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. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of 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:"71236",title:"Wave Spectra in Dusty Plasmas of Nuclear Fusion Devices",doi:"10.5772/intechopen.91371",slug:"wave-spectra-in-dusty-plasmas-of-nuclear-fusion-devices",body:'A partially or fully ionized gas containing neutral atoms, electrons, ions and with the addition of dust particles is known as dusty plasma (DPs). This additional component (dust particle) increases the complexity in the behavior of the system, and thus refers to this system as “dusty” or complex plasma [1]. Nowadays the term “complex plasma” is commonly used in literature to differentiate the dusty plasma. Due to embedded dust particles in the plasma, these dust particles create changes in the charge composition. The new physical processes were introduced in the system such as the recombination of plasma on the particle surface and effects associated with the degeneracy, fluctuations of particle charges which also change the energy and transport phenomena. That’s the way the DPs became a new type of non-Hamiltonian systems. The presence of dust particles in the complex plasma is vital for the collective processes. These micro size dust particles create very low-frequency wave mode, which represents charge particle oscillations against the quasi-equilibrium background of ions and electrons. Generally, a dynamical time scale related to the dust component is in the range of 10–100 Hz. Recently the dusty plasma becomes an interesting field for researchers, technologist and scientists [2, 3]. The initial challenge of fusion in nuclear devices is to confine ionized isotopes of hydrogen atom known as plasma, increase the plasma pressure to initiate and sustain the fusion reaction [4].
Dusty plasma is classified on the basis of density, temperature, potential, and thermal energy. For classification of dusty plasma first, we define Coulomb coupling parameters. Coulomb coupling defines as “the ratio of average potential energy to the average thermal energy of neighboring charged particles” and mathematically written as
Weakly coupled dusty plasmas have higher average thermal energy than average potential energy due to neighboring charged particles. WCDPs have a high temperature, low density and value of Coulomb coupling parameter less than 1
In strongly coupled dusty plasma (SCDP), the average potential energy of neighboring charged particles is dominated on the average thermal energy of the same charged particles. This type of dusty plasma has high charged particles density and low temperature. The SCDPs speedily become emerging fields from last three decades. Due to the presence of dust particles in the atmosphere, the dusty plasma becomes a very significant research field of astrophysical plasma and also in nuclear fusion devices [7, 8, 9]. At higher density and low temperature, the SCDPs undergo in crystallization phase. In Coulomb coupling systems, the SCDPs change phase from liquid to crystal phase at specific values of the Coulomb coupling parameters. SCDPs known as a warm liquid at Γ = 5, liquefy at Γ = 80, it becomes cold liquid at Γ = 100, very cold (Γ = 120) and then liquid phase has a limitation at Γ = 137. The SCDP has a crystalline form at Γ = 175 value; it has very high density and very low temperature [10, 11]. SCDPs appear in many astrophysical objects such as neutron star crusts, white dwarf interiors, supernova cores, and giant planetary interiors. Charged particles in dusty plasma are also found in many physical systems such as condensed matter systems of liquid metals and molten salts, cryogenic traps, electrons trapped on the free surface of the helium. DPs play a very important progressing role in laboratory experiments. Nowadays, recently the dusty plasma plays a very significant role in the nuclear fusion devices for plasma confinement and to control the fusion reaction [39]. For basic experiments in the laboratory, the DPs shows very interesting phenomena such as melting and formation of crystals, collective modes excitation with reference to dust component. Nanostructure layered, and colloidal suspension of dusty plasma was also investigated in these set of references [12, 13, 14].
Dust is present everywhere in the space and environment. Dust particles are much larger than electrons, ions, and neutral particles. Their sizes of dust particles vary from hundreds of millimeters to 10 nm and the mass of dust particle is approximately 7.53 × 10−10 kg. Their dynamic behaviors are easily observed through a CCD camera because of temporal and spatial scale. These dust particles are mostly negatively charged; however, sometimes they have positive charged also, that depends upon the charging phenomena. The large shielding clouds are created for balance the ion thermal current and electron thermal current. Charging phenomena of dust particles are photoionization, electron bombardment, sputtering, etc. The amount of charge at dust particles depends upon shape and size. Most often, they have spherical shape; however, sometimes they are of the form of rod type and or irregular shape [12, 15]. Dust particles are exposed to ion and electron currents from discharge plasma that’s why reached quickly in dynamical equilibrium. The electric charge on the dust particle depends upon their radius (size) and shape, and the charge amount is in the order of 103–104 electrons. Increasing the charge on the particles increases the electrostatic repulsion between them in a confined system and may lead to crystallization [16]. The dust particles are strongly coupled due to high electric charges and unable to move easily so that they look like a solid and liquid phase in the DPs. The phonon spectra in the DPs are easily calculated due to the thermal motion of dust particles [17]. The motion of dust particles generates the longitudinal and transverse waves in the dusty plasma. Due to the complex behavior of a dust particle in DPs, it becomes an independent field for researchers whose study dusty plasma with strong correlation. The charged dust particles are highly susceptible to the different forces in the plasma such as the electric field, neutral and ion drag and can serve as sensitive diagnostic tools [18].
DPs are found in the ionosphere, that’s a lower part of the earth. Noctilucent clouds (NLs) are composed of ice and dust from manmade pollution and heavy clusters. In the space environment, the examples of dusty (complex) plasma are Jupiter rings, were first observed in 1779, comets, planetary rings and spokes, Saturn’s rings and Neptune. The size of dust particles in Saturn ring varies from micron to sub-micron. The radial spokes also consist of micron and submicron sized dust charged particles that are electrostatically levitated. The presence of dust particles in the atmosphere at the altitudes in the range of 80 and 90 km was observed during polar summer mesopause [19]. The presence of dust particles was observed in the nuclear fusion devices, both Tokamaks and stellarators. Due to the presence of dust particles in these devices may disturb the performance and stop working on it. Nowadays, study of dust particles in fusion devices becomes very important. The charging mechanisms of dust particles in these devices are also investigated by Liu et al. [20]. It becomes very necessary for operational Tokamak or other fusion devices to study and found waves and transport properties of dusty plasma. Thermal conductivity, diffusion coefficient, shear viscosity in dusty plasma and charging mechanisms of dust particles in nuclear fusion devices are also needed to investigate [21]. The dust particles are also observed in radio frequency (RF) device and direct current (DC) glow discharge tube and Z-Pinch device etc. Under the laboratory condition, the plasma crystals are observed in different devices such as in RF, DC, thermal plasma, nuclear-induced dusty plasma over wide range of plasma parameters [22].
Fusion energy is a source of energy for a future generation which is almost inexhaustible. Currently, it is an undefeatable challenge for engineering and thermophysical researchers. The basic challenge to achieve the fusion energy is “to achieve a rate of heat emitted by fusion plasma that exceeds the rate of energy injected into the plasma”. The central expectations are focused on two fusion reactor devices, one is Tokamak and the other is stellarator. Today the whole world community is working for nuclear fusion device, which is known as Tokamak. Fusion energy is investigated and comes closest to the explosion. These devices consist of a ring-type magnetic field used to confine the plasma. Tokamak plasma is confined by an electric current flowing in plasma, and in the stellarators, a magnetic field of very complicated shape used to confine plasma stationary. The Tokamak work only in the pulsed mode without auxiliary facilities and stellarators is suitable for continuous operation. The most effective magnetic field configuration is toroidal in the shape of the doughnut. The Tokamaks, stellarators and the reversed field pinch (RFP) are commonly under developing fusion nuclear devices based on toroidal confinement configuration. The Z-pinch is also nuclear fusion device in which is a strong electrical current in plasma to generate X-rays. The magnetized target fusion, referred to as a MIF (magneto-inertial fusion) system, is also currently in progress. In these nuclear devices, a magnetic field is applied to confine the plasma with the help of electromagnetic or mechanical linear implosion. A compression heating is provided with laser hot dense magnetized plasma which is created in the plasma focus (PF) devices. The PF devices belong to the family of dynamic noncylindrical Z-pinch. If in this device deuterium is used as gases then DD fusion reaction takes place [23, 24, 25, 26, 27].
The working conditions of nuclear fusion devices are such that the fuel of these devices must be heated up to heat fuel in nuclear fusion devices heat in the range of 100 × 108 K temperature, at this temperature the fuel is in the plasma state. The temperature of the plasma is very high, and materials are vaporized that contact with it, that’s why plasma must be confined kept in the magnetic fields. In the Tokamak reactor fuel is use in the range of grams (g), so it is a very safe device. The solid impurities are known as “dust” were also found and investigate Scrape-Off Layer transport that is a key element of edge physics research program. For safety precautions against the dust particles, it is very significant for engineers to predict where the quantity of dust particles increases. To resolve the dust transport problem in fusion devices it is necessary for physicists to develop a fully accurate dust transport code (DTC) [28]. It is also required to calculate the plasma parameters from geometrical relations and engineering constraints of nuclear fusion Tokamak device. Plasma density (n), pressure (p), temperature (T), energy confinement time, β (normalized plasma pressure) as a function of α (minor radius of plasma) are the basic main plasma parameters. In addition, some plasma parameters such as plasma current, bootstrap fraction and kink safety factor are required for a plasma physicist in order to understand Tokamak process. The reactor demands toroidal current I to achieving high energy confinement time (very large) for ignition [3, 26]. There are several techniques used for heating plasma in Tokamak. The most common technique use to heat plasma is Ohmic heating, neutral beam injection, RF heating. The fusion plasma has such as high temperature so that they emit little visible light [29, 30].
To understand dynamical information and basic properties of gas, liquids, and solids, it is compulsory to study the basic two phenomena such as phase transition and waves [31]. Dust particles in SCDPs support longitudinal (compressional) waves, also known as dust acoustic waves (DAWs) and transverse waves (shear) [12]. The propagations of longitudinal modes are faster than the transverse mode in the crystalline phase of dusty plasma. The WCDPs does not sustain the transverse wave and only sustain longitudinal waves. The compressional electrostatic waves and DAWs have low-frequency modes due to a larger mass of dust particles. In order to study the thermal motion of dust particles through MD simulation and it was found that cut off wave number is calculated for transverse mode near the solidification phase of dusty plasma [32]. The generalized hydrodynamics (GHD) model of the equation is predicted by the existence of transverse wave mode in the liquid and strong coupling regimes and dispersion properties of longitudinal modes [18]. Investigation of dusty longitudinal waves (DLWs—dust lattice waves) in two-dimensional bi-crystal in an arbitrary direction and it was found that hybrid modes have both components along with transverse and longitudinal directions. The hybrid modes become purely transverse to longitudinal waves for the angle of propagation is 0 or π/2 [33]. Background of the colloidal suspension liquid exerts large friction on the motion of charged particles than the background of dusty plasma gases. Due to low friction between charged particles in the gas phase of dusty plasma waves damped slowly. The complex (dusty) plasma the current correlation functions of complex (dusty) plasma are classified into the longitudinal current and transverse current, also known as longitudinal and transverse (shear) wave’s mode. In the classical fluids, when k approaches zero then longitudinal modes known as acoustic modes. Strongly coupled plasma in the liquid phase supports shear maintained transverse mode. In SCDPs, when k approaches zero then transverse modes are also considered approach as acoustic modes [34] (Figure 1).
Directions of longitudinal and transverse waves in DPs relative to the direction of wave numbers vectors. The direction of the wave vector shows that direction of CL(k, t) is along the wave vector and direction of CT(k, t) is perpendicular to the wave vector (k) [35].
The uniform liquid phase does not support transverse modes of waves. The reason for this is to ignore the migration of diffusion damping. For isotropy liquid, the transverse mode approaches the same Einstein frequency ωE as a longitudinal mode, when the wavenumber k approaches to infinite. The current correlation functions of DPs are studied theoretically, numerically and experimentally. The results are in good agreement with the theoretical prediction, in support of simulation measurements and also verified by experiments [13, 15, 23].
An MD simulation is a tool that studies the microscopic model in a macroscopic system, and this model is quantified in terms of intermolecular interaction and the molecular structure. The results are obtained with accuracy through different simulation techniques (algorithm) and compared with theoretical and experimental results. Simulations are also used to study the wave properties of complex models at the microscopic level which cannot be investigated by experiments [36]. There are several computational techniques that have advantages and also disadvantages with their respective fields. Monte Carlo (MC) and molecular dynamics (MD) simulations are influential tools for the study of transport properties of dusty plasma. Transport properties can also be calculated by Langevin dynamics (LD), MC, path integral MC (PIMC) and MD methods. The disadvantage of the MC technique is that it cannot evaluate the transport properties of dynamical systems and cannot solve and apply the equations of motion [37].
In this chapter the EMD Simulations are performed for a selected system, having the number of particle N = 500 with apply periodic boundary condition (PBCs) on the cubic box in three dimensions coordinates directions. These particles are placed in a cube volume V and interact with each other by pairwise Yukawa potential is given:
Q is the charge, on dust particles, ε0 is permittivity of free space, λD is the Debye length which accounts for the screening of the interaction by other plasma species. The dimensionless plasma parameters have fully characterized the system under study. One is Coulomb coupling parameter and defines as Γ = Q2/4πε0awskBT (already defined in Section 1.1), where a is the Wigner-Seitz radius and is define as aws = (3/4nπ)1/3 with n is the dust particle density, T is the temperature of the system and kB is Boltzmann constant. The screening parameter is and defines as κ ≡ aws/λD. In an EMD technique, Newton’s motion equation is, m(d2r/dt2) = Fi = ΣjFij, integrated numerically for N Yukawa particles with mass m positioned at ri, velocity vi and acceleration ai in the volume (V) of simulation box of particle i (i = 1, 2, 3…..N) is exerted a force on other particle j and it is given as Fi = Σj Fij and i ≠ j. The EMD is performed in the microcanonical ensemble (NVT) for constant volume and temperature [38]. In this chapter, the EMD has been used to investigate the time-dependent current correlation functions [CL(k, t) and CT(k, t)]. The dimension of the simulation box is Lx, Ly, LZ. The periodic boundary condition is used to minimize the surface size effect and applied to the simulation box. The main calculation is performed for N = 500 particles at κ = 4.5 and 5.5, plasma coupling parameters Г (temperature of the Yukawa system) varies from 1 to 100 and wavenumbers k = 0, 1, 2, and 3. The simulation time step is taken as ∆t = 0.001 to allow computing the important data for sufficient 425,000 simulation run. EMD method is reported of the current correlation of SCDPs over sufficient domain of plasma parameters of Debye screening (4.4 ≤ κ ≤ 5.5) and Coulomb coupling (1 ≤ Γ ≤ 100).
The SCDPs support both longitudinal and transverse waves. The experimental importance of time-dependent correlation function is that the spectroscopic technique an example of this technique is neutron scattering. Investigate microscopic dynamical quantities through the MD approach and then comparison by Fourier analysis of the simulation result. The local density gives information about the atom’s distribution. There is also possible to analyze the motion of atoms. The Fourier component of Particle current or momentum current for a single atomic particle in MD unit is given as.
where vj and rj are the velocity and position of a jth particle, by using the Fourier transformation of particle current becomes as for a given wavenumber vector (k).
The correlation function of the current vector component is defined as
For the isotropic fluid under consideration of symmetry above equation can be expressed in term of longitudinal current correlation and transverse current correlation in the relative direction of k, where k is the wave vector and equal to multiple of integers k = 2π/L and L is the size of the simulation box. Wave vector k becomes equal to k = 2π/L (k0, k1, k2, k3), kj ϵ Z, j = 0, 1, 2, 3; L is the length of simulation box and V = L3.
Here x, y, and z are integers.
By putting k = kz the time-dependent longitudinal current correlation becomes as.
Where Nm represents the number of particles, vi and vj are the velocity of the ith and jth particles, <….> gives the statistical average of particle current. Longitudinal current correlation function explains the direction of the waves along the wave vector (wavenumber) and a transverse direction perpendicular to k.
The longitudinal current correlation also related to the dynamical structure factor.
In Eq. (9), the dynamical structure factor and longitudinal current correlation contain the same physical information of the systems. Transverse current and longitudinal current also explain the wave spectra in 3D SCDPs. In our EMD simulation model, the current correlation function is the only function of wavenumber and time (k, t). Through this mathematical model of current correlation, we checked out variation in frequency and peak amplitude of transverse and longitudinal waves in SCDPs for at Г, κ, N, and k [13, 18, 31, 34].
In this section, we describe the consequences of extensive MD simulations methodology work, carried out to explore the current correlation functions (compressional and share waves) of 3D Yukawa liquid via the EMD simulation technique. CL (k, t) (Eq. 7) and CT (k, t) (Eq. 8) is simulated at an extensive combinational range of parameters (Γ, κ, N and k). The CL(k, t) and CT(k, t) which are normalized by plasma frequency (ωp) has been extensively used for prior studies of SCDPs but while here we are investigating its correspondence with time (t). The information waves spectra for nuclear fusion device conditions is generated from simulation goes, that prediction which is true for frequency spectra, current correlation function [CL(k, t), CT(k, t)] simulation are executed for higher screening strength of spherical charged dust particles (κ = 4.5 and 5.5) and Coulomb coupling parameters (inverse of plasma temperature) parallel closely same experimental plasma state (κ, Γ). This was executed to facilitate comparison with presented simulation results and available data of recent and earlier.
In this section, we present our EMD simulation results and their discussion of wave spectra from the current correlation function in the longitudinal and transverse wave’s modes. The specific attention in this chapter is given to CL(k, t) and CT(k, t) for a different combination of plasma parameters which are investigate the behavior of transverse and longitudinal waves in 3D SCDPs. Explanation qualitatively features of the longitudinal (compressional or sound) waves in 3D complex (dusty) plasma shown in Figures 2 and 3. Here our EMD outcomes we compute the CL(k, t) for κ = 4.5 and 5.5 for a number of particles (N = 500) and in the direction of wave vector numbers (k = 0, 1, 2 and 3). We determined properties of longitudinal waves in SCDPs at a different combination of plasma parameters (κ, Γ), the results have plotted the magnitude of CL(k, t) against simulation time (t). In our EMD simulations result, the effect about plasma temperature is observed on the magnitude, wavelength, frequency, and damping phenomena of waves in SCDPs. Figure 2 consists four panels which covering from non-ideal to the liquid and then liquefy state of dusty plasma. The panel (a) of both Figures 2 and 3 represent the results of longitudinal wave spectra in the non-ideal state of dusty plasma at κ = 4.5, 5.5 respectively. It is observed from first panel of these two figures that collective modes of wave spectra are highly damped due to high temperature of dust particles confirmed good agreement with earlier published worked by Nunomura due to collisions between particles [29] and Shahzad et al., for low screening strength [2]. The modes of these waves in amplitudes of longitudinal waves increased at higher wavenumber (k = 3) and also at lower wavenumber has low peak amplitude mode clearly seems from results plotted in Figures 2 and 3. The damping of waves at a higher temperature would attribute this to viscous/collision and Landau damping. The effects of Γ on the propagation of waves in SCDPs are observed from four panels of Figure 2. The frequency modes are increases and amplitude decreases of CL(k, t) with increasing Γ. With increasing Γ the thermal effect decreases in the magnitude and the correlation effect clearly seems. Figures 2 and 3 show the wave’s spectra of longitudinal mode at different values of coupling which covering the non-ideal phase dusty plasma and also liquefy state. It is observed that the damping effect in wave’s mode decrease with decreasing the plasma temperature. Here higher damping at Γ = 1 and at Γ = 5 comparatively low damped and then oscillate very at a low magnitude for Γ = 5. If we further increase the coupling values the damping decrease and longitudinal waves propagate in form of sinusoidal. At Γ = 20 and 100 waves in longitudinal modes properly propagate and with the passage of time, the magnitude of waves decreases we can observe Figures 2 and 3 (d) at Γ = 100 in the liquid liquefy phase.
EMD simulations results of wave spectra of the longitudinal mode against the simulations time (t) for κ = 4.5 covering from non-ideal to liquid states ((a) Γ = 1, (b) Γ = 5, (c) Γ = 20 (d) Γ = 100) of 3D SCDPs and N = 500 for higher wave number (k = 0, 1, 2 and 3).
EMD simulations results of wave spectra of the longitudinal mode against the simulations time (t) for κ = 5.5 covering from non-ideal to liquid states ((a) Γ = 1, (b) Γ = 5, (c) Γ = 20 (d) Γ = 100) of 3D SCDPs and N = 500 for higher wave number (k = 0, 1, 2 and 3).
There is a slight effect of screening strength (κ) on longitudinal wave’s mode in plasma with respect to damping and propagation phenomena. The frequency and amplitude of waves in SCDPs are high at lower values of κ when we increase screening strength at the same N, k and Г amplitude and wavelength are gradual decreases with respect of κ. The magnitude of CL(k, t) 0.2332, 0.0379, 0.0057 and 0.0014 for Γ = 1, 5, 20 and 100 respectively at κ = 4.5 and k = 0. When we increase value of screening κ = 4.5 to = 5.5 then the magnitude of CL(k, t) at k = 0 increase 0.2244, 0.0508, 0.0105 and 0.0016. In this chapter, we have simulated at an N = 500 for the same combination of parameters (N, κ, Г).
In this part, we have investigate CT(k, t) through EMD simulations for 3D SCDPs coulomb system in the classical ensemble (NVT) for N = 500 particles. The charged particles are interacting with each other via Yukawa pair potential. We have analyzed the behavior of the wave’s spectra in the transverse (shear wave) direction in SCDPs, using Eq. (8), by EMD simulations technique. It is found that our results is calculated for CT(k, t) are in a good agreement using the EMD algorithm over an extensive range of plasma parameters and selecting a number of particles. We have ensured that the presented results of CT(k, t) are in satisfactory agreement with prior known simulation, theoretical and numerical results. In our MD simulation result, the effect of plasma temperature is observed on amplitude, wavelength, frequency, and propagation of waves in SCDPs.
Figures 4 and 5 demonstrate the Simulation results which are obtained for CT(k, t) of SCDPs plasma using EMD simulations at k = 0, 1, 2 and 3, Г = 1, 5, 20 and 50 for κ = 4.5 and 5.5. The given simulation results of CT(k, t) spectra are compared with increasing and decreasing sequences of κ, Г, and k. It is observed that the magnitude of transverse current waves has increases behavior with increasing wave numbers. We have observed from further EMD simulations calculation that magnitudes of this wave have decreasing behavior for a high number of particles. The absence of any structure of strongly coupled plasma in the liquid phase the shear waves are not propagated through it. Wave’s in SCDPs are highly damped at high plasma temperature or low values of coulomb coupling. In Figure 4 plotted four penal of shear waves which covering from non-ideal to liquid state of dusty plasma.
EMD simulations results of wave spectra of the transverse mode against the simulations time (t) for κ = 4.5 covering from non-ideal to liquid states ((a) Γ = 1, (b) Γ = 5, (c) Γ = 20 (d) Γ = 100) of 3D SCDPs and N = 500 for higher wave number (k = 0, 1, 2 and 3).
EMD simulations results of wave spectra of the transverse mode against the simulations time (t) for κ = 5.5 covering from non-ideal to liquid states ((a) Γ = 1, (b) Γ = 5, (c) Γ = 20 (d) Γ = 100) of 3D SCDPs and N = 500 for higher wave number (k = 0, 1, 2 and 3).
The panel (a) of both Figures 4 and 5 represent the results of CT(k, t) spectra in the non-ideal state of dusty plasma. It is observed from these figures that collective modes of wave spectra are highly damped at higher plasma temperature and damping of transverse waves reduce with corresponding to decrease the plasma temperature. In the non-ideal state of dusty plasma, the transverse current wave’s mode is highly damped as compare to longitudinal modes especially at higher wave’s number in our case. CT(k, t) spectra having increasing behavior in the magnitude for wave vector. The damping of waves at a higher temperature would attribute this to viscous/ collision and Landau damping that confirmed from earlier published work of [2, 29]. The effects of Г on the propagation of waves in SCDPs are observed from four panels of Figure 2. The frequency modes are increases and amplitude decreases of CL(k, t) with increasing Г. With increasing Γ the thermal effect decreases in the magnitude and the correlation effect clearly seems. Values of CT(k, t) at different parameters as for k = 0(1), CT = 0.2558(0.9016), 0.0454(0.1749), 0.0141(0.0449), 0.0017(0.0071), and for the case k = 2(3) as CT = 2.3453(3.9837), 0.4218(0.7214), 0.0810(0.1260), 0.0144(0.0229), at Г = 1, 5, 20, 100 respectively for κ = 4.5. With the comparison of Figures 4 and 5, we have observed that there is a slight difference occurs when we increase the screening of dust charged particles. The propagation and damping behavior of waves in transverse direction remain nearly same with increasing screening. For particular wave’s number (k), magnitudes of transverse waves have increasing behavior with increasing screening strength.
The EMD simulations are used to investigate the CL(k, t), and CT(k, t) for 3D SCDPs over an extensive range of plasma parameters κ, Г, and k (wavenumber), N (number of particles in simulation box) and k. The first involvement of the presented simulation is that it delivers an understanding of propagation and damping phenomena of waves in SCDPs. In general, the amplitude, frequency of waves is analyzed. The presented simulation specifies that the waves are highly damped at high temperature Г = 1, frequently propagates at intermediate and high value of Г = 20–100. This investigation shows that the values of frequency and amplitude depend on κ, Г, N, and k. It has been shown that the presented EMD method and earlier EMD techniques have comparable performance over the wide range of plasma points, both yielding reasonable results for correlation parameters. New simulations give more reliable and excellent data for the CL(k, t), and CT(k, t) for a wider range of κ and Г are (4.5, 5.5) and (1, 100). The existing simulations deliver more reliable data for existence for waves in SCDPs. In the absence of structure in dusty plasma, the shear wave does not support. The sound wave frequently propagates at medium and higher values of Г in SCDPs. It is suggested that the presented EMD technique based on Ewald summation described here can be used to explore other ionic and dipolar materials. It is very interesting what other types of interaction potentials support correction parameters of strongly coupled plasma and how its strengths depend on the range of new interaction potentials.
The authors thank the National Advanced Computing Centre of National Centre for Physics (NCP), Pakistan and National High-Performance Computing Center (NHPCC) of Xian Jiaotong University, P.R. China for allocating computer time to test and run our MD code.
equilibrium molecular dynamic
screening strength
Coulomb coupling
wavenumber (wave vector)
number of particles
dusty plasma
strongly coupled dusty plasmas
three dimensional
longitudinal current wave
transverse current wave
complex dusty plasmas
weakly coupled dusty plasmas
noctilucent clouds
reverse field pinch
magnetized target fusion
plasma focus
magneto-inertial fusion
Boltzmann constant
plasma density
plasma pressure
plasma temperature
normalized plasma pressure
the minor radius of the plasma
plasma toroidal current
dust acoustic wave
periodic boundary condition
radiofrequency
discharge current
dust transport code
molecular dynamic
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
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