Modeling Fault Propagation Paths in Power Systems: A New Framework Based on Event SNP Systems With Neurotransmitter Concentration

Received December 29, 2018, accepted January 7, 2019, date of publication January 14, 2019, date of current version February 6, 2019. Digital Object Identifier 10.1109/ACCESS.2019.2892797 Modeling Fault Propagation Paths in Power Systems: A New Framework Based on Event SNP Systems With Neurotransmitter Concentration TAO WANG 1,2, (Member, IEEE), XIAOGUANG WEI 3, TAO HUANG 1,2,4, (Member, IEEE), JUN WANG 1,2, HONG PENG 5, MARIO J. PÉREZ-JIMÉNEZ 6, AND LUIS VALENCIA-CABRERA6 1School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China 2Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu 610039, China 3School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610039, China 4Department of Energy, Politecnico di Torino, 10129 Turin, Italy 5School of Computer and Software Engineering, Xihua University, Chengdu 610039, China 6Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla, 41012 Sevilla, Spain Corresponding author: Tao Huang ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 61703345, Grant 51877181, and Grant 51607146, in part by the Key Fund Project of the Sichuan Provincial Education Department under Grant 18ZA0459, in part by the Key Scientific Research Fund Project of Xihua University under Grant Z17108, and in part by the Young Scholars Reserve Talents Support Project of Xihua University. ABSTRACT To reveal fault propagation paths is one of the most critical studies for the analysis of power system security; however, it is rather difficult. This paper proposes a new framework for the fault propagation path modeling method of power systems based on membrane computing. We first model the fault propagation paths by proposing the event spiking neural P systems (Ev-SNP systems) with neurotransmitter concentration, which can intuitively reveal the fault propagation path due to the ability of its graphics models and parallel knowledge reasoning. The neurotransmitter concentration is used to represent the probability and gravity degree of fault propagation among synapses. Then, to reduce the dimension of the Ev-SNP system and make them suitable for large-scale power systems, we propose a model reduction method for the Ev-SNP system and devise its simplified model by constructing single-input and single-output neurons, called reduction-SNP system (RSNP system). Moreover, we apply the RSNP system to the IEEE 14- and 118-bus systems to study their fault propagation paths. The proposed approach first extends the SNP systems to a large-scaled application in critical infrastructures from a single element to a system-wise investigation as well as from the post-ante fault diagnosis to a new ex-ante fault propagation path prediction, and the simulation results show a new success and promising approach to the engineering domain. INDEX TERMS Spiking neural P system, membrane computing, fault propagation path, fault propagation relationship, power system. I. INTRODUCTION Membrane computing (P systems) [1]–[3] is one of the important lines of artificial intelligence (AI) and aims at taking ideas for computing from the structures and the functioning of living cells as well as tissues or higher order structures organized by the cells. There are, basically, three main types of P systems: cell-like P systems [4], [5], tissue-like P systems [6]–[8] and neural-like P systems [9]. In the past decades, the research on neural-like P systems, in the framework of membrane computing, mainly focused on spiking neural P systems (SNP systems) [10]–[12], where the specific type of cells from which we took inspiration were the spiking neurons. Generally speaking, the SNP systems are a kind of distributed and parallel neural computing devices motivated by the behavior of neurons transferring information with each other by identical electric impulse (spikes) [12], [13]. The SNP system aims at incorporating specific ideas from spiking neurons into membrane computing and are considered as a combination of spiking neural networks (SNNs) and P systems. Thus, the SNP systems have the characteristics of 12798 2169-3536 2019 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. VOLUME 7, 2019 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems spiking neurons, P systems and SNNs, which makes the SNP systems suitable for dealing with practical engineering problems with desirable performance [14], such as combinatorial optimization problems [15], image processing [16], intrusion detection [17], knowledge representation and reasoning [18]–[21], fault diagnosis [22]–[28], etc. Among them, the research about fault diagnosis has been the hottest topic with increasing fruitful results, and many variants of the SNP systems were proposed for designing the SNP systembased fault diagnosis methods of different power systems, including the modeling method for electric locomotive systems [23], the fault section estimation method for traction power supply systems of high-speed railways [24], the fault equipment diagnosis method for metro traction power systems [25] and the fault diagnosis methods for electrical power systems [26]–[28], especially from a fuzzy perspective [29]. Although these research works advanced the development of the SNP system-based fault diagnosis methods for power systems, weak points w.r.t. both the SNP system models and the related applications have been observed after intensive studies: 1) Even the models can diagnose faults, but they cannot reveal the mechanism of the fault propagation and can only focus on a single element after a failure. The SNP systembased model is built for each suspicious element, which cannot macroscopically reveal the failure propagation mechanism of the whole power network before a fault. 2) The solution is still derived by manual reasoning procedure and is difficult to be computerized due to the drastically increased complexity of the logic relationships among all power elements and the associated protections with the growth of the system size. 3) The models have very limited topological flexibility thus cannot easily be applied to emerging power systems, characterized by high penetration of renewable and distributed generation, a large number of connected micro-grids, highly integrated multi-energy systems, etc., which require demanding topological flexibility of the models for fault diagnosis. Therefore, it is essential to have a new framework that can take full advantages of the SNP system while extending its flexibility and applications to a larger and more complex system. On the other hand, modeling fault propagation paths is a vital application which forms the fundamental basis for all security related analysis and guarantees the secure operation of the power system, including revealing fault propagation mechanism, analyzing weaknesses of the system, applying suitable countermeasures, etc. Methods for analyzing the dynamic characteristics of power systems, such as oscillations and transients, angular stability, frequency stability, via some real-time simulation platforms, have been focused in [30]–[33]. The dynamic methods mainly study the transient propagation process of fault propagation paths and focus on the fault features of electronic or rotating devices over time. Therefore, the dynamic methods can accurately reveal the operational characteristics of components under contingency within short time, especially under a couple of minutes. However, the dynamic methods are not suitable for studying the fault propagation paths of complex large-scaled systems over long time due to the high complexity of calculation. Therefore, static methods are widely employed to analyze the fault propagation paths due to its efficiency, simplicity and scalability [34]–[37]. In general, the static methods use the power flow results to study the overload in lines of the system with statistical/probabilistic methods to model the fault propagation paths. The ideas are that the fault characteristics of components, such as, lines, protections, etc., are regarded as satisfying some certain distributions and thus the statistical/ probabilistic models such as, Markov chain [38], [39], Monte Carlo [40], can be adopted to analyze the fault propagation characteristics. Besides the pure power system analysis methods, the complex network theory, especially suitable for large-scaled grids, is also one of the popular static methods. For instance, the OPA [41]–[43] and CASCADE [44], [45] models are proposed to analyze the self-organized criticality of grids. In addition, by analyzing some statistical indices of complex network from the perspective of topological structures of electrical networks, many electrical networks are proved to be small-world networks [46], [47]. It indicates that when one or more components fail, the faults can fast spread to other components due to the features of high clustering coefficient and short characteristic path of small-world networks. Further, many studies demonstrate that the electrical networks also have scale-free features [48], [49], which indicates that the networks are vulnerable under deliberate attacks but robust under random attacks. Although complex network theory has an advantage to model large-scale grids, the theory mainly focuses on the topological structure of girds and many studies neglect the operational and physical features to some extent. To reveal features in different fault operations, propagation graphs, such as, cascading fault graph [49]–[51], risk graph [52], influence graph [53], [54] and interaction graph [55], [56], are employed to analyze the fault propagation relationships among lines and propagation paths of fault. These approaches investigate the occurrences of different paths involved in the fault propagations by simulating several cascading events. Particularly, the propagation graphs not only consider topological features but also the physical and operational features, thus provide a valuable research prospect. However, all of the above-mentioned approaches are deriving the fault propagation paths from the entire system point of view without considering many detailed aspects of the element; therefore, sometimes it cannot reflects the real reactions of each element in the network, especially their response against the propagated fault. Furthermore, the traditional methods are based on the reductionism, thus the non-linear reactions, i.e. the complexity and the self-organizational features of the entire system have been lost in the modeling procedure. In addition, the majority of the methods is lack of VOLUME 7, 2019 12799 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems the visualization of the fault propagation, even though some give holistic system-wise information with graphs. Therefore, a more sophisticated framework for identifying the propagation paths is needed to better the analysis per se as well as the final representation of the results with a more intuitive and vivid fashion. Among many different modern methods, the SNP systems, inspired by the neurophysiological behavior of neurons sending electrical impulses (spikes) along axons, is an apt option to model the behaviors of the study objects at both the system and element levels. It conceptualizes and computerizes the real world problem from a holistic way. As each single component can be model as a proposition neuron with dedicated features and functions, the non-linear and complex response with its peer and the entire system can be captured. Inspired by the propagation graphs, the SNP systems are employed to model fault propagation paths in transmission electrical networks. In the SNP systems, connective relationships among neurons can reflect temporal adjacent information; therefore, we can reveal the fault temporal features among lines by the aid of neurons. The spike can dynamically imitate the information transfer process; therefore, we can employ the spike to draw the fault propagation process among lines of a transmission network. In addition, the SNP systems have a strong ability of graphical modeling, knowledge reasoning and parallel computing, which can reveal the propagation mechanism among lines intuitively and vividly. To reformulate a new framework based on the SNP system for studying the fault propagation paths, we propose the event-spiking neural P systems with neurotransmitter concentration (Ev-SNP systems) from the perspective of overload mechanism in electric networks. After improving the Ev-SNP systems to RSNP systems to reduce the computational burden, two standard benchmarks are used to verify the effectiveness of the proposed models. This paper mainly focuses on how to employ the SNP system to model fault propagation paths; therefore, we adopt a general simplified way to study the propagation mechanism from the perspective of overload mechanism. The main contributions of this paper are: (1) Due to the similarity between the spike transmission among different neurons through synapses and the fault propagation in the power systems, we innovatively model the fault propagation in the transmission network through the spike transmission in the SNP system. The Ev-SNP systems with neurotransmitter concentration are proposed to achieve such a goal, where the neurotransmitter concentration is employed to describe the probability and gravity degree of fault propagation among synapses. The higher the concentration is, the higher the probability is. (2) Targeted to the untraceable calculation burden of the Ev-SNP systems, we devise a reduction-SNP system (RSNP system) to combat the dimension disaster of such system by constructing single-input and single-output neurons. (3) We first extend the application of the SNP systems from a locality to a holism, from a single element to a systemwise investigation, from the post-ante application to a new ex-ante framework. This new framework not only can take full advantages of the SNP system, but also can model large and complex system with good topological flexibility. The remainder of this paper is organized as follows. Section II introduces the definition and computation configurations of the Ev-SNP systems. The RSNP systems are proposed to model the fault propagation paths in Section III. In Section IV, the RSNP system is applied to the IEEE 14- and 118- bus systems with the analysis of their effectiveness. Conclusions are finally drawn in Section V. II. EVENT-SPIKING NEURAL P SYSTEMS The SNP system [18] can be viewed as directed graphs composed of neurons and synapses, where the neurons are the vehicle for knowledge (information) representation, storage, and calculation while the synapses are used for knowledge (information) transmission. Three important ingredients including objects, spiking (firing) rules and forgetting rules are contained in the neurons which can be viewed as vertexes of the directed graphs. By contrast, the synapses can be regarded as edges of the directed graphs. To make the SNP system suitable for modeling fault propagation of power systems, we propose a new variation of the SNP system based on the works in [27] and [28], called event spiking neural P systems with neurotransmitter concentration (Ev-SNP systems). Definition 1: An event-spiking neural P system with neurotransmitter concentration (Ev-SNP system, for short) of degree (s,k) with s,k≥1 is a tuple 5=(A,Q,syn,I,O) (1) where (1)A= {a}is a singleton alphabet (ais called spike and represents a fault); (2)Q= {σ1, . . . , σs, σs+1, . . . , σs+k}is a set whose elements are called neurons. Qp= {σ1, . . . , σs}is the proposition neuron set. Each proposition neuron σi(1 ≤i≤s) represents a transmission line (the i-th line) and is denoted by σpi (the i-th proposition neuron). It is of the form (εi,ri), where (a) εirepresents an event that the i-th transmission line corresponding to σpi faults; (b) ridenotes its spiking (firing) rule of the form E/a→aεi, being Ea regular expression over {a}. The firing rule rican be applied if and only if it receives one spike. For the fault propagation path modeling, it means that if a proposition neuron receives one spike, then the rican be applied and will produce a new spike. This new spike a indicates that the event associated with the neuron σpi (i.e., εi) happens, i.e., the i-th transmission line faults. Qr= {σs+1, . . . , σs+k}is the rule neuron set. Each rule neuron σs+j(1 ≤j≤k) is denoted by σrj (the j-th rule neuron) and it can be of two different types: AND-rule neuron (denoted by ⊗-neuron) and OR-rule neuron (denoted by ⊕-neuron). The rule neuron σrj is of the form (cj, ηj,rj), where 12800 VOLUME 7, 2019 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems FIGURE 1. Neurotransmitter concentration. (a) Illustration of neurotransmitter in a biosystem. (b) The neurotransmitter concentration in a simple Ev- SNP system. (a) cjis a natural number expressing the number of the presynaptic neurons of σrj; (b) ηjis a real number in [0, 1] representing the neurotransmitter concentration of σrj. For the fault propagation path modeling, the ηjis used to express the fault propagation probability among the synapses that connect to the postsynaptic neurons of σrj, as shown in Figure 1; (c) ridenotes its spiking (firing) rule of the form E/a(ε1, ..., εcj)→aεcj+1, where ε1, . . . , εcjrepresent the cj presynaptic neurons of σrj, respectively, εcj+1represents the postsynaptic neuron of σrj and Eis a regular expression over {a}. For an AND-rule neuron σrj, its firing condition is E=acj. It means that the firing rule rican be applied if and only if it receives cjspikes from its presynaptic neurons at a same time. For the fault propagation path modeling, it means that if a ⊗-neuron receives cjspikes, then the rican be applied and will produce a new spike. If this new spike ais transmitted to a postsynaptic proposition neuron, then the new spike indicates that the event associated with the neuron σp(cj+1) (i.e., εcj+1) will happen with a probability ηj, i.e., the (cj+1)-th transmission line will fault with a probability ηj. If the new spike ais transmitted to a postsynaptic ⊕-neuron, then the new spike indicates that the event εcj+1can be easily effected by spikes transmitted from different synapses with different probabilities. For an OR-rule neuron σrj, the firing rule rican be applied if and only if it receives at least one spike from its presynaptic neurons. For the fault propagation path modeling, it means that if a ⊕-neuron receives at least one spike, then the rican be applied and will produce a new spike. This new spike a indicates that the event associated with the neuron σp(cj+1) (i.e., εcj+1) happens, i.e., the (cj+1)-th transmission line faults. It is noted that the neurotransmitter concentration ηj of an OR-rule neuron σrj is always equal to 1, which demonstrates that a fault must be propagated to other lines once the OR-rule neuron is active. (3)syn = {1,...,s+k} × {1,...,s+k}provides the arcs of a directed synapse graph such that (l,l)/∈syn, for 1 ≤l≤s+k, and if (s+j,s+j0)∈syn then (s+j0,s+j), /∈syn for any 1 ≤j,j0≤k, that is, there is no cycle only formed by two rule neurons. If (l,l0)∈syn then we say that neuron σlis a presynaptic neuron of σl0and we also say that neuron σl0is a postsynaptic neuron of σl; (4)I⊆Qpand O⊆Qpare the input (proposition) neuron set and the output (proposition) neuron set, respectively. In the Ev-SNP system, the proposition neurons characterize the propositions whose knowledge information is carried by the spikes in the associated proposition neurons. The rule neurons are employed to reason the proposition neuron information and then generate a new spike transferred to one or more postsynaptic neurons. Each rule neuron has associated a presynaptic neuron set (whose elements can be proposition neurons or rule neurons) and a postsynaptic neuron set (whose elements can be proposition neurons or rule neurons). At any instant, a proposition neuron contains one spike and a rule neuron can have two different states: active or inactive. Aconfiguration Ctof an Ev-SNP system at an instant t is a s+t-tuple, i.e., n1,t,...,ns,t,m1, . . . ., mk, describing the number of spikes ni,tassociated with proposition neuron σpi at that moment together with the state mj∈ {0,1} (active/inactive) of each rule neuron σrj at instant t. The initial configuration C0is given by the number of spikes initially associated with each proposition neuron encoding the input information, and all rule neurons are initially inactive. Let us consider an AND-rule neuron σrj with αpresynaptic neurons and βpresynaptic neurons. If the neuron σrj receives exactly αspikes from its presynaptic neurons at an instant t, and all presynaptic neurons are active at that moment t, then the state of σrj becomes active. If the state of σrj is active at the instant t, then it will produce one spike into each postsynaptic neuron at the instant t+1 and the state of σrj becomes inactive at that moment. Let us consider an OR-rule neuron σrj with αpresynaptic neurons and βpresynaptic neurons. If the neuron σrj receives at least one spike from its presynaptic neurons at an instant t, then the state of σrj becomes active. If the state of rule neuron σrj is active at the instant tthen it will produce one spike into each postsynaptic proposition neuron at the instant t+1 and the state of σrj becomes inactive at that moment. Given an Ev-SNP system 5, we denote C⇒5C0meaning that configuration Cyields to configuration C0in one transition step by applying the rules in the proposition neurons and the active rule neurons in C. A configuration is a halting configuration if no rule of the system is applicable and all rule neurons are inactive. A computation is a (finite or infinite) sequence of configurations such that: (1) the first term of the sequence is the initial configuration of the system; (2) each non-first term of the sequence is obtained from the previous configuration by applying spiking rules of the system in a maximally parallel manner with the restrictions previously mentioned; and (3) if the sequence is finite (called halting computation), then the last term of the sequence is a halting configuration. VOLUME 7, 2019 12801 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems All the computations start from an initial configuration and proceed as stated above; and only halting computations give a result, which is encoded by the spikes present in the output neurons from Oassociated with the halting configuration. III. ANALYSIS OF FAULT PROPAGATION PATHS USING THE SNP SYSTEMS A. GENERATION OF FAULT PROPAGATION PATHS In electric transmission networks, the trip of one or more lines will cause load redistribution in the entire network, which may lead to other lines overloaded, i.e. the flow over a line exceeds its capacity. Therefore, a fault propagation path can be described from the perspective of the overload mechanism as a set of overloaded lines that are tripped in turn due to the load redistribution. It should be noted that although the power flow over the network is complex power; however, the main transmitted power is the active part and the reactive is more related with a local problem. Therefore, in this paper, we use DC model to simplify the over load procedure. FIGURE 2. Distribution function of an element tripping probability. 1) CANDIDATE LINES we employ a probabilistic model to represent the tripping probability of a single element. The selected candidate lines, according to the tripping probability, are listed into the next contingency set (failure events), and all the lines inside the set will be tripped in the next step. The distribution function of the probabilistic model of a single element is shown in Figure 2. (1) If the power flow over the line iunder the x-th contingency Pi x−fo is less than or equal to its limit Pi M, then pPi x−fo ≤Pi M=0 (2) (2) If Pi x−fo is more than or equal to 1.2 times of Pi M, then pPi x−fo ≥1.2Pi M=1 (3) (3) If Pi x−fo is in between the aforementioned (1) and (2), then pPi M<Pi x−fo <1.2Pi M=5(Pi x−fo Pi M −1) (4) 2) POWER REGULATION when a network component fails, it may cause a power imbalance; therefore, some measures need to be taken to adjust the outputs of generators and loads to re-establish a new power balance of the system. In this paper, we adjust both generators’ outputs and loads. We use the minimal load curtailment as the objective (5) and employ the DC-OPF in every step to adjust the power injection and withdrawal at each node in the transmission network, if needed: fx=min1x(5) s.t.Px=Bxθx(6) Pmin h≤Phx ≤Pmax h,h=1,2,...,NG(7) where 1xrepresents the load shedding percentage in the contingency x;Pxis the net active power injection; Bxis the susceptance matrix; θxis the phase angle of bus voltages; Pmin hx and Pmax hx represent the lower and upper bound of the output of generator h, respectively; Phx represents the output of generator hduring contingency x;NGrepresents the total number of generators. Based on the above models, the generation of fault propagation paths can be described as algorithm 1. Algorithm 1 Simulation Process of Fault Propagation Paths Input: Electrical network information Output: Load shedding, propagation step, the number of tripped lines. Begin Step 1: Initialization: Choose initial fault lines based on Monte Carlo method. Step 2: WHILE Step 3: Power flow calculation: Calculate the power flow using DC power flow. Step 4: Overload line detection: Form a set of overloaded lines. If exists, go to Step 5; otherwise BREAK. Step 5: Contingency set generation: Apply equations 3 and 4 to the set in Step 4 to define the next contingency set. Step 6: Load shedding calculation: Calculate the minimum load shedding by DC-OPF. Step 7: Candidate line tripping: Cut off the candidate lines in the next contingency set. Step 8: END WHILE End B. THE SNP SYSTEM-BASED FAULT PROPAGATION 1) FAULT CHAIN We employ fault chain theory to describe the fault propagation paths. A fault propagation can be represented as a fault chain E Fy=L1→L2→ · · · → LMy(8) where Lx(x=1,2,...,My) represents the set of tripped lines in the contingency x. Therefore, the set of contingency 12802 VOLUME 7, 2019 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems events can be represented as F=nE F1,E F2,...,E FYo(9) 2) THE Ev-SNP SYSTEM-BASED FAULT CHAINS For a temporal fault relationship Lx→Lx+1(x=1,2,..., My−1) in E Fy, take Lxand Lx+1as the input proposition neuron set and output proposition neuron set, respectively. Denote the temporal fault relationship as rx. Definition 2: The knowledge representation of the aforementioned relationship can be represented as Lx+1=Lx⊗rx(10) Define the neurotransmitter concentration wxof the ⊗-neuron as wx=|Lx+1|/   E Fy  |Lx| |L1|(11) where |Lx|is the quantity of tripped lines in the contingency x.|L1|is the quantity of initial fault lines.    E Fy  is the length of the fault chain y. In equation (11), the less of both    E Fy  and |L1|demonstrates the fault chain can be trigger with higher possibility. Meanwhile, the less |Lx|with more |Lx+1|demonstrates fault propagation is more gravity from the contingency xto x+1. The neurotransmitter concentration wxcan reveal the gravity degree that the tripped lines in the contingency xpropagate faults to candidate lines in the contingency x+1. Therefore E Fyis described as LMy=L1⊗r1⊗r2· · · ⊗ rMy−1(12) Definition 3: If Lx0∈E F1∩E F2∩ · · · ∩ E FY0(Y0≤Y) and Lx0=Li xi⊗rx0(i=1,2,...,Y0), ⊕-neurons are employed to describe as Lx0=hL1 x1⊗r1 x1,L2 x2⊗r2 x2,· · · ,LY0 xY0⊗rY0 xY0i⊕rx0(13) According to the definitions 2 and 3, the Fcan be described as an Ev-SNP system consisted of proposition neurons and rule neurons. Definition 4: If there exist two knowledge representations Lx+1=Lx⊗rxin    E Fy  ,Lx0+1=Lx⊗rxin    E F0y  (E Fy6= E F0y), and L0=Lx+1−Lx0+1(Lx0+1⊂Lx+1), the two knowledge representations are merged as Lx+1∩Lx0+1=Lx⊗rx(14) In addition, we employ    E F0y  and  L01 to define the neurotransmitter concentration w0 xof the ⊗-neuron as w0 x= Lx0+1 /   E F0y  |Lx| L01 (15) where Lx∈E Fyand L0 1∈E F0y. Equation (15) is the same meaning with equation (11) . Definition 5 If there exist two knowledge representations Lx+1=Lx⊗rxin    E Fy  ,Lx+1=Lx0⊗rx0in    E F0y  (E Fy6= E F0y), and L0=Lx−Lx0(Lx0⊂Lx), the spikes of the L0cannot fire the ⊗-neuron; therefore the two knowledge representations are merged as Lx+1=Lx⊗rx. C. REDUCTION OF THE Ev-SNP SYSTEMS Although the Ev-SNP system can properly describe fault propagation paths, the dimension will become very large, leading to a difficulty in describing large-scaled power systems with increasing faults. To reduce the dimension of the Ev-SNP systems, a reduction-SNP system (RSNP system) is proposed by using single-input and single-output AND-rule neuron (⊗-neuron). Thus, equation 10 can be expressed as: Li=Lj⊗rij x(16) where Li∈Lx+1(i=1,2,...,|Lx+1|), Lj∈Lx(j= 1,2,...,|Lx|). The neurotransmitter concentration wxof the ⊗-neuron remains as equation 11. Definition 5: If there exist Li1=Lj1⊗ri1j1 x1in E F1,Li2= Lj2⊗ri2j2 x2in E F2,...,LiY0=LjY0⊗riY0jY0 xY0in E FY0(Y0≤Y, E F16= E F26= · · · 6= E Fy) and Li1=Li2= · · · = LiY0and Lj1=Lj2= · · · = LjY0, the knowledge representations are merged as Li=Lj⊗rij (17) where Li=Li1and Lj=Lj1. The neurotransmitter concentration wij of the ⊗-neuron is wij =wi1j1 x1+wi2j2 x2+ · · · + wiY0jY0 xY0(18) According to the above simplification, the RSNP system has advantageous characteristics as follows. (1) The RSNP system avoids employing the OR-rule neuron (⊕-neurons) and its dimension can effectively be reduced, which enhances the ability of visual modeling and reduces neurons of the system. (2) By representing the Ev-SNP system by single-input and single-output rule neurons, the fault propagation relationship between single lines in an RSNP system can be highlighted, which helps to reveal the fault propagation mechanism of power systems. (3) Through the simplified yet neat graph generated by the RSNP system, the modeling process can be more straightforward. IV. CASE STUDY In this section, the proposed model is applied to the IEEE 14-bus and 118-bus systems. The computational work is conducted in MATLAB, running on a laptop. The laptop (Compaq, v3646TU) is with Intel R CoreTM 2 Duo CPU T7250 @ 2.00GHz CPU, 2.00G RAM, and 64bit windows 7 operating system. We employ the Monte Carlo method to generate the set of initial contingent events. For each fault propagation path, no more than three lines are randomly selected as the initial triggering lines. A. IEEE 14-BUS SYSTEM We first employ the Ev-SNP system Q1to model the fault propagation paths of 20 chains to show modeling mechanism of the proposed method, as shown in Figure 3. It is noted VOLUME 7, 2019 12803 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems FIGURE 3. Ev-SNP system of 20 fault propagation chains. that Figure 3 is used to simply show how the Ev-SNP system would look like in detail with a lower number of simulation, i.e. Figure 3 is not a ‘‘converged’’ graph with stable propagation paths. Y=(A,Q,syn,I,O) where (1)A= {a}is a singleton alphabet (a is called spike and represent a fault); (2)Q={Q1,Q2}is a neuron set, where proposition neuron set is Q1={σ1, . . . , σ13}=σp1, . . . , σp13and rule neuron set is Q2={σ14, . . . , σ30}={σr1, . . . , σr17}, where σ14, . . . , σ25 are ⊗-neurons and σ26, . . . , σ30 are ⊕- neuron; i.e. s=13, t=17, m=30; (3)syn={(1,23), (2,20), (3,19), (4,21), (4,23), (5,20), (5,21), (5,25), (6,16), (6,18), (6,19), (7,15), (8,14), (8,23), (8,24), (9,25), (10,19), (10,20), (10,21), (11,17), (11,22), (11,24), (14,7), (15,26), (16,26), (17,29), (18,28), (19,30), (20,30), (21,29), (22,27), (22,28), (23,27), (24,28), (25,12), (25,13), (26,8), (26,130), (27,5), (28,9), (29,6), (30,11)}; (4)I=σp1, σp2, σp3, σp10,O=σp12, σp13. It is manifest that the Ev-SNP system can reveal the fault propagation paths and the ⊗-neuron can indicate the condition of triggering line faults in the next contingency. For instance, when lines 5 and 9 (i.e., proposition neurons σ5and σ9) are tripped, they emit a spike individually to the ⊗-neuron σ25, leading to the triggering of lines 12 and 13 (i.e., proposition neurons σ12 and σ13). Therefore the temporal fault propagation relationship among lines can be simply traced by the Ev-SNP system. In addition, the neurotransmitter concentrations of ⊗-neurons can reflect the possibility of activating ⊗-neurons. The propagation path σ7→σ15 → σ26 →σ8can be triggered more easily and seriously than others due to the higher neurotransmitter concentration in ⊗-neuron σ15. Meanwhile, line 11 (i.e., proposition neuron σ11) is more easily affected by a fault because it can receive spikes from more ⊗- or ⊕- neurons than others. In addition, it is noted that when a ⊕- neuron has multiple presynaptic ⊗-neurons, it demonstrates that the event associated with the postsynaptic neurons of the ⊕- neuron can be effected by the faults from different propagated paths. For example, the line 6 (i.e., proposition neuron σ6) has a presynaptic ⊕-neuron (i.e., rule neuron σ16), which demonstrates the line 6 can be affected by a propagated fault from two different combinations: a) line 11 (i.e., proposition neuron σ11) and b) lines 4, 5 and 10 (i.e., proposition neuron σ4, σ5, σ10). Although the Ev-SNP system can reveal the fault propagation paths, the dimension of the Ev-SNP system becomes very high with the increasing of simulated events (i.e. fault chains); therefore, the modeling of the Ev-SNP system is very difficult for large-scaled networks. On the other hand, if we reduce the number of triggering contingent events, the model cannot comprehensively investigate fault propagation paths and reveal fault propagation relationships among lines. Therefore, we further proposed the RSNP system to analyze the fault propagation paths. FIGURE 4. RSNP system with 1000 cascading chains. Figures 4 and 5 give the results of the RSNP system with 1000 and 2000 fault chains, respectively, with ⊗- neurons omitted for the sake of space. Based on the neurotransmitter concentration wof ⊗- neurons, the fault propagation relationships among lines can be divided into four levels: very high risk (w≥0.5), high risk (0.3≤w<0.5), moderate risk (0.1≤w<0.3) and low risk (w<0.1). The fault propagation relationships for the very high risk, high risk and moderate risk, except 4-5 and 4-10, are the same with 1000 and 2000 fault chains. Therefore, we can assert that when the simulation reaches 1000 cascading fault chains, the main fault propagation paths are credibly discovered. In addition, the fault propagation paths 1-2 and 1-7 are very high risk, which indicates lines 2 and 7 will fail with high probability once line 1 fails; therefore line 1 is very high vulnerable line that needs prioritized protection. Similarly, line 4 12804 VOLUME 7, 2019 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems FIGURE 5. RSNP system with 2000 fault chains. TABLE 1. Vulnerable lines and fault propagation path in 2000 contingency events. is also a vulnerable line due to the propagation relationships among line 4 and other lines. Line 3 is a moderate vulnerable line, compared with lines 1 and 4. Although its risk level is not very high, it appears in many moderate risk propagation path, which makes it difficult to forecast the propagation in the next contingency; therefore line 3 should also be closely monitored during the normal operation. In summary, the results are shown in Table 1. B. IEEE 118-BUS SYSTEM To demonstrate the ability of the RSNP system on a large system, we employ the IEEE 118-bus system to simulate 2000 fault propagation paths, as shown in Figure 6. For the sake of clarity, only the paths over high risk are given. There are mainly two groups of paths: small group with line 71 as the origin of faults and a large group with lines 3, 8, 36, 54, 96, 104, 107 and 108 as the sources of faults. In the small group, lines 70, 75, 76 and 81 fail with high probability once line 71 trips. In the large group, lines 36, 54 and 96 can only spread fault but are not easily affected by a propagated fault. By contrast, lines 3, 8, 104, 107 and 108 can spread faults as well as be affected by a fault. In addition, there are two main propagation paths in the large group: 36-3 and 36-8-(104, 107, 108). Therefore, strengthening the protection of the lines in the main propagation paths can effectively reduce or even avoid the fault propagation paths. FIGURE 6. Fault propagation relationship over moderate risk with 2000 fault chains. In summary, the Ev-SNP system and its simplified one (i.e., RSNP system) can reveal the fault propagation mechanism and fault temporal relationships among lines efficiently and intuitively due to the advantages of the SNP in terms of graphical modeling and parallel knowledge representations and logic reasoning. V. CONCLUSIONS The fault propagation path identification is fundamental to the power system analysis and secure operation of it as it forms the basis of all following countermeasures. To make the SNP system suitable for modeling fault propagation paths of power systems, this paper proposes an Ev-SNP system with neurotransmitter concentration representing the firing possibility of rule neurons. The Ev-SNP systems can reveal the fault propagation mechanism of fault propagation paths and temporal relationship among lines. In addition, the neurotransmitter concentrations of ⊗-neurons are introduced to reflect the possibility and gravity degree of fault propagation among lines. Moreover, the RSNP system is proposed because the original Ev-SNP systems is only suitable for the set of less contingency events. The proposed RSNP system can intuitively and effectively identify fault propagation paths and vulnerable lines. In this paper, we extended the membrane computing application in power systems from a static single local element fault diagnosis to the dynamic relationship among multiple elements in a system wide scale. The proposed method has been successfully applied the SNP system to a large power system application in terms of investigating the fault propagation paths through a reasonable abstraction of the protection scheme of the element in the power systems. The proposed method has the following advantages over the previous tools: 1) modeling the problem in such a holistic and systematic way that the complexity and non-linear feature can be captured, i.e. the self-organizational and butterfly effects can be naturally considered in the model; 2) modeling VOLUME 7, 2019 12805 T. Wang et al.: Modeling Fault Propagation Paths in Power Systems the fault propagation through a visualization fashion and an analogy between the fault propagation in the electricity grid and the spike transmission in the neuron networks. This analogy allows us to use the synaptic connections among neurons to describe the logical and temporal relationship among power equipment, thus the fault propagation can be modeled as the natural response of a biological system. So, the proposed method has an intuitive illustration based on a strictly mathematical expression, a good description for the adjacent fault relationships between branches, and an understandable graphical model-building process. In the future, our work is further to develop the software of SNP systems to analyze the fault propagation characteristics online of real power networks. In addition, SNP systems can be improved to model fault propagation paths of large complex power systems, such as smart grids with distributed generations, microgrids, etc. REFERENCES [1] G. Páun, ‘‘Computing with membranes,’’ J. Comput., Syst. Sci., vol. 61, no. 1, pp. 108–143, 2000. [Online]. Available: http://www.tucs.fi [2] G. Páun and M. J. Pérez-Jimínez, ‘‘Membrane computing: Brief introduction, recent results and applications,’’ Biosystems, vol. 85, no. 1, pp. 11–22, Jul. 2006. [3] T. Wang, G. X. Zhang, and M. J. Pérez-Jimínez, ‘‘Fuzzy membrane computing: Theory and applications,’’ Int. J. Comput. Commun, vol. 10, no. 6, pp. 861–892, Oct. 2015. [4] G. Páun, G. Rozenberg, and A. Salomaa, The Oxford Handbook of Membrance Computing. New York, NY, USA: Oxford Univ. Press, 2010. [5] G. Zhang, M. Gheorghe, L. Pan, and M. J. Pérez-Jimínez, ‘‘Evolutionary membrane computing: A comprehensive survey and new results,’’ Inf. Sci., vol. 279, pp. 528–551, Sep. 2014. [6] C. Martín-Vide, G. Păun, J. Pazos, and A. Rodríguez-Patón, ‘‘Tissue P systems,’’ Theor. Comput. Sci., vol. 296, no. 2, pp. 295–326, 2003. [7] B. Song, C. Zhang, and L. Pan, ‘‘Tissue-like P systems with evolutional symport/antiport rules,’’ Inf. Sci., vol. 378, pp. 177–193, Feb. 2017. [8] H. Peng, J. Wang, P. Shi, M. J. Pérez-Jimínez, and A. Riscos-Núñez, ‘‘Fault diagnosis of power systems using fuzzy tissue-like P systems,’’ Integr. Comput.-Aided Eng., vol. 24, pp. 401–411, Jan. 2017. [9] T. Wang et al., ‘‘Application of neural-like P systems with state values for power coordination of photovoltaic/battery microgrids,’’ IEEE Access, vol. 6, pp. 46630–46642, 2018. [10] T. Wu, A. Pˇ aun, Z. Zhang, and L. Pan, ‘‘Spiking neural P systems with polarizations,’’ IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 8, pp. 3349–3360, Aug. 2018. [11] L. Pan, G. Paun, G. Zhang, and F. Neri, ‘‘Spiking neural Psystems with communication on request,’’ Int. J. Neural Syst., vol. 27, no. 8, p. 1750042, 2017. [12] D. Díaz-Pernil and M. A. Gutiérrez-Naranjo, ‘‘Semantics of deductive databases with spiking neural P systems,’’ Neurocomputing, vol. 272, pp. 365–373, Jan. 2018. [13] H. Peng et al., ‘‘Competitive spiking neural P systems with rules on synapses,’’ IEEE Trans. NanoBiosci., vol. 16, no. 8, pp. 888–895, Dec. 2017. [14] T. Pan, X. Shi, Z. Zhang, and F. Xu, ‘‘A small universal spiking neural P system with communication on request,’’ Neurocomputing, vol. 275, pp. 1622–1628, Jan. 2018. [15] G. Zhang, H. Rong, F. Neri, and M. J. Pérez-Jiménez, ‘‘An optimization spiking neural P system for approximately solving combinatorial optimization problems,’’ Int. J. Neural Syst., vol. 24, no. 5, p. 1440006, 2014. [16] D. Díaz-Pernil, F. P. Cantillana, and M. A. Gutiérrez-Naranjo, ‘‘A parallel algorithm for skeletonizing images by using spiking neural P systems,’’ Neurocomputing, vol. 115, pp. 81–91, Sep. 2013. [17] R. K. Idowu, R. C. Muniyandi, and Z. A. Othman, ‘‘The prospects of using spiking neural P system for intrusion detection,’’ Int. J. Inf. Netw. Secur., vol. 2, no. 6, pp. 492–498, 2013. [18] J. Wang, P. Shi, H. Peng, M. J. Pérez-Jiménez, and T. Wang, ‘‘Weighted fuzzy spiking neural P systems,’’ IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 209–220, Apr. 2013. [19] J. Wang and H. Peng, ‘‘Adaptive fuzzy spiking neural P systems for fuzzy inference and learning,’’ Int. J. Comput. Math., vol. 90, no. 4, pp. 857–868, Apr. 2013. [20] T. Wang, J. Wang, H. Peng, and H. Wang, ‘‘Knowledge representation and reasoning based on FRSN P systems,’’ in Proc. WCICA, Beijing, China, 2011, pp. 849–854. [21] T. Wang, J. Wang, H. Peng, and Y. Deng, ‘‘Knowledge representation using fuzzy spiking neural P system,’’ in Proc. BIC-TA, Beijing, China, 2010, pp. 586–590. [22] T. Wang, S. Zeng, G. Zhang, M. J. Pérez-Jiménez, and J. Wang, ‘‘Fault section estimation of power systems with optimization spiking neural P systems,’’ Sci. Technol., vol. 18, no. 3, pp. 240–255, 2015. [23] T. Wang, G. Zhang, and M. J. Pérez-Jiménez, ‘‘Fault diagnosis models for electric locomotive systems based on fuzzy reasoning spiking neural P systems,’’ in Membrane Computing (Lecture Notes in Computer Science), vol. 8961. Cham, Switzerland: Springer, 2014, pp. 361–374. [24] T. Wang, G. Zhang, M. J. Pérez-Jiménez, and J. Cheng, ‘‘Weighted fuzzy reasoning spiking neural P systems: Application to fault diagnosis in traction power supply systems of high-speed railways,’’ J. Comput. Theor. Nanos., vol. 12, no. 7, pp. 1103–1114, Jul. 2015. [25] Y. Y. He, T. Wang, K. Huang, G. Zhang, and M. J. Pérez-Jiménez, ‘‘Fault diagnosis of metro traction power systems using a modified fuzzy reasoning spiking neural P system,’’ Romanian J. Inf. Sci. Technol., vol. 18, no. 3, pp. 256–272, Jan. 2015. [26] G. Xiong, D. Shi, L. Zhu, and X. Duan, ‘‘A new approach to fault diagnosis of power systems using fuzzy reasoning spiking neural P systems,’’ Math. Problems Eng., vol. 2013, May 2013, Art. no. 815352. [27] T. Wang, G. Zhang, J. Zhao, Z. He, J. Wang, and M. J. Pérez-Jiménez, ‘‘Fault diagnosis of electric power systems based on fuzzy reasoning spiking neural P systems,’’ IEEE Trans. Power Syst., vol. 30, no. 3, pp. 1182–1194, May 2015. [28] H. Peng et al., ‘‘Fault diagnosis of power systems using intuitionistic fuzzy spiking neural P systems,’’ IEEE Trans. Smart Grid, vol. 9, no. 5, pp. 4777–4784, Sep. 2018. [29] X. Xie, D. Yue, H. Zhang, and Y. Xue, ‘‘Fault estimation observer design for discrete-time Takagi–Sugeno fuzzy systems based on homogenous polynomially parameter-dependent Lyapunov functions,’’ IEEE Trans. Cybern., vol. 47, no. 9, pp. 2504–2513, Sep. 2017. [30] J. Yan, Y. Tang, H. He, and Y. Sun, ‘‘Cascading failure analysis with DC power flow model and transient stability analysis,’’ IEEE Trans. Power Syst., vol. 30, no. 1, pp. 285–297, Jan. 2015. [31] H. T. Ma and B. H. Chowdhury, ‘‘Dynamic simulations of cascading failures,’’ in Proc 38th North Amer. Power Symp. (NAPS), Carbondale, IL, USA, Sep. 2006, pp. 619–623. [32] S. Poudel, Z. Ni, T. M. Hansen, and R. Tonkoski, ‘‘Cascading failures and transient stability experiment analysis in power grid security,’’ in Proc. IEEE Power Energy Soc. Innov. Smart Grid Technol. Conf. (ISGT), Minneapolis, MN, USA, Sep. 2016, pp. 1–5. [33] C. Luo, J. Yang, Y. Sun, J. Yan, H. He, and M. Liu, ‘‘A cascading failure simulation model considering frequency dynamics and power flow distribution,’’ in Proc. North Amer. Power Symp. (NAPS), Charlotte, NC, USA, Oct. 2015, pp. 1–6. [34] S. Mei, Y. Ni, G. Wang, and S. Wu, ‘‘A study of self-organized criticality of power system under cascading failures based on AC-OPF with voltage stability margin,’’ IEEE Trans. Power Syst., vol. 23, no. 4, pp. 1719–1726, Nov. 2008. [35] B. A. Carreras, V. E. Lynch, I. Dobson, and D. E. Newman, ‘‘Complex dynamics of blackouts in power transmission systems,’’ Chaos, Interdiscipl. J. Nonlinear Sci., vol. 14, no. 3, pp. 643–652, Sep. 2004. [36] R. Kinney, P. Crucitti, R. Albert, and V. Latora, ‘‘Modeling cascading failures in the North American power grid,’’ Eur. Phys. J. B-Condensed Matter Complex Syst., vol. 46, no. 1, pp. 101–107, 2005. [37] J. Bialek et al., ‘‘Benchmarking and validation of cascading failure analysis tools,’’ IEEE Trans. Power Syst., vol. 31, no. 6, pp. 4887–4900, Nov. 2016. [38] M. Rahnamay-Naeini, Z. Wang, N. Ghani, A. Mammoli, and M. M. Hayat, ‘‘Stochastic analysis of cascading-failure dynamics in power grids,’’ IEEE Trans. Power Syst., vol. 29, no. 4, pp. 1767–1779, Jul. 2014. 12806 VOLUME 7, 2019