Resilience based importance measure analysis for SoS
PAN Xing1,*,WANG Huixiong1,YANG Yanjing1,2,and ZHANG Guozhong3
1.School of Reliability and Systems Engineering,Beihang University,Beijing 100191,China;2.China Railway Signal and Communication Corporation,Beijing 100070,China;3.Institute of Systems Engineering,China State Shipbuilding Corporation,Beijing 100036,China
Abstract: In a system of systems(SoS),resilience is an important factor in maintaining the functionality,stability,and enhancing the operation effectiveness.From the perspective of resilience,this paper studies the importance of the SoS,and a resilience-based importance measure analysis is conducted to provide suggestions in the design and optimization of the structure of the SoS.In this paper,the components of the SoS are simplified as four kinds of network nodes:sensor,decision point,influencer,and target.In this networked SoS,the number of operation loops is used as the performance indicator,and an approximate algorithm,which is based on eigenvalue of the adjacency matrix,is proposed to calculate the number of operation loops.In order to understand the performance change of the SoS during the attack and defense process in the operations,an integral resilience model is proposed to depict the resilience of the SoS.From different perspectives of enhancing the resilience,different measures,parameters and the corresponding algorithms for the resilience importance of components are proposed.Finally,a case study on an SoS is conducted to verify the validity of the network modelling and the resiliencebased importance analysis method.
Keywords: system of systems(SoS),resilience,network modelling,importance measure analysis,operation loop.
1.Introduction
In recent years,resilience has aroused interest of more and more researchers from different domains,especially the complex systems[1]and infrastructures[2].The term‘resilience’originates from Greek‘resilere’,which means‘bounce back’[3].In 1970s,Holling[4]first defined‘resilience’in the study of ecology as the ability of an ecosystem to recover to the original balanced status after environmental change or human activities.Since then,the term‘resilience’is used to depict the ability that a system or entity can restore its performance or function after a disruptive event,and this definition is already used in different subjects,e.g.,ecology[4,5],sociology[6,7],economics[8,9],and engineering[10,11].
In engineering discipline,the resilience of a system is defined as the ability that a system can resist collapse and recover its performance characteristic after a disruptive event,e.g.,system malfunction,being attacked[12].The resilience of a system is strongly related to reliability(the ability to perform the intended function),robustness(the ability to resist malfunctions or external attacks)and recoverability(the ability to recover from malfunctions or external attacks),and thus affects the performance of the system.In system of systems(SoS)theory,resilience is the ability that an SoS restores its weakened performance with its characteristics and structure.For an SoS consisting of several systems,resilience is an important guarantee that a system can remain stable and perform its expected functions.Among the member systems of the SoS,each component has different degrees of importance,which requires a resilience importance measure(RIM)for the SoS,and hence provides a direction for the design and optimization of SoS.
In the research on design and optimization of SoS,the abstraction of the SoS into a network is commonly used in SoS modelling.The last decades have witnessed intensive studies on the network components and the connection among them,especially in the research of network centric warfare(NCW).In NCW,the evolution of an SoS is simplified as the process of decision making and execution.Specifically,the entities of ally are divided into three categories:sensors,decision points,and influencers.Meanwhile,considering the antagonism in systematic operation,the entities of the opponent are abstracted into targets[13,14].As a matter of fact,the above categorization is constructed on the basis that the operation of an SoS can be seen as a process of decision and action.Therefore,in other domains,e.g.,critical infrastructure,transportation system,the components are also abstracted into the above four categories.In this study,we follow the above categorization,and use the above four categories of components in the modelling and research on resilience and importance measure analysis.
When analyzing the resilience importance of an SoS,it is a basic method to build a resilience model that analyzes the disruptive events and the influence.As a matter of fact,all systems will encounter different kinds of interference in life circle,thus preventive or resistant strategies are used to guarantee the operational capacity and resistibility against the interference,e.g.,intentional attacks,natural disasters,human accidents,and malfunctions.Previous studies have offered insight into these strategies in different types of SoS,e.g.,protecting critical system elements[15],operating in degraded mode[16],providing redundancy[17],deploying false targets[18],and launching preventive strikes[19].The abovementioned strategies improve the survivability of the SoS by different approaches,while the mechanisms of these strategies can be intelligibly subdivided into three categories:predicting the adverse event,deferring its occurrence,and improving the reliability of the SoS[20].Take the false target deployment as an example,when an approaching threat is detected(predicted),the system will deploy a false target so that the adverse event is excluded(deferred).However,it is sometimes costly and unrealistic to prevent or evade the adverse event;therefore,we hope that the SoS can quickly restore its performance from the disturbing event.Resilience,therein,is a measure to depict such ability of an SoS.Many devastating events have proved the significance of resilience in an SoS,because it is the key factor to diminish the impact brought by the adverse events[21,22].Generally,there are three characteristics that build up the resilience of the SoS:independence of component functions,redundancy of intersystematic functions,and self-reconstructive or evolutional functions[23].Among the above three characteristics,resilience is based on the redundancy of inter-systematic functions,while resilience is accomplished by the selfreconstructive and evolutional functions.In an SoS,when a subsystem failure causes malfunctions,the performance of the SoS can be recovered by recombining the other components.In a word,there are many factors that enable the SoS to resist and restore from adverse events and performance decrease,including redundancy,and reliable or effective recovery measures.This manuscript will focus on the disruptive events,divide the process of performance change into different phases,and study the factors that affect the resilience of SoS.
Analyzing the resilience importance of an SoS is one aspect of importance analysis.Another important issue in the reliability and risk analysis of complex system is the recognition of uncertainty.When analyzing the uncertainty,the measure of importance is to recognize the uncertain component parameters that pose the biggest impact on the overall performance with the help of sensitivity analysis[24].Importance measurement is an efficient tool to recognize the important input parameters and regulate the uncertainty of system output.Thus,analysts can depict the most influential or critical risk context by importance measure analysis,and,therefore,optimize the design of system and improve the logistic strategy.
In order to determine the importance of system components,previous studies proposed a variety of indexes measuring components importance.The most commonly used indices are Birnbaum measure,failure critical index(FCI),the Fussell-Vesely measure,risk achievement worth(RAW),risk reduction worth(RRW),etc.[25].In addition,there are derivatives of parameters in system risk,such as the likelihood ratio gradient[26].Moreover,research on the uncertain importance measure(UIM)method of parameters combined with the probabilistic risk assessment(PRA)method in reliability models is a priority in the field of reliability and safety analysis[27-30].Meanwhile,other UIM indices are researched from different prospectives in reliability,such as time-independent[31,32]or cost-based importance measure[33,34],and logistic support process[35].
The literature above indicates that it is important to conduct importance measure analysis on the system nodes and analyze the design and optimization of the system from the perspective of resilience[36].Previous studies on importance measure analysis of system or SoS focused on the influence of the accession of a new SoS member,while neglecting the influence of resilience.Focusing on the improvement of resilience,this article proposes a modeling method of SoS network,in which the observe-orientdecide-act(OODA)loop is used to quantify the performance of the system,and the trend of performance change is studied to build the resilience model.Furthermore,an analytical method is proposed to measure the resilience and provide different resilience-based importance indices.
2.OODA-based resilience model
2.1 Two types of resilience model
There are two different types of SoS resilience: timeirrelevant model(i.e.,quotient resilience model),and timerelevant model(i.e.,integral model)[37].Fig.1 illustrates the trends of SoS performance under the two resilience models,whereφ (t )represents the SoS performance at timet .
首先，随着我国经济的快速发展，能源的消耗量在持续增加，对于能源的供应提出了更高的要求，能源危机是我国经济发展中必须要面临的潜在危险。太阳能不仅是一种新型的清洁能源，而且其获取方便，随着太阳能发电技术的不断成熟，在应用中越来越广泛而普遍，是未来新能源发展的重要方向。车载光伏远程控制空调可以减少传统的化石能源的消耗，而且我国的汽车数量在逐年增长，数量庞大，太阳能在其次制冷中的应用对于节能减排有很大的帮助。
(i)Quotient resilience model
In the resilience process,the occurrence of a disruptive eventej will decrease the performance of SoS,while recovery measure can restore the performance.In the quotient resilience model,as shown in Fig.1(a),the resilience of the SoS is defined as the ratio of the restored performance to the lost performance.Equation(1)provides a concise expression of SoS resilience:
where
represent the eigenvalues of matrix
Fig.1 Trends of SoS performance in the disturbing event under two resilience models
Equation(1)exhibits the ability that an SoS can bounce back to the normal performance after a disruptive event.If the SoS can restore its performance completely,i.e.,φ restored(t ) =φ lost(td ), we say the SoS has complete resilience;if the SoS does not restore at all,i.e.,Recovery(t )=0,we say the SoS has no resilience.In this model,resilience is defined as the ratio of restored performance to the lost,thus it is called the quotient resilience model.
In the quotient resilience model,when performance indicatorφ (t ),time of disruptive eventej ,start time of recoveryts ,and end time of recoverytf are determined,the SoS resilience,equivalent toR (t )in(1),can be represented as
whereRQ (t|ej )represents the quotient resilience of the SoS at timet after the disruptive eventej ,φ (t|ej )represents the SoS performance at timet ,andφ (td|ej )represents the SoS performance after the disruptive eventej .
从表8和表9可以看出:(1) 黄铁矿和脉石矿物的比热容存在一定差异，黄铁矿比热容小，脉石(SiO2)比热容大；(2) 黄铁矿和脉石的热膨胀系数存在明显差异，但均随着温度升高而增大。当微波选择性加热黄铁矿时，其与脉石之间的温度差异大，黄铁矿的温度高，其所产生的热膨胀大，而脉石矿物不吸收微波，其脉石的温度低，所产生的热膨胀小，因此，黄铁矿和脉石矿物之间由于热膨胀差异显著容易产生应力集中，并对矿石矿物产生破坏，形成微裂隙降低整体矿石矿物强度。
The denominator in(2)represents the reduced performance of the SoS,and less reduced performance means better SoS resilience;the numerator represents the performance recovered by the recovery measure.Therefore,the denominator and numerator respectively represent the influence of the disruptive event and the recovery measure.
(ii)Integral resilience model
本项目在现有激光焊接装配技术的基础上开发了新型多功能激光焊接试验平台，用于激光焊接设备日常焊接工艺试验。焊接试验平台在满足现车手动压紧和气缸自动压紧功能的基础上完成电磁吸附压紧功能开发，并进行激光焊接试验，分析了电磁场对激光焊缝的影响，研究电磁吸附式工装用于奥氏体不锈钢车体激光焊接的可行性。
The quotient resilience model can only depict the resilience of the SoS at a certain time,while neglecting the overall effect of SoS resilience during the whole attackrecovery process.Integral resilience model,however,measures the SoS resilience in the process by calculating the accumulated effect of performance over time.Fig.1(b)illustrates the triangular integral resilience model,which evolves from Fig.1(a).The resilience triangle,which is the dash area in Fig.1(b),illustrates the performance loss after the disruptive event.In the integral model,the performance lost in the resilience process can be mathematically represented as
whereRL represents the performance lost in the resilience process,andQ (t )=100φ (t )/φ (t 0)represents the quality of SoS performance.When there is no adverse event,Q (t )is assigned to be 100.
In the integral model,a smaller resilience triangle indicates better SoS resilience.To enhance the resilience,the resilience triangle,or the value ofRL should be small.For example,increasing the performanceand reducing the time cost in the recovery will result in higher resilience.
In the integral resilience model,the retention amount of performance,instead of the lost amount of performance,is used to depict the SoS resilience.This can be mathematically represented as
whereRI (t|ej )represents the integral resilience at timet after the disruptive eventej ,φ (t 0)represents the expected performance with no disruptive event,andφ (u|ej )is the SoS performance at timet .
The numerator in(4)is the integral of SoS performance to time during the disruptive event,while the denominator represents the integral of expected SoS performance with no disruption.The area under the performance curve indicates the accumulated effect of SoS performance over the time.
2.2 Performance measurement of SoS
2.2.1 OODA loop of SoS
According to the above resilience model,a performance indicatorφ (t )must be defined in the analysis of SoS resilience.Previous studies on the network-based SoS model also defines various measures of network performance.Network efficiency and largest connected component(LCC)are two common performance measurements in the study of complex networks.To some extent,these traditional measurements reflect the performance of the SoS,while they are limited to some certain aspects of the performance[38].For example,network efficiency focuses more on the ability of the network to transmit information,while LCC neglects the functionality of other connected groups of nodes.In NCW,the operation loop provides a description for the operational process of a weapon SoS by extracting the decision and operation loop in the operational process.More operation loops indicate more alternatives in decision-making,thus will provide higher fighting capacity.Therefore,the number of the operation loop is a comprehensive and popular indicator to measure the performance of an operational SoS.
whereRhNni represents the overall resilience when the new nodeni does not fail;Rh represents the normal situation.Combining(11)and(14),we obtain
As a description of decision and action,the operation loop is also significant in complex network:nodes in a network can be classified as four types:sensor,decision point,influencer and target.When these four types of nodes constitute a loop,the corresponding decision and action are finished.Further,the number of OODA loops illustrates the operational efficiency of the network.Therefore,in this study,we take the number of OODA loops as the performance indicatorφ (t )in the resilience analysis.
2.2.2 OODA loop approximate algorithm
The above assumptions and settings are extracted from an imitative operational context and might not accord with the real operational situation,but they are applicable and feasible in the study of resilience.Substituting matrixA into(9)at any time during the resilience process,we can calculate the number of operation loops and plot the trend of the SoS performance,as shown in Fig.7.
通过例4进一步推导可知，若极限式中有幂指函数地f(x)g(x)，常用换底公式eg(x)lnf(x)将其化为指数函数进行处理。
An SoS can be simplified as a heterogeneous network consisting of specific components and connection,including sensors,decision points,influencers,and targets.Assume an operational network withN entities,the topology,i.e.,connections between entities,can be explicitly represented as an adjacency matrixA =[aij ]N×N ,where
李：近年来，高校虽然进行了一些管理体制的改革，并取得了一定成效；但仍存在高校行政权力和学术权力失衡等问题。集中表现为：学术权力没有得到应有的重视和有效利用，学术权力与行政权力的界限模糊，二者常常交织在一起，造成分工不明，责任不清。集中表现为“三化”，即“学术权力行政化”“行政权力学术化”“大学管理‘机关化’”。高校行政权力和学术权力之间的冲突凸显，强行政、弱学术，这些也是引起学术腐败和教育腐败的重要原因。
We use a simplified SoS example to illustrate the calculation of operation loops.The SoS consists of decision nodeD 1,sensor nodesS 1andS 2,influencer nodesI 1andI 2,and target nodeT 1.This operational network can be represented as matrixA in Fig.2.The number of operation loops can be estimated with matrixA .
Fig.2 Simple example of SoS and its corresponding adjacency matrix
In Fig.2,it is evident that the number of loops that pass throughD 1 with the length of 4 is 4.While in largescale SoS networks,the number of operation loops is large.Since the operation loop is a special case of the closed loop,it is assumed that a large number of closed loops indicates a large number of operation loops.Therefore,the number of closed loops can be used as an estimation to the number of operation loops.
Let
represent the number of closed loops that pass through nodevi with the length ofk ,i.e.,the number of closed paths that start atvi and end atvi withk steps.In matrixAk ,the matrix multiplication ofA s,each element
equals the number of paths from nodei to nodej withk steps,therefore the elements on the primary diagonal in matrixAk equals the closed loops that pass throughvi with the length ofk ,i.e.,
For the network in Fig.2,we obtain
The number of operation loops with the length ofk can be calculated[41]with
whereR (t ) represents the resilience of the SoS,φ restored(t )represents the restored performance at timet ,i.e.,the difference between performance at timet under recovery strategies and the minimum performance under the disruptive eventej ,andφ lost(t )represents the maximum performance loss,i.e.,difference between original performance beforeej and minimum performance underej .
Therefore,the number of all operation loops in the SoS can be calculated by summarizingSk
To estimate the number of operation loops,we adopt the approximate algorithm proposed by Tan et al.[42],in which a weighted sumck =1/k !is used to eliminate the recalculation of loops with multiplied lengths as well as the influence of the length of the loops.Mathematically,
For a large-scale network,the value ofS′ is very large.Therefore,we rescaleS′ [42]as
The above equation provides an approximate method with simple parameters in the matrix,while depicting an important aspect of the performance of the SoS network.Equation(9)is not the exact number of operation loops,though,the weighted sum of closed loops considers the structural properties of the SoS,and hence is more preferable in computational efficiency.Therefore,in this study,
is used as a comprehensive indicator to measure the performance of an operational SoS.
2.3 Resilience using OODA loop
In this study,the integral resilience model is used to measure the resilience of the SoS.Considering the number of operation loops during the degeneration and recover process,the resilience of the SoS under a certain disruptive eventej can be represented as
whereR ((t 0,t )|ej )represents the integral resilience of the SoS at timet after the disruptive eventej .
represents the number of operation loops with no disruptive event.
represents the number of operation loops at timet during the mission.
Integral resilience measures the accumulated performance of the time,and the ability of the SoS to perform the expected functions in a time period.In order to measure the accumulative effect of the SoS performance,the resilience at the end of the mission is measured to depict the resilience of the SoS.
Therefore,on the basis of(10),we propose a definition of overall resilienceRh :the ability of an SoS to maintain its expected functionality after a disruptive eventej .It can be measured by integral resilience at time interval[t 0,tm ]:
where
represents the number of operation loops after the disruptive eventej ,and
represents the number of operation loops at timet under the mission circumstance.
综上所述，小学教师在开展语文教学的过程中，将生本理念巧妙的应用到阅读教学中是非常必要的。在此模式下，不仅可以让学生对文章有一个更加深刻的理解，使其品味人生百态、产生思想共鸣，也能使学生逐渐的发觉阅读的魅力所在，使其喜欢阅读、爱上阅读，为后续学习的开展打下有利基础。虽然，在当前，部分教师在教学实践中还并未认知到这一点，但我确信只要在大家的一致努力下，一定能改进这些不足，为生本教育的实现助力。
3.Importance measure analysis
As introduced above,the mechanisms of strategies to improve the survivability of the system can be divided into three categories:predicting the adverse event,deferring its occurrence,and improving the reliability of the SoS.The three categories actually represent the strategies conducted in three phases:before,in,and after the disruptive event.Accordingly,in analysis of importance of the SoS components,the measurement changes along with the phase of the strategies.In this section,the importance measures in three phases are introduced respectively.
3.1 Pre-event importance measure
Pre-event importance measure is to analyze the impact of addition of a node before the occurrence of the disruptive event.As shown in Fig.3,the two curves respectively represent the trend of SoS performance in the resilience process with and without the addition of a new nodeni .The solid line represents the original trend of SoS performance in the resilience process
while the dashed line represents the trend after the addition of a new node
When the start time and end time of eventej (te ,td ),as well as the start time and end time of recovery measure(ts ,tf )are determined,we obtain the overall resilience of the SoS at timetm and the pre-event importance indicator.
Fig.3 Trends of SoS performance in pre-event importance measure
Therefore,pre-event RIM can be mathematically represented as
whereRn +ni represents the overall resilience towards the disruptive event with new nodeni ;Rh represents the situation without new nodeni .Combining(11)and(12),we obtain
3.2 Middle-event importance measure
Middle-event importance measure is to analyze the impact that a new node is added to the network during the disruptive event.As shown in Fig.4,the three curves respectively represent the SoS performance after eventej in three different states of new nodeni :the normal status,no node failure,and no failure on nodeni .The corresponding symbol of the performance under three states is marked in the figure.Note that when failure occurs in fewer nodes,the recovery time will decrease,i.e.
Fig.4 Trends of SoS performance in middle-event importance measure
According to the definition of integral resilience,the dash area in Fig.4 represents the accumulative effect of the SoS performance when nodeni does not fail.Given the ending time of recovery measure
,the overall resilience of the SoS at timetm can be calculated to measure the middle-event importance.
一是以学讲话促进秋收生产。垦区各地牢记总书记努力当好维护国家粮食安全“压舱石”，把“中国粮食，中国饭碗”抓在手上、记在心中的嘱托，以讲话精神为动力，全力加强秋收工作，确保颗粒归仓、保质保量。集团成立了以总经理为组长的秋收工作领导小组，逐级分片包点，明确任务、落实责任，科学指导、靠前指挥、昼夜抢收。截至10月25日，垦区4300余万亩耕地秋收工作全面告捷，预计全年总产430亿斤以上，创历史新高。截至11月5日，垦区高标准完成4300余万亩耕地“黑色越冬”工作。水田备苗床土400多万立方米，秋做床1.3亿多平方米，旱田秋起垄1300多万亩，为2019年农业生产奠定了良好基础。
Therefore,middle-event RIM can be mathematically represented as
The OODA loop is a decision cycle known as the Boyd cycle,first proposed by U.S.Air Force Maj.John Boyd[39].Although the OODA loop originates from the combat process,this theory has been widely used in military,business,public administration,etc.,especially in military command and control.In OODA theory,the military operation is divided into four basic activities:observe,orient,decide,and act,and an operation loop is a closed-loop consisting of the four activities:detect the sensor(S)and trace the target(T),and transmit the target information to the decision point(D)[40];the decision point makes decision and issues the order after analyzing the target information and the operational situation;influencer(I)takes action,after which sensor(S)will detect the target again to affirm the attack;finally,the decision point will decide whether a second action is needed.In modern military theory,an operation is a cyclic process consisting of observation,judgement,and action.Owing to the difference between the reliability and capacity of each component,the performance of different operation loops is not the same,however,a network with more operation loops is able to process more information and launch more strikes in operations,and have more alternatives when some of the SoS components fail.Therefore,we propose that the number of operation loops is suitable for the measurement of the SoS performance.
3.3 Post-event importance measure
Post-event importance measure is to analyze the impact that a set of failure nodes are recovered after the disruptive event.As shown in Fig.5,the three curves respectively represent the SoS performance after eventej in three different situations of recovery:recover all nodes,recover nodeni only;and no recovery.The corresponding symbol of the performance under three situations is marked in the figure.Note that
According to the definition of integral resilience,the dash area in Fig.5 represents the accumulative effect of the SoS performance when nodeni is recovered.
Fig.5 Trends of SoS performance in post-event importance measure
Accordingly,after-event RIM can be mathematically represented as
whereRhrni represents the overall resilience when the failure nodeni is recovered;Rh represents the normal situation.Combining(11)and(16),we obtain
说她啰嗦她还不承认，关于她去北京旅游的事，念叨了起码几十遍，每一个细节都翻来覆去地描述，以至于她一说登长城，我就能接着说：“我知道，你去的那回，好多外国人背着小孩在那登长城。”她听不出我的言外之意，还喜滋滋地补充：“是啊，都是粉嘟嘟的外国小毛头，太好看了。”我心里很不以为然，哪个种族的娃娃还不都是粉嘟嘟的啊。
4.Case study
4.1 Simple example
In this study,the process of resilience and importance measure analysis is examined by a simple but integrated case study,in which the sensors,decision points,influencers,and opponent targets are extracted from an operation SoS.Table 1 illustrates the types,numbers,and symbols of nodes in the network.Table 2 illustrates the probability that two nodes of different types are connected.For example,PTS =0. 4 represents that the probability that a sensor node is connected to a target node is 0.4.Hence,the SoS network,which is generated with the model parameters in Table 1 and Table 2,is exhibited in Fig.6.Note that different colors of nodes represent the node types.
Table 1 Nodes in the network
Table 2 Probabilities of connection between node types
Fig.6 SoS network in the case study
4.2 Resilience analysis
According to(11),the overall resilience of the SoS during the disruptive events can be obtained in the following.
5.建立科学的干部考评指标体系。按照科学发展观的总体要求，按照黄蓝两大战略推进规划，科学制定干部考评指标体系。对各级各部门承担的目标任务，纳入政务督查，实行规划推进落实责任制。
2018年，进入全球汽车零部件供应商100强的中国零部件企业已有6家，创历史新高，进入500强的企业就更多了。进入百强的6家中国企业分别是延锋（排名第16）、海纳川（排名第65）、中信戴卡（排名第71）、德昌电机（排名第79）、五菱工业（排名第80）和敏实集团（排名第92）。
Without loss of generality,we set the start time of the missiont 0=0,and the duration of the mission is 18 time units,therefore,tm =18.At timete =4,the SoS is subject to attack untiltd =7.When the attack begins,the abovementioned nodes will fail in sequence at random times in the attack duration.Meanwhile,the last failure occurs at the end of the attack duration.In the simulation,we generateNf- 1 random numbers in time interval(te ,td )to define the start times of each node failure:td 1=4. 8,td 2=6. 1,andtd 3=7.
Before the recovery begins,we assume the preparation time is 3 time units,thusts =10.During the recovery,the sequence of recovering each node depends on the types of the nodes.The priority of recovering different types of nodes follows D ＞S ＞I .When recovering nodes of the same type,nodes that possess higher degrees will be recovered first.Also,the recovery time,depicted as mean time to repair(MTTR),of each node varies with its type,as shown in Table 3.
Table 3 MTTR of network nodes
As mentioned above,the sequence of recovering each node in evente 1followsD 1＞S 1＞I 1,and the recovery of each node finishes at time:tr 11=12,tr 12=13. 6,andtr 13=tf 1=15.Likewise,for evente 2,the recovery follows:S 8＞S 4＞I 5,tr 21=11. 6,tr 22=13. 2,andtr 23=tf 2=14. 6.
In this study,the number of operation loops is calculated by an approximate algorithm based on eigenvalue of the network adjacency matrix.When the scale of the SoS is very large,the connectivity of the network will be complicated,thus in this study,the number of closed loops is used as an approximation of the number of closed operation loops,i.e.,ignoring the sequence of nodes in an operation loop.
Fig.7 Performance of the SoS during the disruptive events
In this case study,without loss of generality,we assume there are two disruptive events:e 1,e 2.Whene 1occurs,nodesS 1,D 1,I 1 will fail,i.e.,
;whene 2occurs,nodesS 8,S 4,andI 5 will fail,i.e.,
Each node in the network has only two states:normal or malfunction,and the status of each node will change in the presence of attack or recovery.
这也意味着，教材尤其是高校教材绝不是抄抄写写的简单工作，而是一件复杂而有创造性的工作。甚至，编教材有时候比科研还难，因为写论文只需要明确表达自己的观点就够了，编写教材需要综合考虑多方面的因素，如读者的情趣与能力，教学的要求与师生的要求以及与相关学科的关系等。
The above results show thatRh 2 is greater thanRh 1,which indicates that the SoS is more resilient against evente 2.There are two reasons for the difference.In the attack process,evente 1 leads to the failure of decision pointD 1,which is the key node in the SoS.Since there are only three decision points in the network,any component failure in decision points will result in large quantities of disconnected operation loops and sharp decrease of performance.On the other hand,in the recovery process,the recovery of decision points costs more time.Hence the SoS will remain in low performance for longer,and cannot perform the expected functions of command and control.From(13),(15)and(17),it is evident that if the SoS is in low performance for long,the SoS is less resilient.
The overall resilience duringe 2:
The overall resilience duringe 1:
4.3 Importance measure analysis
In this section,we conduct the RIM analysis on evente 1 in the network shown in Fig.6.After the disruptive event,nodesS 1,D 1,andI 1 fail,by which we can measure the resilience importance of each node.
4.3.1 Pre-event RIM
In pre-event RIM,we add a node to the network and trace the performance of the SoS during evente 1.The new node can be a sensor(S ),decision point(D ),or an influencer(I ),and the trends of performance in three conditions are respectively shown in Fig.8.
随着国家“一带一路”战略的实施，中国走向世界的步伐加快，文化的交流也显得尤为重要，国际学术文化、知识资源的相互交流与促进将成为科技发展的重要推动剂。党的十九大报告中指出推动文化事业和文化产业发展，推进国际传播能力建设，讲好中国故事，展现真实、立体、全面的中国，提高国家文化软实力。而国家学术文化、科技文化的发展能够展现国家科技的实力与水平。
Fig.8 Trends of SoS performance when different types of new nodes are added
The overall resilience of the SoS,and the pre-event resilience importance are illustrated in Table 4.
Table 4 Resilience and importance in pre-event analysis
4.3.2 Middle-event RIM
Following the steps introduced in Section 3.2,we obtain the trends of SoS performance in different conditions in middle-event RIM,as shown in Fig.9.
Fig.9 Trends of SoS performance when encountering different types of failure
The overall resilience of the SoS,and the middle-event resilience importance are illustrated in Table 5.
Table 5 Resilience and importance in middle-event analysis
4.3.3 Post-event RIM
Following the steps introduced in Section 3.3,we obtain the trends of SoS performance in different conditions in post-event RIM,as shown in Fig.10.The overall resilience of the SoS,and the middle-event resilience importance are illustrated in Table 6.
Fig.10 Trends of SoS performance under different recovering measurements
Table 6 Resilience and importance in post-event analysis
4.3.4 RIM of different nodes
In Fig.11,the resilience importance of different types of nodes in the network is illustrated with a histogram.As shown in the histogram,the importance index of decision pointsD is high in pre-event,middle-event,and postevent.This is because the decision points possess high degrees and that all operation loops are subject to a small number of nodeD s.Once more decision points are added to the network,the number of operation loops will considerably increase the number of operation loops.Hence,the performance of the operational network will increase and the SoS can maintain its performance during the disruptive events.If the decision points can maintain the expected functions during disruptive events,the SoS will lose only a small quantity of operation loops.Meanwhile,if the failed decision points are recovered in priority,the SoS will restore its performance rapidly.In summary,our results of importance measure analysis indicate that the decision points are vital for an SoS as well as for the improvement of resilience.In all the three types of RIM analysis,i.e.,pre-event,middle-event,and post-event RIM,the decision points should be addressed more.
从表2可以看出，通过改进混合GA-PSO的配网重构，网损由重构前的202.677 1 kW下降到129.830 9 kW，同时最低节点电压由重构前的0.913 4 p.u提升到0.938 8 p.u。说明电压的质量得到了改善，论证了改进混合GA-PSO的有效性。将本文得到的结论与文献[9]的结果进行比较，更能说明这种优势。
Fig.11 Resilience importance of different types of nodes
5.Conclusions
Since the structure and quantity of member systems in an SoS are complex,it is important to design and optimize the SoS from the perspective of resilience importance,so that the SoS will be more resilient when confronted with uncertain disruptive events.In this study,we propose an NCW-based model to describe the components and connections of an SoS,in which the performance is measured by the number of operation loops.In the resilience analysis,we conduct RIM,focusing on the attack and recovery process of the SoS.From the perspective of resilience,we propose and examine the resilience indicator of SoS and the corresponding algorithms.Specifically,the resilience situations are classified into three types according to the phases of performance change under disruptive events:pre-event,middle-event,and post-event.The difference of the three conditions is also the source of reasons for resilience:pre-event resilience depends on component redundancy,middle-event resilience depends on SoS reliability,and post-event resilience depends on the recovery of the SoS.Besides,other source of SoS resilience,e.g.,SoS evolution,reconfiguration,task-reorganizing,are to be examined in the future research.
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DOI: 10.21629/JSEE.2019.05.10
Manuscript received March 06,2019.
*Corresponding author .
This work was supported by the National Natural Science Foundation of China(71571004).
Biographies
PAN Xing was born in 1979.He received his B.S.degree in mechanical engineering,and Ph.D.degree in systems engineering from Beihang University(BUAA),Beijing,China,in 2000,and 2005,respectively.From 2005 to 2009,he was an assistant professor with the School of Reliability and Systems Engineering, Beihang University, Beijing,China.Since 2009,he has been an associate professor.From 2012 to 2013,he was a visiting scholar at the Department of Systems and Industrial Engineering,University of Arizona,Tucson,USA.His research interests include reliability engineering,systems engineering,and system risk analysis.
E-mail:panxing@buaa.edu.cn
WANG Huixiong was born in 1995.He received his B.S.degree in safety science from School of Reliability and Systems Engineering,Beihang University,in 2018.He is currently a master student in the same school.His interests of research include systems engineering and complex network.
E-mail:wanghuixiong@buaa.edu.cn
YANG Yanjing was born in 1991.He received his B.S.degree in material engineering from Yanshan University in 2015,and M.S.degree in industrial engineering from Beihang University in 2018.Since 2018,he has been an assistant engineer at China Railway Signal and Communication Corporation.His research interests include signal system engineering and system and systems engineering(SoSE).
E-mail:yangyanjing@crscu.com.cn
ZHANG Guozhong was born in 1979.He is a senior engineer with the Institute of Systems Engineering,China State Shipbuilding Corporation.His research interest is system of systems engineering(SoSE).
E-mail:18911990023@189.cn
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