SCIENCE CHINA Technological Sciences • Article •
April 2016 Vol.59 No.4: 604–617 doi: 10.1007/s11431-016-6025-2
Complex network theory-based condition recognition of electromechanical system in process industry WANG RongXi, GAO JianMin, GAO ZhiYong*, GAO Xu & JIANG HongQuan State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China Received September 26, 2015; accepted December 21, 2015
In order to recognize the different operating conditions of a distributed and complex electromechanical system in the process industry, this work proposed a novel method of condition recognition by combining complex network theory with phase space reconstruction. First, a condition-space with complete information was reconstructed based on phase space reconstruction, and each condition in the space was transformed into a node of a complex network. Second, the limited penetrable visibility graph method was applied to establish an undirected and un-weighted complex network for the reconstructed condition-space. Finally, the statistical properties of this network were calculated to recognize the different operating conditions. A case study of a real chemical plant was conducted to illustrate the analysis and application processes of the proposed method. The results showed that the method could effectively recognize the different conditions of electromechanical systems. A complex electromechanical system can be studied from the systematic and cyber perspectives, and the relationship between the network structure property and the system condition can also be analyzed by utilizing the proposed method. complex network, condition recognition, phase space reconstruction, complex electromechanical system Citation:
Wang R X, Gao J M, Gao Z Y, et al. Complex network theory-based condition recognition of electromechanical system in process industry. Sci China Tech Sci, 2016, 59: 604617, doi: 10.1007/s11431-016-6025-2
1 Introduction In the modern process industry, a production system can be viewed as a distributed and complex electromechanical system that consists of a collection of mechanical equipment, chemical reactors, and computerized monitoring systems. Because of the specificities of the technological flow of the process industry, any mistake or disturbance in the flow will be exponentially amplified in the following processes, which could even trigger major production accidents. Therefore, recognizing the operating condition of the production system and detecting abnormal states are meaningful goals to prevent safety accidents, and they are also the long-term research directions of industry and academia.
The traditional condition-recognition methods can be divided into three types: experience-based, model-based, and data-based methods. Independent of experience and strict analytic models, data-based methods are the mainstream approaches for condition recognition in the modern process industry. Statistical approaches have been widely applied for condition recognition and feature extraction in the last few decades. Principal component analysis (PCA) [1, 2] can discover the pattern of the operating conditions and remove the noise from a monitored time series using dimension reduction. Similar to PCA, the independent components are extracted to reduce the data dimensions for pattern recognition in an independent component analysis [3]. However, the methods mentioned above have a strong linearity assumption, which greatly limits their applicability. To solve the nonlinear problem of observer data, ker-
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nel-based techniques have been successfully developed to tackle the nonlinear problem in recent years, such as kernel principle component analysis (KPCA) [4, 5] and kernel independent component analysis (KICA) [6, 7]. The basic idea is that the mapped data are analyzed using conventional linear statistical analysis techniques in the feature space, which is equivalent to a nonlinear analysis in the original input space. GuangFei et al. [8] proposed a synthetic criterion for early chatter recognition by integrating the standard deviation and one-step autocorrelation function. Kai et al. [9] presented a general framework to quantitatively assess the operating condition of an electromechanical system based on meshing scientific data visualization and digital image processing. They then transformed the condition recognition into a method for quantizing the abnormal degree by calculating the difference between the benchmark color spectrum and observed color spectrum. However, their work was incomplete because these methods ignored the coupling relationships among the components of a production system in the process industry and locally analyzed the complex system. A complex production system is essentially joined together by discrete pieces of electromechanical equipment to form a coupled, highly corrective distributed and complex electromechanical system network for transmitting power, energy, and control information. Thus, complex electromechanical systems and networks are naturally coincident. Complex networks are capable of offering the comprehensive statistical characteristics of a dynamic system in monitoring the time series of the process industry from a new angle. Since the groundbreaking works of Watts and Strogatz regarding small-world networks [10] and Barabasi and Albert regarding scale-free networks [11], real-world phenomena have begun to be studied from the perspective of actual networks and network theory. Through researching the topological, static, and dynamic properties of the real world, complex network theory has illustrated the essences of complex systems. Most research on complex networks involves two main steps: the representation of the problem as a complex network followed by the analysis of its topological features, as obtained by a set of measurements. Given the measurements, it is possible to identify different categories of structures. Numerous review articles and books on complex networks, which the readers may find useful to consult, have already appeared in the literature. Boccaletti et al. [12] reviewed the major concepts and results achieved from a study of the structure and dynamics of complex networks and summarized the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology and from statistical mechanics to medicine and engineering. The authors focused on the problem of neglecting all of the extra information regarding the temporal- or context-related properties of interactions and offered a comprehensive review of both the structure
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and dynamic organization of graphs made from the diverse relationships between their constituents. Furthermore, several relevant issues have been examined, including a full redefinition of the basic structural measures and how the multilayer nature of a network affects its processes and dynamics [13]. Further research has provided meaningful results in the field of complex network molding and analysis based on a time series. Zhang and Small [14] discovered that a complex network could be used to analyze the dynamic characteristics of a non-linear time series; since then, dynamic analyses based on complex networks have received widespread attention. Zhang et al. [15] used the methods of complex networks to characterize different time series and showed that the constructed network inherited the main properties of the time series. He also illustrated that different time series with different periodicities had different complex network characteristics. Mehraban et al. [16] researched the coupling relationship between different time series based on a penetrable visibility graph. Zhou et al. [17] proposed an improved visibility graph method, i.e., a limited penetrable visibility graph to establish a complex network from a time series, and tested its anti-noise performance. By transforming a time series into a weighted and directed network, Gao and Jin [18] and Sun et al. [19] described the dynamic characteristics of a chaotic system. Gao and Jin [20] proposed a reliable method for constructing complex networks from a time series with each vector point of the reconstructed phase space represented by a single node and the edge determined by the phase space distance. The detailed processes for constructing a weighted and directed network have been proposed in addition to distinguishing different time series using complex network properties such as the betweenness and clustering coefficients. Recently, the complex network theory has been introduced into the study of recognition. Wu et al. [21] proposed a novel approach to address edge detecting issues in image processing based on the local dimension of a complex network. Gonçalves et al. [22] established a complex network of video frames by mapping each pixel of the video into a node of the complex network, and obtained a feature vector by calculating the spatial and temporal average degree for dynamic texture recognition. Tang et al. [23] constructed complex networks from traffic time series by considering each day as a cycle and each cycle as a single node. They then analyzed some statistical network properties to recognize the traffic flow pattern. Gai et al. [24] proposed a complex network-based air flow prediction method to analyze the working characteristics of complex compressed air networks. A literature review showed that the previously mentioned research promoted the application of complex networks in time series analyses. It could also be used as the research foundation for recognizing the condition of an electromechanical system based on network properties. However, the
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effectiveness of numerical methods is limited. These achievements cannot be used to directly recognize the conditions of a complex electromechanical system systemically; most of the literature focused on complex network modeling and analysis from a single time series. A novel method for recognizing different operating conditions is proposed that combines phase space reconstruction with complex networks, followed by the transmission of each operating condition (phase point of the reconstructed phase space) in the reconstructed condition-space into a single node utilizing the quantitative geometrical features of the multi-dimensional operating condition. Furthermore, a complex network is constructed based on a limited penetrable visibility graph with each phase point (system condition) of the reconstructed phase space represented by a single node, and the different operating conditions of a complex electromechanical system are determined by analyzing the statistical features of this complex network. The outline of this paper is as follows. Section 2 discusses the complex network models and definitions of the statistical properties. Section 3 introduces the mathematical model of the phase space reconstruction and proposes new qualitative and quantitative methods for representing the reconstructed phase points. Section 4 illustrates the network modeling based on a limited penetrable visibility graph for the condition network of a complex electromechanical system. Section 5 presents a real case to explain the processes of the proposed method and verify the validity of this work.
2 Complex network models and statistical properties 2.1
Models of complex network
There are many complex systems in the real world. If the physical significance of the entities is ignored, the entities can be abstracted as nodes, and if links are used to describe the relationships between them, a complex system can be modeled by the network into regular and random networks. From the late 1950s to the beginning of the 21st century, random networks have been accepted as the best type of network to describe real systems. Recently, however, investigators have found that many real networks have different topological structures than previously researched networks. Most social, biological, and technological networks display substantial non-trivial topological features, with connection patterns between their elements that are neither purely regular nor purely random [25–27]. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, a community structure, and a hierarchical structure. In the case of directed networks, these features also include reciprocity, triad significance profiles, and other features. In contrast, many of the network mathematical models that
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have been studied in the past, such as lattices and random graphs, do not show these features. These networks are called complex networks. Two well-known and much studied classes of complex networks are scale-free networks and small-world networks, the discovery and definition of which are canonical casestudies in the field. Both are characterized by specific structural features, including power-law degree distributions for the former and short path lengths and high clustering for the latter. In this work, the statistical properties of small-world networks were applied to analyze the conditions of a complex electromechanical system. A network is called a small-world network by analogy with the small-world phenomenon. In 1998, Watts and Strogatz [10] introduced a model of the small-world phenomenon in a static graph. From a regular ring lattice, they randomly rewired the edges of this graph with a probability p varying from 0 (i.e., leading to a regular network) to 1 (i.e., leading to a random graph)[28]. During this process a new network between the regular network and the random network was constructed with short path lengths and high clustering. It is often useful to consider a matricial representation of a network. A network of size N can be completely described using the adjacency matrix A, an N×N square matrix whose entry aij(i,j=1,...,N) is equal to 1 when the link lij exists, and zero otherwise. The diagonal of the adjacency matrix contains zeros. It is thus a symmetric matrix for undirected networks [12]. 2.2
Statistical properties of complex networks
There are many quantitative properties that can be used to describe the topology of an undirected network based on the complex network theory. Typically, the properties of a complex network are characterized by the following basic quantities. For a network of size N, they are as follows [12]: Degree ki: for a certain node i, ki is defined as the number of edges rooted in the node. Degree distribution P(k): P(k) is defined as the probability distribution of the nodes with degree k, or P(k) is the percentage of nodes with degree k in the network. Cumulative degree distribution Pcum(k): Pcum(k) is defined as the probability distribution of the nodes with a degree greater than k, which can be described by the following formula: Pcum (k )
N 1
P(k ).
(1)
k k
Average shortest path length L: this quantity is applied to reflect the degree of separation, or how small the network is, with the formula, L
1 D(i, j ), n(n 1) / 2 1 i , j n ,i j
(2)
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where n is the node number, D(i,j) is the distance between i and j, and D(i,j) is the smallest number of edges between i and j. Clustering coefficient C: this coefficient is used to characterize the degree of tightness of the neighbor nodes, and C is defined as follows: C
2 ki 1 n , n i 1 ki (ki 1)
(3)
where ki is the degree of node i, and ki is the number of linked edges of the neighbors of node i. The degree distribution is one of the most important and basic topological properties, and it completely determines the statistical properties of uncorrelated networks. Commonly, the first step for analyzing a real network is to research the degree distribution. Different types of network can be distinguished based on the different degree distributions. A scale-free network is a network whose degree distribution follows the power-law distribution P ( k ) ~ k , where is the power-law exponent and 2
