Parallels between materials science test and protracted subsystem disruptions.
\r\n\tIn the book the theory and practice of microwave heating are discussed. The intended scope covers the results of recent research related to the generation, transmission and reception of microwave energy, its application in the field of organic and inorganic chemistry, physics of plasma processes, industrial microwave drying and sintering, as well as in medicine for therapeutic effects on internal organs and tissues of the human body and microbiology. Both theoretical and experimental studies are anticipated.
\r\n\r\n\tThe book aims to be of interest not only for specialists in the field of theory and practice of microwave heating but also for readers of non-specialists in the field of microwave technology and those who want to study in general terms the problem of interaction of the electromagnetic field with objects of living and nonliving nature.
",isbn:"978-1-83968-227-8",printIsbn:"978-1-83968-226-1",pdfIsbn:"978-1-83968-228-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8f6a41e4f5ce0e9c48628516d7c92050",bookSignature:"Prof. Gennadiy Churyumov",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10089.jpg",keywords:"Electromagnetic Wave, Microwave Energy Application, Electromagnetic Energy Generation, Intelligent Microwave Heating, Microwave Organic Chemistry, Microwave Reactor, Microwave Discharge, Microwave Plasma, Microwave Drying System, Tissue Microwave Heating, Measurement Automation, Industrial Microwave Process",numberOfDownloads:224,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 3rd 2020",dateEndSecondStepPublish:"July 24th 2020",dateEndThirdStepPublish:"September 22nd 2020",dateEndFourthStepPublish:"December 11th 2020",dateEndFifthStepPublish:"February 9th 2021",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Prof. Gennadiy I. Churyumov is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology and a senior IEEE member.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"216155",title:"Prof.",name:"Gennadiy",middleName:null,surname:"Churyumov",slug:"gennadiy-churyumov",fullName:"Gennadiy Churyumov",profilePictureURL:"https://mts.intechopen.com/storage/users/216155/images/system/216155.jfif",biography:"Gennadiy I. Churyumov (M’96–SM’00) received the Dipl.-Ing. degree in Electronics Engineering and his Ph.D. degree from the Kharkiv Institute of Radio Electronics, Kharkiv, Ukraine, in 1974 and 1981, respectively, as well as the D.Sc. degree from the Institute of Radio Physics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine, in 1997. \n\nHe is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology. \n\nHe is currently the Head of a Microwave & Optoelectronics Lab at the Department of Electronics Engineering at the Kharkiv National University of Radio Electronics. \n\nHis general research interests lie in the area of 2-D and 3-D computer modeling of electron-wave processes in vacuum tubes (magnetrons and TWTs), simulation techniques of electromagnetic problems and nonlinear phenomena, as well as high-power microwaves, including electromagnetic compatibility and survivability. \n\nHis current activity concentrates on the practical aspects of the application of microwave technologies.",institutionString:"Kharkiv National University of Radio Electronics (NURE)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"24",title:"Technology",slug:"technology"}],chapters:[{id:"74623",title:"Influence of the Microwaves on the Sol-Gel Syntheses and on the Properties of the Resulting Oxide Nanostructures",slug:"influence-of-the-microwaves-on-the-sol-gel-syntheses-and-on-the-properties-of-the-resulting-oxide-na",totalDownloads:94,totalCrossrefCites:0,authors:[null]},{id:"75284",title:"Microwave-Assisted Extraction of Bioactive Compounds (Review)",slug:"microwave-assisted-extraction-of-bioactive-compounds-review",totalDownloads:12,totalCrossrefCites:0,authors:[null]},{id:"75087",title:"Experimental Investigation on the Effect of Microwave Heating on Rock Cracking and Their Mechanical Properties",slug:"experimental-investigation-on-the-effect-of-microwave-heating-on-rock-cracking-and-their-mechanical-",totalDownloads:28,totalCrossrefCites:0,authors:[null]},{id:"74338",title:"Microwave Synthesized Functional Dyes",slug:"microwave-synthesized-functional-dyes",totalDownloads:21,totalCrossrefCites:0,authors:[null]},{id:"74744",title:"Doping of Semiconductors at Nanoscale with Microwave Heating (Overview)",slug:"doping-of-semiconductors-at-nanoscale-with-microwave-heating-overview",totalDownloads:45,totalCrossrefCites:0,authors:[null]},{id:"74664",title:"Microwave-Assisted Solid Extraction from Natural Matrices",slug:"microwave-assisted-solid-extraction-from-natural-matrices",totalDownloads:25,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. 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by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"67463",title:"Quantum Algorithms for Fluid Simulations",doi:"10.5772/intechopen.86685",slug:"quantum-algorithms-for-fluid-simulations",body:'\nIn recent years, the field of quantum computing [1] has developed into an active and diverse field of research, and significant progress has been made in a number of important areas. For a relatively small number of applications, quantum algorithms have been developed that provide a significant speedup relative to classical methods. Shor’s algorithm for factoring composite integers and Grover’s algorithm for quantum search were key developments in establishing quantum computing. More recently, significant progress has been made in the area of quantum chemistry and quantum physics. Beyond those two fields, only recently have quantum computing applications appeared in other areas of science and engineering, e.g., work in computational electromagnetics [2, 3], mixing in turbulent flow [4], and computational fluid dynamics [5]. More general applications have been developed which take advantage of the unique capabilities of quantum computing platforms, e.g., methods for the solution of linear systems of equations [6] and Poisson equation [7].
\nIn recent years significant progress has been made in designing and constructing quantum computers. Currently available quantum computers are relatively small-scale and have become known as noisy intermediate-scale quantum (NISQ) computers. These machines have a limited number of qubits (expected to increase to 50–100 in coming years), a limited connectivity between these qubits, a small set of available quantum gates, and typically very little or no quantum error correction.
\nThis chapter describes results of a recent investigation aiming to assess the potential of quantum computing and suitably designed algorithms for future computational fluid dynamics application, particularly for NISQ-type quantum hardware. In this work, the quantum circuit model is used for a “universal” or “digital” quantum computer, i.e., work on adiabatic quantum computing is not considered here. In the absence of the required quantum hardware, large-scale parallel simulations on parallel classical computers are required in developing such algorithms. In this work the recently developed quantum simulator [5] included in the MΦC multi-physics CFD framework is used [8, 9].
\nIn the near future, the most likely scenario for the introduction of quantum computing hardware is through the quantum coprocessor model, i.e., where a quantum processing unit (QPU) is loosely coupled to a classical computer with one or more CPUs [10]. In current designs, the quantum processor requires storage at low temperatures in a cryostat leading to a distinct physical separation between the classical and quantum hardware. Coupling takes place by exchanging classical information. In application of this hybrid quantum/classical approach, the quantum processor acts like a coprocessor with the quantum processor dealing with selected computationally demanding tasks. The quantum processor receives information from the CPU, and this is used to initialize the quantum state in the quantum processor. During the quantum simulation, the quantum state is transformed by application of quantum gates in quantum circuits. Then measurement operations are used to extract classical information from this quantum state, and this is subsequently passed to the CPU. Since in quantum mechanics a measurement leads to the (partial) collapse of the quantum state, in the hybrid classical/quantum approach, typically multiple realizations of the quantum state are needed to obtain classical information with acceptable levels of noise and uncertainty. It is important to recognize that, since initializing a particular quantum state in quantum computer can be a significant challenge, this hybrid approach can only be expected to lead to significant computational speedups in case the quantum simulation is significantly faster for the selected problem than conventional solution methods.
\nAs an example of this hybrid classical/quantum approach, the author introduced a quantum computing application in which the vortex-in-cell method was used to solve the incompressible-flow Navier-Stokes equations in a regular domain [5]. In this algorithm, the Poisson solvers dominating CPU time requirements are based on the quantum computing equivalent of the fast Fourier transform, i.e., the quantum Fourier transform. In this chapter, this algorithm and its application to example flow problems are investigated further. Specifically, the effect of applying an approximate QFT instead of the full QFT is analyzed for different levels of approximation or truncating of rotation gates in the quantum circuit implementation.
\nThe second part of this chapter describes a more recent investigation into the development of quantum algorithms relevant for computational aerodynamics based on modeling at the kinetic level. The key innovation in these developments is the design targeting execution of the algorithm fully on the quantum processor. In particular, at the start of the simulation, multiple quantum states in the quantum processor would be initialized. Then, the quantum algorithms would perform a series of unitary transformations. Only at the end of the simulation would measurements be performed to extract the required classical. A key question this study aims to answer is for which applications this approach is feasible.
\nThis chapter is organized as follows. Section 2 describes key principles of quantum computing relevant to the quantum algorithms for fluid simulations described here. Section 3 describes the hybrid quantum/classical implementation of the vortex-in-cell method along with a number of example applications. The quantum discrete-velocity algorithm for kinetic flow modeling is described in Section 4. Conclusions and future research directions are presented in Section 5.
\nThe fundamental unit of quantum computation is the qubit [1]. Whereas a classical bit is confined to existing in either the 0 or 1 state, a qubit can be in a state of superposition, i.e., it exists in both states simultaneously. Upon measuring the qubit, the quantum state collapses to either of these two states, and the qubit is no longer in a state of superposition. The state of a qubit is defined through a pair of complex numbers [1]. A collection of \n
In the present work, the quantum circuit model of quantum computing is used. In this case, the unitary operations on a quantum state allowed by quantum mechanics are represented by a series of quantum (logic) gates acting on the quantum state. A quantum logic gate is an elementary quantum computing device that performs a fixed unitary operation on selected qubits in a fixed period of time. Written in a matrix form, unitary means that the determinant of the transformation equals one.
\nThe quantum state |Ψ> for a register with \n
We will now describe how this quantum state can be used to represent the storage space required for the computational problems of interest here, representing discretized partially differential equations. As a first step, consider a function f discretized on a (regular) mesh with N mesh points. It follows that \n
However, it is important to stress that the quantum state vector only represents the likelihood that upon measurement the quantum state collapses into a particular state [1]. In other words, with \n
The ability to implement the quantum Fourier transform (QFT) efficiently on a quantum computer is of paramount importance for many quantum algorithms. Figure 1 shows the “standard” quantum circuit implementation of the QFT for an example register with 6 qubits. In Figure 1, “H” represents the one-qubit Hadamard gate, and the “Rk” gates are controlled rotation gates over an angle defined by index “k,” i.e., \n
Quantum circuit for QFT on six-qubit register.
A particular challenge is presented by the controlled rotation gates particularly those involving small angles. The QFT can be implemented approximately by removing all rotation gates with angles smaller than a certain threshold value, resulting in the approximate QFT (AQFT). In particular for fault-tolerant implementations, this is desirable as it greatly reduces the gate count. In the following, we define the approximation or “band-limiting” in the AQFT as follows. The rotation gates are eliminated above a limit value “k,” i.e., for an angle smaller than 2\n
The vortex-in-cell method is a well-studied hybrid particle-mesh method for incompressible flows and is particularly well suited for flows in regular domains such that efficient Poisson solvers can be used. In the present work, the Fourier analysis approach to solving the problem in a fully periodic domain is used, using the QFT for the required discrete Fourier transforms.
\nThe vortex-in-cell (VIC) method solves the incompressible-flow Navier-Stokes equations, transformed into the Helmholtz equations for vorticity evolution [5]:
\nIn simulations using the VIC, the flow evolves through a (large) number of time steps. During each of these time steps, the velocity field is recomputed using solutions of three Poisson problems for stream function \n
This part of the VIC is of particular interest here, as it represents the part that would be performed by the quantum processor in the quantum coprocessor model.
\nFigure 2 shows an example of a VIC simulation of two leapfrogging vortex rings, i.e., flow structures of fundamental importance in fluid mechanics. The lower vortex ring is stronger than the ring above it, and it will therefore convect upward faster, leading to the interaction of the vortex rings as shown. The iso-surface represents vorticity strength, i.e., a direct indicator of the “strength” of the considered vortex. Results are compared for two different meshes, 1283 and 2563, to highlight the dependency of the solution on the chosen mesh size. Also, in the shown simulation, no quantum errors were simulated, and the full QFT was used. If we now replace the QFT with the AQFT, the results shown in Figures 3 and 4 are obtained. In Figure 3, the “k” limit in the QFT is set to five for both meshes, showing that for the finer mesh this leads to unacceptable errors, while the coarser-mesh simulation still produces worthwhile results. If the “k” limit for the finer mesh is increased from 5 to 6, i.e., more controlled rotation gates are included in the AQFT circuit, the simulation on the finer mesh can also be made to produce similarly useful results. These example results show what level of approximation in the QFT is tolerable for application of the VIC method. For other QFT-based CFD solvers, a similar sensitivity study would need to be conducted.
\nVortex-in-cell simulation of leapfrogging vortex rings. Effect of mesh refinement is shown. “Noiseless” simulation using full QFT.
Vortex-in-cell simulation of leapfrogging vortex rings. For two different mesh resolutions, the effect of applying AQFT is shown (“k” limit is 5).
Vortex-in-cell simulation of leapfrogging vortex rings. For 1283 mesh, AQFT (“k” limit 5) is used. For 2563 mesh, “k” limit in AQFT is 6.
In computational fluid dynamics, the most widely used methods involve solving the Navier-Stokes equations for a continuum fluid, i.e., where fluid density, velocity components, and energy in each location in the computational domain are to be found from conservation equations. The vortex-in-cell method used in the previous section employs the Navier-Stokes equations in a transformed form involving vorticity rather than velocity, but importantly it still uses the continuum flow assumption.
\nAn alternative approach to the Navier-Stokes-based modeling is a description of the flow at a more detailed level, i.e., at the kinetic level [11]. Instead of governing equations for mass, momentum, and energy conservation, the flow is now described by the Boltzmann equation governing a particle distribution function in state space (or 3D velocity space for a 3D monatomic gas flow) for each location in the considered domain [10].
\nFor a monatomic gas, the distribution function \n
The distribution function defines for each point \n
The key advantage of this approach is that non-continuum flows, i.e., flows for which the density is so low that we cannot assume it to act as a continuum, can also be modeled. However, the main problem is the large computational cost when using a direct discretization approach due to the high dimensionality, i.e., for a 3D flow problem, we have a six-dimensional solution space (or seven when including time).
\nA further main challenge is the cost of evaluating the collision term. For the free-molecular flows considered here, the collision term can be discarded, and we will use the collisionless Boltzmann equation instead.
\nIn the discrete-velocity method used here, the velocity space is discretized using a uniformly spaced Cartesian mesh. A maximum molecular velocity magnitude is defined, \n
The discrete-velocity approach used here has a number of characteristics facilitating a quantum implementation:
A uniformly spaced Cartesian mesh is used for the spatial discretization as well as for the discretization of the velocity space.
In case solid objects are present in the computational domain, these are rectangular, and its edges align with the mesh lines in the mesh. Specifically, solid bodies can be defined by “tagging” selected groups of cells in the mesh.
A constant velocity-space discretization is used in each point in space, i.e., the velocity-space boundaries defined earlier as well as the number of discrete velocities are identical in each cell.
The convection part of the Boltzmann equation (i.e., the second term on the left-hand side in the shown equation) along with gas-solid interactions determines the time evolution of the distribution function in the absence of interparticle collisions.
The time-integration method used here is based on the reservoir technique [13], such that during the time integration the convection step always exactly involves the distribution function defined in a cell of computational mesh to move to a cell that is a nearest neighbor. This is commonly referred to as “streaming” of data.
In the examples shown, problems are restricted to two-dimensional flows. For this case, the collisionless Boltzmann equation originally defined for 3D can be reduced to two kinetic equations for two reduced distribution functions:
\nwhere the velocity and coordinate vectors are now defined in 2D.
\nInspired by the previous work on the Dirac equation [14], an efficient quantum circuit implementation of the streaming operations in x- and y-direction on a Cartesian 2D mesh can be created as shown in Figure 5. The example shows how a 12-qubit register is used for a discretized function with 6 qubits defining the indices of 64 grid points in x- and y-coordinate directions. The circuits with the filled circles define streaming to “right” neighbors, i.e., when each control qubits has the |1> state, the target qubit gets negated (“X” symbol used here). Similarly, the left streaming operation employs multiple-control NOT gates with target qubits being negated when each control qubit is in state |0>.
\nQuantum circuit implementation of “left streaming” and “right streaming” in x- and y-direction on a 64 × 64 Cartesian mesh.
For the streaming operations in quantum discrete-velocity method, further extensions are needed. First, the quantum register needs to be extended relative to that shown in Figure 5 to account for the storage of two discretized reduced distributions defined in the discretized velocity space. The additional distribution function is accounted for using an additional qubit termed “g” in the following quantum circuit diagrams. The number of additional qubits needed for the discrete-velocity mesh clearly depends on the number of discrete velocities used. For example, for a 16 × 16 discrete-velocity mesh, we add 8 qubits (four for each direction in state space). The qubits are denoted by the “u0,”…, “v0,”… qubits in the quantum circuits. Finally, to account for solid objects, we use an additional qubit (“BC” in the diagrams) set to |1> to denote a cell within fluid and |0> for a cell within a solid. For a 64 × 64 Cartesian mesh and a 16 × 16 discrete-velocity mesh, Figure 6 shows the quantum circuit used to simulate the free-molecular flow around a rectangular body, for which the evolution of the flow field starting from an initial uniform flow is shown in Figure 7. The key feature in the quantum circuits shown in Figure 6 is the extended number of control qubits, i.e., beyond the checks on the qubits corresponding to spatial coordinates, control qubits also involve the “BC” qubit (fluid/solid flag) as well as the qubits related to the discrete-velocity indices. This least feature originates from the need to stream only data associated with selected discrete velocities in the used time-integration method.
\nQuantum circuit implementation of streaming operations for discrete-velocity method (with 16 × 16 velocity mesh). Two-dimensional domain with 64 × 64 Cartesian mesh.
Discrete-velocity simulation of Mach 2 flow around rectangular body. 64 × 64 Cartesian mesh, velocity-space mesh with 64 × 64 discrete velocities.
A key aspect of the flows simulated by the quantum algorithm described here are gas-solid interactions. In this work, specular-reflection boundary conditions are assumed. This means that upon hitting a solid surface, a gas particle will bounce off this surface with the wall-normal velocity component effectively getting reversed and the tangential-flow component being preserved. In Figure 7, the time evolution of a Mach 2 free-molecular flow is shown, starting from an initial flow at uniform velocity everywhere.
\nAs can be seen in Figure 7, the specular-reflection boundary conditions gradually make the velocity vectors align with the solid wall such that there is no longer a flow into the solid body. This is the physically correct behavior. In the quantum register, the indexing for the velocity-mesh data is designed such that a change of the sign of the discrete velocity implies a bit negation of qubits representing the index of the considered discrete velocity. As an example, Figure 8 shows the quantum circuit implementation of the specular reflection for a rectangular body. In this case, only 16 × 16 discrete velocities are used for clarity. It can be seen that control qubits are used to “select” cells in space for which to apply the boundary condition. A further control qubit involves the “BC” qubit representing the solid/fluid flag. The negation operation (“X”) is applied to the qubits representing the velocity-mesh index to create the “change of sign” of the considered discrete-velocity data. This circuit is shown as an illustration of the quantum algorithm design approach used here.
\nQuantum circuit implementation of specular-reflection boundary conditions for rectangular body. 64 × 64 Cartesian mesh, 16 × 16 discrete-velocity mesh.
Although the circuits shown in Figure 6 perform the correct convection operations, an important practical constraint needs to be considered. For the quantum computer implementations achieved so far and those foreseen for the near future, the kind of multi-qubit-controlled NOT operations used here cannot be implemented. A small set of native gates will be available which most likely includes NOT, CNOT, and the Toffoli gate.
\nThe solution around this limitation is the introduction of ancilla qubits, which can be regarded as the quantum equivalence of additional workspace in the memory of a classical computer. Then circuits involving multi-qubit operations involving a large number of control gates can be transformed into circuits with more qubits and a larger number of gate operations, however now with a smaller number of control qubits. As is typical in this context, we assume that the ancilla qubits are initially in |0>, and since these are to be reused multiple times, the transformed circuits need to reset the ancilla qubits of this state at the end of the operations. For the quantum circuits implementing streaming in positive x-direction considered previously, Figure 9 shows the required circuit transformations for the cases of 1, 2, or 3 ancilla qubits. Increasing the number of qubits will make the circuits more demanding to implement, so in the actual implementation there is an important trade-off between gate complexity (i.e., within the set of native gates available) and total number of qubits. For the 64 × 64 two-dimensional mesh and 16 × 16 discrete velocities considered in Figure 6, the transformed circuits show that with three ancilla qubits, the maximum number of control gates is reduced to four in a five-qubit-controlled negation gate. However, for most practical hardware implementations, further transformations would be required.
\nQuantum circuit implementation of streaming operation. Circuit transformations are shown for addition of 1, 2, and 3 ancilla qubits.
This chapter presented two different quantum algorithms with possible applications in computational fluid dynamics. Beyond their very different areas of application, the key differences are the computational model with regard the quantum coprocessor model of quantum computing. The hybrid quantum/classical algorithm for the vortex-in-cell method involves repeated exchanges of information between classical and quantum hardware, i.e., at each time step in the time integration. In contrast, the quantum algorithm implementing a discrete-velocity method for kinetic flow modeling can be performed on the quantum processor for the duration of the simulation, with classical information exchange only required at the start and end of the simulation.
\nThis work addressed a number of key challenges that remain to be investigated further. Firstly, the need for further efficient quantum algorithms as well as a further understanding of how to apply the quantum coprocessor model for this type of flow simulations was investigated. Secondly, the measurement-based extraction of classical information fundamentally changes the way quantum algorithms for CFD application will most likely be used. Finally, obtaining detailed information on the full flow field will be a challenge, so applications for which only certain characteristics of the solution are desired would present a good choice for future applications.
\nA part of the simulation results presented were obtained using the EPSRC-funded ARCHIE-WeSt High Performance Computer (
Digitization is occurring in many industries in many different forms; however, regardless of the application, a common set of enablers are employed. As the proliferation of digital transformation continues, decision makers will need to distinguish between reliability and resiliency in the planning, design, and operation of these subsystems. Tightly coupled common hardware and software platforms potentially increase the breadth of accidental failures as well as the impact of intentional sabotage. Beyond end use applications is an overall reliance on electricity which these digital subsystems require to function. Hardware, software, and electricity form the foundation upon which digitalization rest. The increased interdependence and interconnection can lead to common failure modes of previously isolated subsystems, resulting in increased probability of high impact events. Interconnection results in the establishment of a singular system with all other structures existing as subsystems. Evaluation of subsystems will need to include internally and externally initiated disruptive events. Highly impactful events, sometimes termed black swans, cannot only disrupt subsystems but fundamentally change their structure. Impactful as they are, rarity can make these events prone to underinvestment due to heuristics and biases, most prominently the availability heuristic. A quantifiable metric can aid in our ability to appropriately allocate resources to study, adapt, and mitigate these high impact, low probability events before they unexpectedly fracture the established subsystems we rely on. The avoidance of fracture is central to the application of the modulus of resilience in critical subsystems. The chapter will review the differences between the reliability and resiliency as well as the importance of distinguishing between the concepts. Additionally, ideals related to resilience are identified and expressed in a concise operational definition. The research utilized the progression shown in Figure 1 for the investigation.
\nResearch phases.
Representative plot of event type distribution.
Borrowing concepts from materials science allows for an isomorphic application where analogous structures are leveraged to represent HILP event scenarios. In this chapter, the isomorphic application is presented to provide a method of quantifying resiliency or its absence based on the intended aim of the subsystem. This concept is consistent with select portions of previous literature, but divergent in others. Following a review of previous research, a gap analysis was completed to identify opportunities for new considerations in quantifying resiliency. Lastly, an example in applying the modulus of resilience for critical subsystems is provided to demonstrate the computational process.
\nReliability and resiliency are sometimes discussed in a similar context with respect to subsystem performance; however, they differ conceptually in both the events they measure and the characteristics they quantify. The measures which define reliability provide insights as to the context of the metrics use. Many of the most common reliability metrics utilize mean-based calculations from reoccurring failures over time. These metrics include mean time between failure (MTBF), mean time to failure (MTTF), and mean time to repair (MTTR). These metrics require successive failures in order to quantify subsystem performance. Mean time between failure (MTBF) is used in reliability to provide the number of failures per million hours for a subsystem. Mean time to repair (MTTR) is the time needed to repair a failed subsystem. Mean time to failure (MTTF) measures reliability for a subsystem which cannot be repaired. It is the mean time expected until the first failure of a subsystem. MTTF is a statistical value and represents the mean over a long period of time and a large number of operations. The reliability metrics can effectively represent common cause events which produce reoccurring failures; however, these calculations are less applicable to low probability special cause events. A special cause is something special, not part of the system of common causes. It is detected by a point that falls outside the control limits [1]. Often, subsystems have an allowable level of tolerance to minor disruption preventing sustained impairment in accomplishing the aim of the subsystem. Plotting the number of events by type versus percent of subsystem output disrupted graphically displays the relationship between common cause and special cause events. The allocation of events is closely represented by a pareto distribution Figure 2.
\nResiliency events reside at the tail of the distribution as rare events resulting from extraordinary scenarios. Such events have been produced by multiple failures within a single subsystem as discussed in the book Normal Accidents by Charles Perrow. His work examined failures in highly complex operating environments. The increasing interdependence results in an interconnected ecosystem where a failure in a single subsystem can create failures in multiple subsystems. When interactive complexity is joined with tight coupling, the risk of a system accident is considerably increased. Interconnectedness and complexity among contemporary subsystems is increasing at a rapid pace as technologies develop faster than assessments can be made regarding their risks. As we move away from individual events and account for the larger system, we find the “eco-system accident,” an interaction of systems that were thought to be independent but are not because of the larger ecology [2]. As systems grow in size and in the number of diverse functions they serve, and are built to function in ever more hostile environments, increasing their ties to other subsystems, they experience more and more incomprehensible or unexpected interactions [2]. Common mode failures, first included in analytical models in 1967, can contribute to unexpected actions from complex systems. In addition to common mode failures, proximity and indirect information sources are two additional indications of interconnectedness. Ultimately, the probability of a subsystem being subjected to significant disruption is dependent on the cumulative probability of both internal and external risks. Inevitably, the probability of significant disruption will increase as interdependence increases. While increases in events causing significant disruption are expected, their count is not expected to be significant enough for the application of mean-based reliability metrics. Therefore, resiliency-based metrics are needed which match the periodicity and scale of high impact, low probability events.
\nHILP events require a subsystem to bounce back to normalcy following major disruption. The goal is to regain pre-disruption levels of output as quickly as possible; however, recovery time is not the only metric of importance. The shape of the recovery curve is also of significance. Resiliency aids in defining a disaster response paradigm which differs from previous approaches such as resistance and sustainability by emphasizing return to normal. Nonetheless, the literature frequently uses the concept of resilience to imply the ability to recover or bounce back to normalcy after a disaster occurs [3]. Review of scholarly work related to the resiliency concept identified three main ideals: no assumption that disaster prevention is always possible, recognition of the need to include social variables, and the necessity to include disciplines outside the physical sciences and engineering. The term resiliency has increased in usage over the past decades. A multitude of definitions have been proposed whose interpretations can align with either resistance or sustainability. Although the resilience construct offered advantages in many areas relative to competing paradigms, the ambiguity associated with its meaning and scope hindered consensus. The multiplicity of definitions is a reflection of the philosophical and methodological diversities that have emerged from disaster scholarship and research [4].
\nResilience first came to prominence in the English language in the early 19th century when Tredgold used the term to describe a property of timber [5]. In his essay “On the transverse strength and resilience of timber,” Tredgold tested the properties of timber to be used in ship making. Tredgold cites resilience as the power of resisting a body in motion [5]. The statement is foundational in establishing the concept of resilience as more than recovery but instead as an ability to first withstand an applied force. Furthermore, Tredgold varied the weight and height of objects dropped on the test samples and recorded the effects to different forces on various wood pieces. These effects ranged from no effect, broke to curved. A second reference to the consideration of force can be found in the 1858 work, “On the Physical Conditions Involved in the Construction of Artillery, and on Some Hitherto Unexplained Causes of the Destruction of Cannon in Service,” by Robert Mallet. He states the modulus of resilience of other writers, referred to hereafter, depends, is much greater for gunmetal, and hence a given force produces a greater proportional distortion of form [6]. The modulus of resilience was further formalized by materials science using stress/strain testing.
\nThe range of methods for defining resilience include qualitative, quantitative and probabilistic. A quantitative method can be used to compare outcomes using data from different actual events. A number of researchers have explored quantifying resilience to move beyond qualitative representations. Henry and Ramirez-Marquez [7] proposed a quantitative approach for system resilience as a function of time. The formulation was a ratio of the recovery and losses using a figure-of-merit function. A disruptive event (ej) at time, te, impacts the system until time, td.
\nAs shown, the numerator relates to the recovery until time t and the denominator represents the total loss due to disruption. Hosseini et al. [8] reviewed definitions and measures of system resilience. Their literature review was based on multiple domains including organizational, social, economic, and engineering using papers published between 2000 and April 2015. The major categories of assessment approaches are qualitative and quantitative with quantitative measures further defined as either probabilistic or deterministic.
\nThe intent to analyze protracted subsystem disruptions leads to a focus on quantitative deterministic methods of calculating resiliency. The literature review by Hosseini et al. [8] included 11 deterministic methods of quantification. Bruneau et al. [9] utilized a method of integration based on the degradation in quality of infrastructure during recovery period of Eq. (5). Larger RL values indicate lower resilience while smaller RL imply higher resilience. Hosseini et al. [8] RL is calculated based on the formulation in Eq. (2).
\nZobel [10] proposed a method based on the total possible loss over some suitably long-time interval (T*), percentage of functionality lost after disruption (X), and time required for full recovery (T). An effort was made to analyze different combinations of X and T which result in the same level of resilience as shown in Eq. (3).
\nThis metric is based on a linear recovery making it unrealistic for some scenarios.
\nAlternative methods were proposed by Cox et al. [11] based on economic resilience using the difference in disruption (%∆DYmax) between the expected disruption (%∆Y) and maximum potential disruption (%∆Ymax). Therefore, an estimate of performance degradation is required. Such an estimation may be a challenge to precisely develop; however, the formulation is shown in Eq. (4).
\nAlternatively, Rose [12] considered time effects using a concept of dynamic resilience. The quantification of dynamic resilience is the difference in system recovery with hastened system recovery (SOHR) and without hastened system recovery (SOWR). This calculation is utilized over the total number of time steps (N) considered. The dynamic resilience calculation is shown in Eq. (5).
\nWang et al. [13] explored resilience in information systems based on the number of operations in the enterprise information system (m). The ratio of the demand time (di) and completion time of operation (ci) are weighted by the importance of operation (zi).
\nThe larger the value of the metric the more resilient the system is determined to be. The calculation requires the assignment of a weight and assumes the number of operations is known. When attempting to quantify unknown events the number of operations can be difficult to estimate.
\nChen and Miller-Hooks [14] quantifies the “post-disruption expected fraction of demand that, for a given network, can be satisfied within pre-determined recovery budgets” (Hosseini et al.). The measure was based on transportation networks and compares the maximum demand that can be satisfied before disruption (Dw) and after disruption (dw) for pair (w).
\nOrwin and Wardle [15] considered the instantaneous and maximum disturbance in the quantification of resilience. The maximum absorbable force without upsetting system function (Emax) and effect of the disturbance on safety (Ej) at a given time (Tj) are used to define resilience.
\nFrameworks for local and global resilience were introduced by Enjalbert et al. [16] for modeling system safety in public transportation systems. A safety indication function (S(t)) is used to calculate resilience either instantaneously or over time, representing local and global, respectively. Global resilience is calculated from the time of disturbance (tb) to the end of the disturbance (te). The calculations are as follows:
\nFrancis and Bekera [17] introduced a metric for dynamic resilience. The calculation uses the speed of recovery (Sp), original performance level (Fo), performance level at new stable level (Fr\n) and performance level immediately after disruption (Fd). The speed of recovery variable assumes exponential growth for a maximum acceptable recovery time (tδ), total recovery time (tr) to a new equilibrium state, time to complete initial recovery (\n
Otherwise,
\nCimellaro et al. [18] utilized quality of service to represent resilience. The method uses before disruption quality of service (Q1(t)), post disruption quality of service (Q2(t)), a control time (TLC) and a weighting factor (α) in developing a healthcare resilience metric.
\nAside from the works investigated by Hosseini et al. [8], Dessavre et al., [19] introduced a new model and visual tools adding a stress dimension representing the force and stress of disruptive events. Defining the stress of the events is not a trivial task and completely domain dependent [19].
\nA review of the concepts found in literature was completed for elements consistent with the modulus of resilience. Methods were limited to quantitative approaches which could be utilized with empirical data sets. Although the use of scaling factors was identified in literature [13, 18], such methods are not desired in the development of subsystem-based methods due to the subjectivity associated with them. A ratio-based approach has merit in its ability to normalize event effects and resulting recovery. Area-based calculations using integration are preferred to point calculations based on their ability to compensate for nonlinear restoration curves; however, complexity beyond the resilience triangle [9] would be necessary to capture differences in event magnitude and restoration response in disparate events.
\nThe concept of a yield point was not identified in existing literature. A return to normal operation was typically used to identify the end of the restoration time period; however, this approach does not set the time based on the aim of the subsystem. Evaluations of subsystems beyond a critical point with respect to use of the subsystem output could lead to poor decision-making. One of the main weaknesses of the current resilience metric is that they do not relate the effects of a disruptive event to any of the event characteristics, unlike materials science [19]. Materials science utilizes a change in length for evaluation of stress and strain; however, the difference in recovery response to a common cause and special cause event was not found in the literature review. These distinctions serve to highlight the differences between reliability for normally occurring events and resiliency to low frequency events. Additionally, the need for utilizing subjective variables [10, 11, 12, 14, 15] does not lend well to empirical study.
\nThe ability to normalize responses to different events is beneficial for evaluating the resiliency of different subsystems or different events on the same subsystem. The literature reviewed began analysis of the event from the start of restoration [7] or by treating the entire curve from time of event to the completed restoration as a single integral [16]. This approach can confuse the quantities of force, stress and strain. An equal force can result in different stress and strain based on the subsystem being reviewed. As a result, the descending slope and associated area prior to the start of recovery may prove informative of stress. Strain is more associated with the total area under the curve. The review of literature did not identify a bifurcation of the curve to delineate stress (prior to start of recovery) and strain (total area). Therefore, the assumption of instantaneous loss and exponential recovery [17] are not representative of many empirical cases.
\nIn reviewing the concepts of resilience, a force is applied to a subsystem, the subsystem absorbs a portion of the force, experiences stress, and adapts to recover to a pre-disruption state. These references highlight an importance of considering the stress on the subsystem in determining the resiliency of a subsystem. Three primary points of measure for use in quantifying resiliency were identified including: stress, total area of event and change in length. Stress is a foundational variable of resiliency, as the term resiliency implies a response to a significant disruption. Therefore, only events of significance from a subsystem level are commonly referred to in terms of resilience. Additionally, the ability to compare resiliency events needs some level of normalization based on the associated stress for each event. Force continues to be applied until the subsystem decay ceases, allowing for subsystem assessment and initiation of recovery. The rate of subsystem decay influences the stress applied to the subsystem and the subsystem ability to bounce back. This connection exists due to the role of adaptation in the resiliency process. A slow evolving scenario (i.e., slow subsystem decay) presents the subsystem opportunity to adapt, resist, and recover in ways an acute decay will not. Therefore, when considering the normalization process of resiliency both the decay (i.e., stress proxy) and recovery portion of the resiliency curve must be independently considered. The delayed decay provides an opportunity for improved response from the subsystem.
\nTotal area of recovery best quantifies recovery and resiliency by compensating for the nonlinearity in the response function. As the subsystem attempts to recover, disruptions in the recovery process may cause discontinuities not captured by linear slope calculations. Similarly, time to recovery (i.e., 3 days to recovery) calculations may fail to represent intermediate progress in recovery.
\nConsideration of a failure point based on the aim of the subsystem aids in representing real-world scenarios. Recovery which occurs after a critical point of the subsystem would indicate a lack of resiliency. As an example, if a water subsystem requires 10 days to restore operation post contingency but the consumers of the water can only survive 4 days without water; the subsystem lacks resiliency. Attempts to quantify the subsystem’s resilience should stop at 4 days. Calculations beyond the 4-day time period no longer support the aim of the subsystem or the practical operation of the subsystem.
\nLastly, change in length was included in the materials science calculation of the modulus of resilience. The change in length from the original length to the length under stress could be translated to a subsystem resilience construct to allow consideration of how subsystem recovery under lower stress common cause events and high stress special cause events are related. The consideration of a change in length may aid in joining concepts associated with reliability in the quantification of resilience.
\nComparing these constructs with the reviewed literature results in the identification of conceptual gaps. The resulting resiliency values should reflect the subsystem performance for practical cases. Units are required based on subsystem parameters. The x-axis utilizes units of time, while the y-axis measures the units associated with the aim of the subsystem.
\nThe methods of quantification reviewed begin the process of quantification at the point of recovery or assume no time delta between the initiating event and start of recovery. To support the incorporation of stress in the quantification of resilience, a bifurcation of the event curve is used as shown in Figure 3.
\nBifurcation of event curve.
The use of ratio methods may provide consistency in scenarios of similar characteristics. When disparate characteristics are present, computed values may prove inconsistent with event outcomes. Depending on the event characteristics, either ratio methods or area-based methods may identify a less resilient subsystem response as more resilient. Figure 4 depicts the concept of less recovery time for less disruption. The scenario of Figure 4 is representative of a minor difference in subsystem response and would provide consistent rankings for resilience outcomes in many cases, where less area is representative of increased resilience.
\nRecovery curves with similar characteristics.
Conversely, cases may exist where a longer recovery results from a less impactful initial event. The delayed recovery to a less impactful event could result from many factors including a lack of preparedness, inability to adapt, etc. In such cases, observation would assume that the subsystem which took longer to recover from a less impactful event is less resilient. However, present formulations may suggest the opposite. Figure 5 illustrates this scenario, where the smaller area is not representative of the more resilient outcome.
\nRecovery curves with dissimilar characteristics.
The fracture point should be set based on the aim of the subsystem. For example, if a drinking water subsystem failure requires a 7-day restoration period but 4 days is the survival period without water; the calculation of subsystem resiliency should be limited to a 4-day period. In some cases, the acknowledgement of a fracture point will result in the calculation of resiliency stopping prior to the subsystem returning to pre-disruption output levels. Figure 6 represents a case where the subsystem recovery takes longer than the subsystem failure point.
\nRepresentation of failure point.
Calculations to quantify resiliency which consider values beyond the failure point are theoretical as opposed to practical in nature. The failure point should be given priority in quantifying resiliency.
\nAn operational definition is derived from the combination of literature review and isomorphic adaptation of the modulus of resilience. Hence, resiliency is defined as the ability to limit proportional stain from abnormal stress to less than the subsystem yield point, through the achievement of recovery in less than the subsystem critical timeframes. This definition allows the use of quantitative measures in the calculation of resilience in a deterministic and normalized approach based on concepts from materials science.
\nAn evaluation between two groups can result in an isomorphic application of findings from one structure to another. This mapping between groups can yield opportunities to apply known methodologies in an inter-disciplinary manner. The process of verifying an isomorphism requires the identification of elements in each structure and evaluating their equivalence. If equivalence is identified an opportunity for applying the computational framework may exist. The quantification of subsystem resilience was compared to resiliency as used in materials science. Materials science’s definition of resiliency includes the concepts of per unit volume, maximum energy, and integration from zero to the elastic limit. The modulus of resilience (Ur) is found from the stress-strain curve measured during the tensile test. Stress (σ) in the stress-strain curve is “the applied force per unit original undeformed cross-sectional area of the specimen” [20] as delineated in Eq. (15).
\nwhere F = force; A0 = cross sectional area.
\nYoung’s modulus (E) serves as a measure of stiffness for a solid material. “Because of the difficulty in determining the elastic limit, it is commonly replaced by the proportional limit, which is the stress at which the stress-strain curve is out of linearity” [20].
\nAnd,
\nwhere F = force; A = actual cross-sectional area; ∆L = amount of change in length; L0 = original length of the object.
\n“The modulus of resilience is the strain energy per unit volume absorbed up to the elastic limit for a tensile test and equals the area under the elastic part of the stress-strain curve” [20].
\n“This quantity indicates how much energy a material can absorb without deforming plastically” [20]. Plastic deformation occurs when a material undergoes non-reversible changes in response to applied forces. The use of the stress-strain curve from materials testing is similar to conditions faced by disrupted subsystems regardless of type. Stress is the impact to the material under test, while strain is the resulting effects of the stress.
\nBased on the desire of applying a consistent methodology to quantify resilience regardless of disruption magnitude or subsystem size, the percentage of subsystem disrupted is proposed to achieve a per unit value. The area under the curve will then be integrated from the beginning to end of the disruptive event. Calculus to determine area under the curve is shown in Eq. (19).
\nwhere Ei = Event initial; Er = Event restored.
\nThe area under the curve will then be applied to the maximum percentage of subsystem disrupted.
\nwhere SD = % of subsystem disrupted; Anl = Area under the curve to nonlinearity; At = Total area under the curve; Da = Duration of average disruption; De = Duration of event disruption.
\nProtracted subsystem disruptions create stress and strain due to an inability to complete the subsystem aim. The similarities between tensile strength test used in materials science and the need to measure stress and strain subsystems create an isomorphic relationship. Table 1 shows the parallels between materials science and protracted subsystem disruptions.
\nMaterials science. | \nProtracted subsystem disruption | \nComparisons | \n
---|---|---|
Stress applied | \nPeak percent of subsystem out of service | \nPercent out of service is equivalent to stress | \n
Cross-sectional area | \nArea under curve from the origin to peak subsystem out of service | \nArea from zero to peak subsystem out of service is point where curve loses linearity | \n
Actual cross-sectional area | \nArea under curve for entire disruptive event | \nRepresents total strain experienced by subsystem | \n
Change in length | \nDelta between subsystem’s average duration of disruptions and event disruption duration | \nUse of change in duration accounts for the change in length between average and protracted event | \n
Original length | \nSystem’s average duration of disruptions | \nAccounts for average non-protracted disruptions events | \n
Parallels between materials science test and protracted subsystem disruptions.
The application of the modulus of resilience to a specific subsystem requires the identification of an aim the subsystem exist to accomplish. “Without an aim, there is no system” [21]. The aim should be quantifiable with metrics available for analysis. The data must be accessible in order to serve as the basis for the resilience calculations and will vary based on the subsystem under study. Examples include percentage of successful operations or percentage of end users receiving service. The next section provides an empirical example in applying the modulus of resilience.
\nThe power industry was selected to provide an example for applying the modulus of resiliency using empirical data. The aim of the electric subsystem is to deliver electricity to all end use customers; therefore, data regarding the number of customers out of service can be used to quantify subsystem performance. The use of customers out of service in quantifying subsystem performance was supported by a review of regulatory reliability metrics used by Public Utility Commissions. For major electric utility disruptions, DOE situation reports provide customer outage information for and are publicly available from the DOE website. One of the most prominent events to challenge utilities is hurricane, and as a result, multiple hurricane events have data on the DOE website. Following data collection, plots can be constructed of the electric utility response in restoring customers. The inflection points were identified, and a yield point designated by reviewing disaster preparedness data from the Capital Region Study [22]. The study indicated that 73% of survey respondents had less than 10 days of food stored. Therefore, an event lasting greater than 10 days would most likely result in scarcity from food spoilage and diminished retail capabilities. With a known bifurcation and yield point, analysis can be completed.
\nHurricanes Wilma and Irma presented an opportunity to compare resiliency of separate events in the same region. Following Wilma, the ability of several infrastructures to recover from severe events was reviewed in the Florida region. “[M]ore than $141.5 million has been obligated by FEMA for 119 Hazard Mitigation Grant Program projects to build stronger, safer more resilient communities in Florida” [23]. Florida was once again subjected to a hurricane when Irma came ashore 12 years later. More than six million customers lost power as a result of Irma; compared to 4 million from Wilma. Although more than a decade apart, these two storms provide an opportunity to compare the recoveries following significant investment in resiliency. The comparison of the two resiliency indices can present an opportunity to calculate a cost per unit of resiliency and explore concepts such as diminishing returns or optimization from multi-hazard investment. Multi-hazard resiliency actions would provide an ability to address multiple HILP scenarios with a single investment. A resiliency index for each of the scenarios would be computed in order to create a composite change in resiliency for a given investment. The goal of this composite approach is to provide a means for justifying highly adaptable subsystem structures based on resiliency benefits.
\nThe example demonstrates the process of calculating the resiliency index for a power utility scenario and comparing the response before and after the investment in resiliency. The values shown in Table 2 were extracted from United States Energy Information Administration (EIA) data. The additional data points associated with 0.5 and 1.5 days were included due to nonlinearities in customer outages associated with Hurricanes Wilma and Irma, respectively. Similarly, day 9 for Hurricane Wilma was approximated for the purpose of this analysis. The data required to calculate the change in length was available by collecting System Average Interruption Duration Index (SAIDI) data. SAIDI data provides a basis for the average duration a customer faces and can be compared to the protracted system disruption as a change in length.
\nDay. | \n% Out of service (Hurricane Wilma 2005) | \n% Out of service (Hurricane Irma 2017) | \n
---|---|---|
0 | \n0 | \n0 | \n
0.5 | \n34 | \n20 | \n
1 | \n35 | \n40 | \n
1.5 | \n34 | \n64 | \n
2 | \n31 | \n56 | \n
3 | \n28 | \n40 | \n
4 | \n21 | \n31 | \n
5 | \n18 | \n20 | \n
6 | \n12 | \n11 | \n
7 | \n10 | \n7 | \n
8 | \n9 | \n4 | \n
9 | \n6 | \n1 | \n
Outages for Hurricanes Wilma and Irma.
Following the collection of empirical data, the total area under the curve was calculated by dividing the outage curve into time steps and summing the areas of each time step as shown in Figures 7 and 8, respectively.
\nHurricane Wilma restoration plot.
Hurricane Irma restoration plot.
The study region had a SAIDI of 60 minutes and a protracted outage duration of 12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Wilma is determined as shown in Eq. (7).
\nThe study region had a SAIDI of 57 minutes and a protracted outage duration of 12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Irma is determined as shown in Eq. (22) based on EIA data [24] (Tables 3 and 4).
\nDay. | \n% Out of service (Hurricane Wilma 2005) | \nArea | \n
---|---|---|
0 | \n0 | \n0.085 | \n
0.5 | \n0.34 | \n0.173 | \n
1 | \n0.35 | \n0.173 | \n
1.5 | \n0.34 | \n0.163 | \n
2 | \n0.31 | \n0.295 | \n
3 | \n0.28 | \n0.245 | \n
4 | \n0.21 | \n0.195 | \n
5 | \n0.18 | \n0.150 | \n
6 | \n0.12 | \n0.110 | \n
7 | \n0.10 | \n0.095 | \n
8 | \n0.09 | \n0.075 | \n
9 | \n0.06 | \n\n |
\n | Total area under curve | \n1.758 | \n
\n | Area under curve to nonlinearity | \n0.258 | \n
\n | Maximum % of customers out | \n0.350 | \n
Resiliency index calculation for Hurricane Wilma.
Day. | \n% Out of service (Hurricane Irma 2017) | \nArea | \n
---|---|---|
0 | \n0 | \n0.050 | \n
0.5 | \n0.20 | \n0.150 | \n
1 | \n0.40 | \n0.260 | \n
1.5 | \n0.64 | \n0.300 | \n
2 | \n0.56 | \n0.480 | \n
3 | \n0.40 | \n0.355 | \n
4 | \n0.31 | \n0.255 | \n
5 | \n0.20 | \n0.155 | \n
6 | \n0.11 | \n0.090 | \n
7 | \n0.07 | \n0.055 | \n
8 | \n0.04 | \n0.025 | \n
9 | \n0.01 | \n\n |
\n | Total area under curve | \n2.175 | \n
\n | Area under curve to nonlinearity | \n0.560 | \n
\n | Maximum % of customers out | \n0.640 | \n
Resiliency index calculation for Hurricane Irma.
Change in resiliency is found by Eq. (3).
\nThe determination of a change in resiliency allows for a quantitative measurement related subsystem response. The use of resiliency indices can aid in quantifying the efficacy of resiliency investment.
\nIn this chapter, a comparison to mean-based reliability was contrasted with the use of resiliency calculations for HILP events. Resiliency calculations are required, given the infrequent nature of protracted subsystem disturbances. Following a review of resiliency computations, a gap analysis was used to identify the opportunities for ensuring a resiliency calculation can capture the nonlinearities observed in empirical data. Parallels are provided between the modulus of resilience construct from materials science and an isomorphic application defined. In conclusion, an example is presented for the power utility sector demonstrating the methods of collecting the inputs and completing the computations. These inputs include defining the aim of the system and failure point, data collection, determination of bifurcation point, and the use of reliability data for calculating a change in length.
\nThe ability to calculate resiliency regardless of the subsystem or scenario can assist in the evaluation of resiliency actions already taken or planning for new investment. The ability to compute resiliency on a common base may also offer opportunities to optimize investment based on interconnectedness to the subsystems which yield the greatest improvement. A more integrated approach may lead to increased systemic resiliency as opposed to more common heuristics-based subsystem specific approaches. The proposed method more closely adheres to the ontological and conceptual frameworks associated with initial references of resiliency. Furthermore, subjective inputs are avoided increasing the replicability and repeatability of associated research. By acknowledging a yield point specific to the aim of the subsystem, results from the resiliency index better represent the outcomes of real-world subsystems. Lastly, bifurcating the event curve allows the onset characteristics of the disruptive event to normalize the resiliency performance metric.
\nFurther research on the distribution of events by type will be conducted to validate the anecdotal evidence regarding common cause and special cause events. This additional data will assist in the development of statistics for assessing the correlation between increasing interdependence and HILP events for critical subsystems. In order to test a wider array of empirical data sets, resiliency indexes will be calculated using both historical and future HILP event data. The results of these analyses will be used to continually evaluate the efficacy of the metrics and identify opportunities for enhancements.
\nhigh impact, low probability
\nmean time between failure
\nmean time to failure
\nmean time to repair
\ndepartment of energy
\nFederal Emergency Management Agency
\nenergy information administration
\nsystem average interruption duration index
\nresiliency index
\nLicense
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