonstrates the integration of differential privacy and granular computing to construct methods for safeguarding the privacy of data within the power system. In Section 5, a series of experimental studies are implemented, and conclusions are reached in Section 6.

2. Literature Review

With the development of the economy, the level of urbanization has intensified, and statistically, the urban population has already exceeded 54%. The urban population figure is expected to continue growing, and it is projected that by the year 2050, the urban population will reach 66% [7]. One of the key goals of both smart cities and smart grids is to achieve more intelligent decision making, enhancing the efficiency and reliability of cities and power systems. However, the analysis of electricity data heavily relies on user electricity consumption data, leading to the exposure of a significant amount of sensitive information.

Dalenius proposes a pioneering vision for data security, in which no personal information can be obtained from a database without access to the database itself [8]. However, with the continuous progress of attack methods, even if personal information is not in this database, there is a threat of privacy disclosure. In system modeling, there are generally two sorts of possible attacks on sensitive data. A linkage attack is one where attackers combine information about published data or published models with their existing background knowledge to infer the sensitive information they want. They re-identify the data generators at different granularities, such as an attribute level, a record level, or a table level [9,10]. A probabilistic attack means that the attacker obtains more background knowledge through model parameters and structural information.

Differential privacy (DP) was proposed by Dwork et al. [11]. Differential privacy is unaffected by attackers’ background knowledge, allowing for data utility while remaining invisible. This simplifies the process of data privacy protection. Differential privacy has shown provable privacy guarantees for databases without significant accuracy loss [11]. Differential privacy means that the output of a learning model does not show significant statistical differences while the model is trained on neighboring datasets, in which the samples contain individual privacy. The advancement of differential privacy prevents attackers from inferring sensitive data through model parameters and protects personal information from leakage.

Differential privacy is applied to statistical queries, such as histogram publishing [12,13], streaming data publishing [14,15], graph data publishing [16,17], etc. Castelluccia et al. have introduced a novel Distributed Laplace Perturbation Algorithm (DLPA) that eliminates the need for a trusted third party. They present the first demonstrable private and distributed solution for smart metering, which optimizes utility without relying on a trusted third party, such as an aggregator [18]. Lei et al. propose the development of a new encryption algorithm for offshore wind power systems, focusing on secure image transmission. They have also introduced encryption rules based on Zhou Yi’s Eight Trigrams [19]. Li et al. propose a privacy-preserving, two-party, secure computation mechanism for consensus-based P2P energy trading [20]. In addition, numerous researchers have employed various privacy-protection methods to safeguard the security of power grid data in diverse aspects of the electrical power system. These methods include blockchain technology [21], data encryption [22], anti-synchronization, and the Bogdanov transformation map [23]. The objective is to achieve a secure and efficient smart grid system. Cao et al. [24] propose a framework that combines local differential privacy with federated learning, allowing well-resourced power providers to assist Internet of Things (IoT) users in supplying power. This framework addresses the issue of energy-intensive machine learning model training on most IoT endpoints in the power IoT ecosystem and considers the tradeoffs between local differential privacy, data utility, and resource consumption.

To strike a balance between data utility and data security, we apply granular computation in power data modeling. Granular computing is a methodology that emphasizes the importance of granular structures in knowledge representation. Information granules are indeed the basic data units used in granular computing, which is a methodology for processing, analyzing, and representing complex data.

Information granules can be formed in many ways, depending on the specific problem being addressed and the nature of the data being processed. They can be formed by grouping individual data elements based on their similarity, partitioning the data into smaller subsets, clustering the data into groups [25,26,27], or applying other techniques to group the data into higher-level entities [28,29,30].

By breaking down the data into smaller, more manageable pieces, information granules can help to make complex information more accessible and easier to understand. Granular computing draws on a wide range of mathematical and computational techniques, including fuzzy sets [31], rough sets [32], shadowed sets [33], interval analysis, probability theory, and machine learning. It has applications in a variety of fields, including data analysis [34,35,36,37], image processing [38], decision making [39], and pattern recognition [40].

3. Preliminaries of Differential Privacy and Granular Computing

3.1. Differential Privacy

The notion of differential privacy proposes a privacy model that provides a provable privacy guarantee for individuals [41]. When an attacker attempts to deduce information about an unknown record by querying the difference between a known dataset and another dataset containing an unknown record, the content of the unknown records cannot be inferred from the query results [42]. We consider that $\chi$ is a finite data universe with the size $\left|\chi \right|$.

Definition 1

(Minkowski Distance). The Minkowski distance of order p (where p is an integer) between two sets $\mathit{D}=\{{\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_i\}$ and ${\mathit{D}}^{\prime }=\{{\mathit{x}}_1^{\prime },{\mathit{x}}_2^{\prime },\dots ,{\mathit{x}}_i^{\prime }\}$ is defined as

$$\begin{array}{c}\hfill {\parallel ·\parallel }_p=(\sum _{i=1}^{\left|\chi \right|}\parallel {\mathit{x}}_i-{\mathit{x}}_i^{\prime }{{\parallel }^p)}^{\frac{1}{p}}.\end{array}$$

(1)

We refer to the Minkowski distance with $p=1$, where the number of differing elements between two datasets is one; that is, $\parallel \mathit{D}-{\mathit{D}}^{\prime }{\parallel }_1\le 1$. These two datasets are neighboring datasets.

Definition 2

($(\epsilon ,\delta )$-Differential Privacy [11]). A randomized mechanism M produces $(\epsilon ,\delta )$-differential privacy for every set of outputs S in the output space of the mechanism M, and for all pairs of neighboring datasets of $\mathit{D}$ and ${\mathit{D}}^{\prime }$ such that $\parallel \mathit{D}-{\mathit{D}}^{\prime }{\parallel }_1\le 1$, if M satisfies the following

$$\begin{array}{c}\hfill Pr[M\left(\mathit{D}\right)\in S]\le Pr[M\left({\mathit{D}}^{\prime }\right)\in S]\ast e^{\epsilon }+\delta ,\end{array}$$

(2)

where the dataset $\mathit{D}$ comes from the universe $\chi$.

If $\delta =0$, this definition is the strict $\epsilon$-differential privacy. The parameter $\epsilon$ ($\epsilon \ge 0$) is the privacy budget, which controls the privacy guarantee level of mechanism M. The privacy budget is a measure of the extent of data privacy protection in differential privacy. Therefore, the privacy budget quantifies the level of privacy protection. When the value of $\epsilon =0$, the model has the highest level of privacy guarantee. The attacker cannot infer the desired information through the differential method.

Definition 2 denotes centralized differential privacy; that is, the raw data are collected through the curator, the privacy protection model is added to the collected data, and the perturbed aggregated information is released to the public user, as shown in Figure 1.

Sensitivity is a key measure of differential privacy, which is determined by the query function, indicating the change in the output of the neighboring dataset to a query function. The greater the sensitivity, the greater the impact of the query function on neighboring datasets. When setting the disturbance mechanism, we need to increase the disturbance to make the output results as similar as possible. Global sensitivity is first proposed, which measures the maximum difference between queries of two neighboring datasets. Global sensitivity is defined as follows:

Definition 3

($l_p$–Global Sensitivity [43]). For all $(\mathit{D},{\mathit{D}}^{\prime })$ such that is $\parallel \mathit{D}-{\mathit{D}}^{\prime }{\parallel }_p\le 1$, the $l_p$-global sensitivity of a query $f:\mathit{D}\to {\mathbb{R}}^d$ is defined as

$$\begin{array}{cc}\hfill \Delta f& =\underset{\mathit{D},{\mathit{D}}^{\prime }}{max}{\parallel \mathit{f}\left(\mathit{D}\right)-\mathit{f}\left({\mathit{D}}^{\prime }\right)\parallel }_p,forintegerp>0,\hfill \end{array}$$

(3)

where sensitivity $\Delta f$ is the maximum difference measured by the Minkowski distance in query f when any tuple is in the input dataset. Here, p is a variable element in Minkowski distance, which can take 1, 2, and ∞. Sensitivity is a measure of the maximum change in the query when the dataset changes only one vector.

The implementation of differential privacy relies on the design of privacy mechanisms M, typically achieved by adding noise in data release or data analysis models. This ensures that an attacker cannot compute the input data from the output results of two neighboring datasets. Common methods include the Laplace mechanism, the Gaussian mechanism, and the exponential mechanism, which introduce perturbations at the model input, objective function, or model output. The Laplace mechanism [44] achieves differential privacy by adding Laplace noise to the query or model output results before returning them to the user.

Definition 4

(Laplace Mechanism [11]). Given a function $f:\mathit{D}\to {\mathbb{R}}^d$ over a dataset $\mathit{D}$, mechanism M provides the ε–differential privacy if it satisfies

$$\begin{array}{c}\hfill M\left(\mathit{D}\right)=f\left(\mathit{D}\right)+Lap(\frac{\Delta f}{\epsilon }),\end{array}$$

(4)

where $Lap\left(\frac{\Delta f}{\epsilon }\right)$ is the added Laplace noise and $\Delta f$ is the $l_1$-sensitivity $(p=1)$ that controls how much perturbation is required in the mechanism M. The scale parameter of Laplace noise is represented as $\frac{\Delta f}{\epsilon }$ in differential privacy; that is, the amount of added noise is determined by the sensitivity and privacy budget. As usual, Laplace and Gaussian mechanisms are suitable for handling numerical data, while the exponential mechanism is typically used for non-numerical data.

3.2. Granular Computing and Information Granules

In this section, we introduce the basic concepts of granular computing, including information granule representation and construction. Granular computing is a method of data processing and knowledge representation that aims to abstract and summarize the original data so as to improve the processability and comprehensibility of data. It is based on the concept of “granules”, which divide the original data into different levels of granularity, thus transforming the complex original data into simpler and easier forms.

Information granules are entities that refine and abstract data. We create information granules of different sizes according to practical problems to enhance the ability of computer reasoning and processing information. The mapping relationship of datasets in information granules represents the semantics of data. As a result, information granules often use multiple sets to express their hierarchical structure according to the actual situation. Interval-type information granules are a simple and clear granular expression form, and information is organized together according to different division criteria. The fuzzy set better aggregates and divides the data, hides unnecessary information in the data, enhances the expression of important semantics of the data, and still has the ability to express knowledge in noisy data.

Fuzzy C–Means (FCM) algorithm is often used to cluster data and find a structure in data [45]. Suppose that a finite dataset $\mathit{D}$ is provided, which consists of N input–output pairs, i.e., $\mathit{D}=\{({\mathit{x}}_1,y_1)$, $({\mathit{x}}_2,y_2)$, …, $({\mathit{x}}_N,y_N)\}$, where ${\mathit{x}}_i\in {\mathbb{R}}^d$ and $y_i\in \mathbb{R}$ for $i=1,2,\dots ,N$. The minimized objective function of the FCM is given in the form

$$J=\sum _{j=1}^c\sum _{i=1}^N{u}_{ij}^m{\parallel {\mathit{x}}_i-{\mathit{v}}_j\parallel }^2,$$

(5)

where the vector ${\mathit{v}}_j$ is the center of a cluster (prototype), $j=1,2,\dots ,c$, and the associated partition matrix $\mathit{U}=\left[u_{ij}\right],u_{ij}\in [0,1]$, contains membership degrees, which describe the affiliation of each numerical data point with the cluster center. The parameter m denotes the fuzzification factor, whose value is usually set to 2. The numerical prototypes are computed by

$${\mathit{v}}_j=\frac{{\sum }_{i=1}^N{u}_{ij}^m{\mathit{x}}_i}{{\sum }_{i=1}^N{u}_{ij}^m}.$$

(6)

The associated partition matrix $\mathit{U}=\left[u_{ij}\right]$ comes in the form

$$u_{ij}=\frac{1}{{\sum }_{k=1}^c{\frac{\parallel {\mathit{x}}_i-{\mathit{v}}_j\parallel }{\parallel {\mathit{x}}_i-{\mathit{v}}_k\parallel }}^{2/(m-1)}},$$

(7)

where $\parallel ·\parallel$ is the Euclidean distance. The FCM algorithm updates the numerical prototype $\mathit{v}$ and finally obtains the minimum of the objective function.

In this work, we use the FCM algorithm to extract the semantics of numerical data and generate the basic element of information granules, i.e., numerical prototype $\mathit{V}=\{{\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_c\}$, where c is the number of the numerical prototype in the dataset. Zhu et al. propose hypercube-like information granules [46]. Lu et al. construct cone information granules [47].

The basic cognitive framework of information granules can be expressed as

$$\mathbf{\Omega }=\left\{{\mathrm{\Omega }}_j\right|j=1,2,\dots ,c\}$$

The notation $\mathbf{\Omega }$ represents information granules, which can be encoded using numerical prototypes and the following granular mapping rules:

(1)

Symmetric Allocation of Information Granularity

$$[min(a_j(1-{\psi }_j/2),a_j(1+{\psi }_j)/2),max(a_j(1-{\psi }_j/2),a_j(1+{\psi }_j)/2)],$$

(8)

where the notation $\psi =\left\{{\psi }_j\right|j=1,2,\dots ,c\}$ is a certain allowable level of information granularity in $[0,1]$.

(2)

Asymmetric Allocation of Information Granularity

$$[min(a_j(1-{\psi }_j^{-}),a_j(1+{\psi }_j^{-})),max(a_j(1-{\psi }_j^{+}),a_j(1+{\psi }_j^{+}))],$$

(9)

(3)

Uniform Allocation of Information Granularity

$$[min(a_j(1-\psi )/2,a_j(1+\psi )/2),max(a_j(1-\psi )/2,a_j(1+\psi )/2)],$$

(10)

where $\psi$ is a uniform allocation of information granularity. The collection of its parameters $\mathit{a}$ is called the granularity parameter. In the buildup of the mapping, the values of the parameters are optimized, yielding a vector $a_{opt}$. After finding the numerical prototype of the numerical data, the upper and lower boundaries of the information particle are optimized to reach the Pareto frontier of the bi-objective optimization function. The specific optimization strategy is called the principle of justifiable granularity, which will be described in the next section.

4. Materials and Methods

In terms of security, internal security refers to the protection of individual data points within the information granules, while external security refers to the protection of the information granules themselves from disclosure or manipulation. Differential privacy is a technique for achieving both internal and external security by adding noise to the data in a way that preserves the overall statistical properties of the data while making it more difficult to identify individual data points or extract sensitive information. Our work constructs information granules based on differential privacy. It mainly studies the information granules of fuzzy sets and interval sets. Our work is mainly divided into internal security and external security.

We explore techniques for optimizing information granules to improve their effectiveness and efficiency in problem-solving. Additionally, we provide a framework for analyzing the reasonable granularity of information granules and discuss how it can be used to ensure that the granularity is appropriate for power systems.

4.1. The Principle of Justifiable Granularity: Coverage and Specificity

The principle of justifiable granularity is a concept in granular computing that suggests that information should be divided into small, meaningful units but not so small that the information becomes meaningless or inefficient. In other words, information should be granular enough to be useful but not so granular that it becomes too fine-grained to be practical [27,48,49,50].

The purpose of granular computing is to abstract data information into a basic paradigm of expressing information granules, which can derive fundamental principles based on functional similarity, semantic closeness, and other aspects, making it easier to publish, store, and analyze information [51]. The principle of justifiable granularity can construct information granules with clear semantics and reasonable density and is widely used to optimize information granules and achieve the protocols of allocation in information granules.

The construction of information granules belongs to a mapping method, which transforms numerical data into the form of information granules (interval or fuzzy interval) through a certain mapping relationship. Subsequently, in the optimization process, we strive to transform the allocation problem into a well-defined performance index and framework with clear evaluative functions. Suppose the input and output of the numerical dataset are $\mathit{D}=\{({\mathit{x}}_1,y_1)$, $({\mathit{x}}_2,y_2)$, …, $({\mathit{x}}_N,y_N)\}$, where ${\mathit{x}}_i\in {\mathbb{R}}^d$ and $y_i\in \mathbb{R}$ for $i=1,2,\dots ,N$. The principle of justifiable granularity provides optimization criteria for constructing information granules $\mathbf{\Omega }$, which are the available experimental evidence (coverage) and semantic meaning (specificity) of information granules [48].

Experimental evidence (coverage): The coverage serves to calculate how many numerical data are contained in the information granule $\mathbf{\Omega }$. The numeric evidence accumulated within the bounds of $\mathbf{\Omega }$ has to be as high as possible [52]. The greater the coverage rate, the larger the amount of data information that the information granules contain, indicating that it has pretty good experimental evidence. Generally, we use cardinality to calculate coverage:

$$Cov=card\left\{x_i\right|x_i\in \mathbf{\Omega }\}.$$

(11)

More generally, we map the coverage to an increasing function with respect to this cardinality, say $f_1\left(card\left\{x_i\right|x_i\in \mathbf{\Omega }\}\right)$.

Semantic meaning (specificity): Semantic meaning measures the specific expression ability of information granules. If the information granule contains few data points, then its semantic expression ability is stronger. If the information granule contains a large number of data, the content expressed by the information granule is not specific enough. We take the interval information granule as an example, ${\mathrm{\Omega }}_1=[a,b]$, and the length of the interval can be expressed as $Sp=|b-a|$. The continuous non-increasing function can be reasonably expressed as $f_2$, say $f_2\left(Sp\right)$. The shorter the length of the interval, the better the specificity, and vice versa.

It can be seen from the above that the experimental evidence and semantic meaning are in conflict with each other. In order to optimize the granularity of information and realize the reasonable expression of information granularity, we establish a compromise criterion

$$Q=(\sum _{j=1}^c{f}_1\left(card\left\{x_i\right|x_i\in \mathbf{\Omega }\}\right))\ast f_2\left(\right|b-{a\left|\right)}^{\beta },$$

(12)

where $\mathbf{\Omega }=\left\{{\mathrm{\Omega }}_j\right|j=1,2,\dots ,c\}$ is the notation of information granules. Equation (12) is the basic representation of the objective function with the principle of justifiable granularity, which can be improved according to the method of information granule construction.

4.2. External Security Information Granules (ES-InG) Based on Differential Privacy

In this section, we propose a method of building secure information granules for the first time. The first framework is the construction of external secure information granules (ES-InGs), which is based on the idea of a localized differential privacy framework. This framework discusses the strategy of distribution optimization [53]. In any dataset, the probability density function of differential privacy is optimized by adding random noise, such as Laplace noise or Gaussian noise. The key to ES-InG is to add disturbance to the numerical data to ensure the security of the data before building the information granule, and then to ensure the security of the information granule through optimization such as fuzzification and the principle of justifiable granularity.

As shown in Figure 2, noise is added to the numerical data, and then the numerical data containing noise are expressed in granularity, and the information granules are constructed and optimized. Finally, the safe information granules are provided as input for further analysis and querying. Adding disturbance to numerical data is often used by many privacy protection methods. The problem is that as the number of numerical data increases, more noise is added. As a result, the data availability decreases [54]. The construction of information granules can effectively express the meaning of these data, help users understand these data, and integrate useful data to help users make decisions.

When constructing external security information granules based on differential privacy (ES-InG), the main problems are how to add appropriate noise to meet the demand for differential privacy and how to build reasonable information granules through optimization and feedback to enhance semantic expression. We take one-dimensional data as an example to construct ES-InG.

Given a dataset ${\mathit{D}}_1=\{x_1,x_2,\dots ,x_i\}$, $i=1,2,\dots ,n$, an information granule is constructed in Figure 3a. This is an interval-type information granule, and the interval is $[x_1^{-},x_1^{+}]$. The red point is numerical data in the dataset, but it is not included in the information granule. We add disturbance noise to dataset ${\mathit{D}}_1$ as follows:

$${\hat{x}}_i=x_i+Lap\left(\frac{\mu }{b}\right),$$

(13)

where $Lap\left(\frac{\mu }{b}\right)$ is the Laplace noise, $\mu$ is the expectation of Laplace distribution, and b is the variance of Laplace distribution. Other noise distribution mechanisms can also be applied to data perturbation, the such as Gaussian mechanism and optimal data-independent noise distribution [54,55].

In the construction of information granules, the allocation of information granules can be realized through a variety of information-granule-allocation protocols, according to the diverse types of data. The information-granularity-distribution protocol includes the uniform and symmetrical distribution of information granularity in Figure 4a, the uniform and asymmetric distribution of information granularity in Figure 4b, the non-uniform and symmetrical distribution of information granularity in Figure 4c, and the non-uniform and asymmetric distribution of information granularity in Figure 4d.

According to the above four allocation protocols, the perturbed data can be optimized based on different protocols. Consequently, the perturbed data can be accurately allocated to the information granules, as shown in Figure 3b. Through the optimization of semantic similarity and functional similarity, information granules allow the perturbed data semantics to be accurately expressed.

4.3. Internal Security Information Granules Based on Differential Privacy

In this section, we propose two internal security information granulation frameworks based on differential privacy, which are macro methods. They depend on the data type and the credibility of data storage on the one hand, and the performance of querying or analyzing information granulation on the other hand.

Figure 5 shows that the differential privacy mechanism is added in the granulation process to realize the secure release of information granules. It is in the process of constructing information granules, which is called internal information granule security. The advantage of this framework (IS-InG-I) is that it can design a differential privacy mechanism according to different granulation algorithms and finally realize data security during information granule querying. Compared with the structure of information granules based on external security, internal security information granules can reduce the amount of noise added, represent data semantics more accurately, and enhance data availability. The information granule construction method based on a framework (IS-InG-I) is proposed in [56].

Figure 5b is another framework based on internal security information granules (IS-InG-II). First, numerical data are granulated to obtain interval/fuzzy set information granules $[x^{-},x^{+}]$. Take one-dimensional dataset ${\mathit{D}}_1$ as an example, as shown in Figure 6a. The information granule is $\mathrm{\Omega }=[x_1^{-},x_2^{+}]$. Then, we add perturbation noise to the information granule based on the query function. Note that this is in the form of the information granule; that is,

$$\widehat{\mathbf{\Omega }}\left(x\right)=\mathrm{\Omega }\left(x\right)+Lap(\frac{\Delta f}{\epsilon }),$$

(14)

where the sensitivity $\Delta f$ and privacy budget $\epsilon$ in the Laplace mechanism is set according to the query function and privacy protection requirements. There are many kinds of query functions, including summation, counting, averaging, and various analysis algorithms. The differential privacy mechanism leads to the change in information granules. As shown in Figure 6b,c, the interval/fuzzy set is enlarged or decreased, depending on the noise. Finally, the information granules satisfying differential privacy are obtained.

These are the three information granularity security frameworks proposed in this work. ES-InG, IS-InG-I, and IS-InG-II are built based on the optimization criteria of information granularity and the information-granularity-allocation protocol and can be applied to a variety of big data modeling and data analysis tasks. In the next section, we experiment with the performance of information granules constructed and analyze their applicability.

5. Experiments

The three new frameworks for developing and analyzing granular input spaces and privacy-preserving granular modeling are elaborated on by us. We are concerned with the data security issues related to the analysis of electricity usage in the corporate production process. In addition, building on data security protection, we aim to test the algorithm’s applicability. We use two-dimensional and multi-dimensional data to construct information granules, ensuring secure and reliable publication and analysis of information granules. We set a synthetic discrete dataset, which includes 800 data points. The mean vectors $v_i$ and variance matrices ${\mathrm{\Sigma }}_i$ of data are as follows:

$$\begin{array}{ll}{v}_1=[3,3]& {\mathrm{\Sigma }}_1=\left[\begin{array}{cc}0.8& 0\\ 0& 0.3\end{array}\right]\\ v_2=[6,6]& {\mathrm{\Sigma }}_2=\left[\begin{array}{cc}0.3& 0\\ 0& 0.5\end{array}\right]\\ v_3=[4,4]& {\mathrm{\Sigma }}_3=\left[\begin{array}{cc}1.1& 0\\ 0& 0.7\end{array}\right]\\ v_4=[5,7]& {\mathrm{\Sigma }}_4=\left[\begin{array}{cc}0.2& 0\\ 0& 0.5\end{array}\right]\end{array}$$

Among the information granules based on differential privacy, we use hypercube information granules and hypersphere information granules in our study, as shown in Figure 7.

The multi-dimensional dataset includes actual power consumption data collected within Hitachi Corporation, encompassing electricity usage data for office buildings, production facilities, and cloud computing laboratories from various perspectives and dimensions. It also includes parameters such as the company’s air conditioning intake temperature, set temperature, airflow, and more. The multi-dimensional datasets primarily consist of Steel Industry Energy Consumption (SIEC) [57] and Appliances Energy Prediction (AEP) [58], which are sourced from publicly available datasets in the machine learning repository (http://archive.ics.uci.edu/ml, accessed on 1 September 2023). The characterization of these datasets, including the number of instances and the number of features and output variables, is summarized in

Table 1. Summary of datasets in experiments.

Datasets	No. of Attributes	No. of Instances	Characteristics	Missing Values
Synthetic	2	800	Categorical	No
Hitachi	6	5372 × 12 1	Multivariate	No
SIEC	9	35,040	Multivariate	No
AEP	29	19,735	Time-Series	No

¹ This includes data records from 12 individual electricity meters for various locations, including different floors in the office building, the factory, and laboratories. Please refer to the main text for more details.

. In the experiment, when constructing multi-dimensional security information granules, the FCM algorithm is used, and the information granules are optimized through the principle of justifiable granularity.

Based on the principle of justifiable granularity, the evaluation index of the structure of information granularity depends on coverage, specificity, and its objective function. In the experiments, it is important to pay attention to the relaxation parameter $\beta$ and the number of numerical prototypes used when constructing information granules. Additionally, when constructing information granules that include privacy protection, the amount of noise added is an important experimental indicator.

5.1. Two-Dimensional Dataset

In the synthetic two-dimensional dataset, the actual division of the clusters is four, and we discuss the difference in the information granules produced by different numerical prototypes in the information granules constructed based on differential privacy. In the objective function of the principle of justifiable granularity Equation (12), the value of flexible parameters $\beta$ is also a focus of attention. We compare the difference between information granules constructed based on differential privacy and ordinary information granules through coverage specificity and the objective function Q. As the ES-InG framework adds noise to the original data, we compare the trend of FCM loss function and optimization results in J to determine the accuracy of numerical prototype construction.

We report the number of numerical prototypes in information granules and ES-InG in

Table 2. The effect of cluster number on the construction of information granules.

No. Center	Q	J	$\mathit{Cov}$	$\mathit{Sp}$	${\mathit{Q}}_{\mathit{DP}}$	${\mathit{J}}_{\mathit{DP}}$	${\mathit{Cov}}_{\mathit{DP}}$	${\mathit{Sp}}_{\mathit{DP}}$
2	16.59	26.12	312.15	1.06	5.96	121.12	280.78	0.58
3	9.94	23.16	149.55	1.19	3.68	105.59	207.81	0.74
4	6.12	35.74	116.11	1.37	2.54	101.79	155.2	0.78
5	4.57	32.03	87.47	1.42	1.95	80.58	114.36	0.86
6	3.7	30.29	84.42	1.41	1.56	69.79	92.84	0.95
7	2.94	29.24	64.37	1.48	1.31	59.35	82.45	1.02

. In ES-InG, the privacy budget is set to 0.1, which is a small privacy budget, indicating that the perturbation is relatively large and the degree of protection is high. As shown in Table 2, the loss function J of FCM decreases as the number of numerical prototypes increases, indicating that the more information granules are constructed, the less clustering loss is incurred.

According to the principle of justifiable granularity, the greater the coverage, the more information contained in the information granule, and the lower the specificity. The information granule represents more specific and clear information. This provides us with an optimization standard for evaluating information granule construction. Additionally, we have formulated an objective function Q based on the principle of justifiable granularity. A higher value of Q indicates a stronger semantic expression capability for information granules. The Q value is an objective function that balances the two conflicting factors. Generally, we calculate its maximum value as the local optimal value. Therefore, we compare the construction results of information granules and ES-InG. Table 2 shows that under the premise of different numerical prototypes, the Q value and specificity of ES-InG are better, indicating that the information granules constructed by the ES-InG framework are successful.

Additionally, we compare the impact of the flexible parameter $\beta$ on information granule construction as shown in

Table 3. Effects of balance parameters on information granules.

$\mathit{\beta }$	$\mathit{Cov}$	$\mathit{Sp}$	Q	${\mathit{Cov}}_{\mathit{DP}}$	${\mathit{Sp}}_{\mathit{DP}}$	${\mathit{Q}}_{\mathit{DP}}$
1	158.11	1.9	300.75	208.06	1.96	409.09
2	147.63	1.79	236.68	206.74	1.87	363.39
3	140.35	1.66	163.32	204.67	1.75	277.67
4	132	1.53	91.44	203.32	1.64	186.96
5	116.11	1.37	35.74	200.86	1.51	101.79
6	102.93	1.24	11.94	194.96	1.38	43.36
7	89.56	1.11	2.96	189.12	1.27	16.61

. Compared with ordinary information granules, the coverage and $Q_{DP}$ of ES-InG are better, while the specificity of the two is similar. Therefore, the information granules with the differential privacy mechanism added exhibit good construction results.

Comparing Table 2 with Table 3, we can conclude that the successful construction of information granules in the ES-InG framework is due to the addition of noise, which reduces the discreteness of data with large discreteness, and also due to the reasonable allocation of granularity during granulation, which increased the semantic-expression ability of noisy data.

Figure 8 reports the changing trend of coverage, specificity, and the principle of justifiable granularity Q with the number of information granules and also reflects the impact of the change in flexible parameter $\beta$ on the structure of information granules. There is a conflicting relationship between coverage and specificity, and it is important to balance their results. Additionally, Figure 8 provides relevant reference indicators that can be used to select the number of numerical prototypes and the size of $\beta$ based on the requirements of information granules. The purpose of flexible parameters is also to balance specificity and coverage based on raw data.

As shown in Figure 9, the conclusions drawn from Table 2 and Table 3 are further confirmed, namely, why the granulation of ES-InG is justifiable and how Laplace noise acts on the original data and makes the data more compact through regularization, as shown in Figure 9b,e. Furthermore, with the optimization of $\beta$, the representational ability of data granulation is enhanced, as shown in Figure 9c,f.

5.2. Multi-Dimensional Dataset

We list several multi-dimensional datasets, construct hypercube information granules and information granule models based on differential privacy, and compare the performance of the original information granules and ES-InG. Based on the experiments on two-dimensional information granules, we develop a similar experimental analysis process. Due to the complexity of multi-dimensional spatial data visualization, we use three-dimensional graphs and radar graphs to represent them.

The analysis of the numerical prototype and flexible parameters of the AEP dataset in Figure 10 indicates a decreasing trend in overall coverage, specificity, and objective function as the flexible parameters increase. The dashed line represents differential-privacy information granules, and the solid line represents the original information granules. Figure 10a–c assess the degree and semantic expressive capacity of information granule construction from three perspectives, coverage, specificity, and the objective function, based on the principle of justifiable granularity. Simultaneously, we compare information granules derived from the original data with those obtained after the introduction of differential privacy mechanisms. We also take into consideration the impact of data prototypes and granule allocation on information granule representation. The experimental results reveal that by adaptively adjusting the flexible parameter $\beta$, all three evaluation metrics gradually stabilize in the AEP dataset. Simultaneously, the varying number of numerical prototypes also converges toward a consistent trend.

We use the representational method of a radar chart to visualize the relationship of each information granule. Radar charts are mainly used to show the relative size of multiple variables and the proportion of these variables in the overall data. We put the ES-InG model and the original model results in the same radar graph and put multiple information granules in the same graph to visually compare their differences and similarities. This visualization method allows one to see the relationships between and importance of different information granules.

On the other hand, granular computation, through the analysis of data semantics, helps emphasize the correlation and importance of different data attributes. This allows for a better focus on specific attributes in data analysis. This characteristic is particularly crucial in differential privacy algorithms, as it enables the quantitative allocation of noise to maximize data utility. We construct the information granules and calculate information granularity in the AEP dataset. Blue represents the original information granules, and orange represents ES-InG. We designate the attributes of the AEP dataset as $\mathit{X}=\{X1,X2,\dots ,X27\}$. Figure 11a indicates that among the original information granules, the information granules that cover less information are $X6$, $X9$, and $X26$. In the ES model, the information distribution among each attribute is relatively balanced. This is primarily due to the noise introduced by differential privacy. By comparing the original information granules in Figure 11a, we can adjust the allocation of noise for $X6$, $X9$, and $X26$ to enhance the accuracy of data analysis. In the evaluation of concreteness, the original information granules and ES-InG have similar indexes.

In the SIEC dataset, Figure 12 shows the results of coverage, specificity, and Q with the number of information granules and flexibility parameters. The original data exhibit stronger expressive capabilities than differentially private data with added noise. Through experimental analysis, we have identified that the primary reason for this is a large number of data records and a substantial variety of data features in the SIEC dataset. Consequently, introducing noise into the original data results in a more significant disturbance, leading to a decrease in data expressiveness. To address this issue, on the one hand, we aim to optimize noise allocation as much as possible when perturbing the data. On the other hand, in our future research, we plan to design two additional granule computation frameworks based on differential privacy.

We designate the attributes of the SIEC dataset as $\mathit{X}=\{X1,X2,\dots ,X7\}$. Figure 13 analyzes the importance of each attribute in the semantic representation within the dataset. When the granularity allocation is larger, it signifies that the attribute is more critical in data analysis. In the ES-InG model, the ones that cover more information are $X5$ and $X6$, and the granularity allocation of the original information granules is higher. In the evaluation of specificity, as shown in Figure 13b, the original information granules and ES-InG have similar indexes.

The Hitachi dataset is a self-constructed dataset. We have collected data on power consumption, air conditioning inlet temperature, room temperature, air conditioning status, airflow, date, and timestamp in the office and production areas of Hitachi Corporation. These data span from May 2022 to May 2023 and are sampled every hour. The data have been divided into two equal parts for testing and training. The dataset comprises 12 individual datasets, which include records for the first to fifth floors of the office building, total power consumption, records for various laboratories, and records for the production workshop.

We conducted experiments on all 12 Hitachi datasets. Due to the large number of data, it is challenging to visualize using line plots. Therefore, we used three-dimensional heatmaps to represent the coverage, specificity, and objective function value Q of the ES-InG model. Figure 14 illustrates the impact of numerical prototypes and flexible parameters on the construction of information granules. Here, we introduce Laplace noise into the original dataset and construct information granules to mitigate the impact of noise on the utility of the original data.

We designate the attributes of the SIEC dataset as $\mathit{X}=\{X1,X2,\dots ,X6\}$. Figure 15a illustrates that the attributes $X4$, $X5$, and $X6$ contain a greater degree of granule allocation, making them more important in semantic representation. At the same time, the establishment of original information granules in this dataset does not significantly impact the data’s semantic meaning. However, the ES-InG model provides a clearer representation of the tendencies in the features of attributes $X4$, $X5$, and $X6$. In the evaluation of specificity, as shown in Figure 15b, the original information granules and ES-InG have similar indexes. $X1$ and $X2$ are date and time data, and as a result, they have a relatively low impact on granule allocation and the overall semantic representation in the radar chart when representing the data. However, if we were to utilize time series algorithms, the roles of $X1$ and $X2$ would become more significant. Therefore, in our future research, we will propose more granule computing models based on differential privacy, such as BP neural networks and Long Short-Term Memory (LSTM) networks.

6. Conclusions

In summary, we first introduce the construction method of information granules and the measurement method of information granules. On this basis, we propose three information-granule-construction frameworks based on differential privacy, namely ES-InG, IS-InG-I, and IS-InG-II. This study conducted experiments on ES-InG information granules. Our work improves ES-InG and constructs IS-InG-I and IS-InG-II.

We have constructed privacy-preserving information granules based on differential privacy on top of the construction method for the original information granules. Due to the semantic expressiveness and abstraction of information granules, as well as their fuzzy features, the semantics of the noisy information granules have been improved. Using a radar chart, we have analyzed the interrelationships and importance of the same data information granules. In the experiment, the construction of information granules for two-dimensional data is more accurate. For multi-dimensional data, due to the influence of different dimensions, the constructed results are slightly inferior to the original information granules. Future studies will include two directions: we will propose an optimization method with flexible parameters specifically for multi-dimensional information granules, and we will propose more granule computing models based on differential privacy (IS-InG-I and IS-InG-II), such as BP neural networks and Long Short-Term Memory (LSTM) networks.