Читать книгу: «Cyber-Physical Distributed Systems»

Modeling, Reliability Analysis and Applications

Huadong Mo

School of Engineering and Information Technology, University of New South Wales

Canberra, Australia

Giovanni Sansavini

Institute of Energy and Process Engineering

Department of Mechanical and Process Engineering

ETH Zurich

Zurich, Switzerland

Min Xie

Department of Systems Engineering and Engineering Management

City University of Hong Kong

Hong Kong, China

This edition first published 2021

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Preface

A cyber‐physical system (CPS) consists of a collection of computing devices communicating with one another and interacting with the physical world via sensors and actuators in a feedback loop. Increasingly, such systems are everywhere, from smart buildings to medical devices to automobiles. The emergence of CPSs as a novel paradigm has revolutionized the relationship between humans, computers, and the physical environment. CPSs are still in their infancy, and most recent studies are application‐specific and lack systematic design methodology. As a result, it is challenging to investigate and explore the core system science perspective needed to design and build complex CPSs, which are of great importance in many applications.

Using the underlying theories of systems science, such as probability theory, decision theory, game theory, control theory, data analysis, organizational sociology, behavioral economics, and cognitive psychology, this book addresses foundational issues central across CPS applications, including: (I) System Verification – How to develop effective metrics and methods to verify and certify large and complex CPSs; (II) System Design – How to design CPSs to be safe, secure, and resilient in rapidly evolving environments; (III) Real‐Time Control and Adaptation – How to achieve real‐time dynamic control and behavior adaptation in diverse environments, such as distribution and in network‐challenged spaces; (IV) System of Systems – How to harness communication, computation, and control for developing new integrated systems, reducing concepts to realizable designs, and producing integrated software–hardware systems at a pace far exceeding today's timeline.

In general, this book has four essential topics. Chapters 1 and 2 provide readers who do not have a sufficient background on CPSs with a general introduction, research gaps, and representative CPS applications, including CPS modeling, statistical analysis of CPS performance, probability prediction of CPS state, robust CPS control techniques, and management and optimization of CPS reliability and risk. Chapters 3 and 4 mainly concern the robust control of CPSs by designing optimal control strategies, or resource management to enhance robust performance and improve the reliability index against time delays and packet dropouts, which are the inherent properties of open communication networks. Chapter 5 addresses the data‐driven degradation modeling of aging physical (actuators) and cyber (sensors) components of CPSs, and corresponding optimal maintenance plans to improve the reliability of CPSs. Chapters 6 and 7 investigate the cyber security of CPSs, introduce the general concept of cyberattacks, design vulnerability models, and risk assessment procedures, and develop game‐theoretic mitigation techniques and Bayesian‐based cyberteam deployment strategies.

More specifically, Chapter 1 summarizes the evolution from the traditional physical system to the CPS and provides an overview of dynamic and dependent behaviors to be addressed in the subsequent chapters of the book. The introduction discusses some important and recent challenges in improving traditional physical systems in terms of CPSs, popular research trends in evaluating the impacts of CPSs on society, and opportunities for enhancing the performance of realistic applications, which are primarily network control systems. The detailed properties, requirements, and vulnerabilities of utility systems are also introduced. The reasons why the proposed modeling techniques work is important in a field that would be difficult to deal with if the cyber and physical domains were treated separately.

In Chapter 2, readers acquire the basic knowledge to be used in data‐driven statistical modeling, the estimation of the probabilistic CPS state, and a comprehensive framework for conducting reliability analysis of CPSs. In addition, this chapter introduces how to use to historical data to validate the performance of the proposed CPS model, and how to use performance indexes to facilitate the resilient design of CPSs. Moreover, it also demonstrates a real‐time test platform for various industrial applications and the standard procedures for improving real‐time criteria.

Chapter 3 focuses on the stability of CPSs, where decision makers perform dynamic control and adaptation based on real‐time data from sensors. It provides two examples of the design of the controller parameters for robust system performance. The first example illustrates the development of adaptive control for wide‐area measurement power systems, where communication delays are predicted to provide delay compensation for additional frequency stability. In addition, the integration of control theory, power engineering, and statistical estimation is discussed. The second example is an extension of wide‐area measurement power systems from a dedicated communication network to open communication networks, where occurrences of communication delays and packet dropouts result in the failure of the power management system from renewable energy resources. Explicit and implicit methods are then designed for system integration, analysis, and improvement.

Chapter 4 illustrates a system‐of‐systems framework for the reliability of distributed CPS accounting for the impact of degraded communication networks. This is quite different from the focus of Chapter 3, which mainly covers the stability of CPSs from a control perspective. Based on the collected dataset, the degradation path of open communication networks is described in terms of stochastic continuous time transmission delays and packet dropouts. A distributed generation system with open communication infrastructure is used as an example, which is a multi‐area distributed system that is more complicated than the single‐area power system presented in Chapter 3. An optimal power flow model is proposed to generate consecutive time‐dependent optimal operation scenarios for a distributed CPS. Quantitative analysis is carried out to evaluate the effect of networked degradation on the reliability indexes of CPSs, e.g., energy not supplied and operation cost. A prediction method for reconstructing missing data is proposed to mitigate the influence of packet dropouts, which is universal and applicable to most current industrial applications.

Chapter 5 models the functional dependence between stochastic aging actuators and sensors within their operating environments. This dependence is considered in the time domain, causing a distinct degradation status in the actuators and sensors. Reliability modeling of the stochastic effects and effective maintenance activities are discussed for different types of CPSs, including the cooling system in a nuclear power plant, a one‐area energy system with a single generation group, and a multi‐area energy system with several different generation groups.

Chapter 6 explores the concepts, principles, practices, components, technologies, and tools behind risk management for cybersecurity of CPSs, providing practical experience through a realistic case study that focuses on the methodologies available to identify and assess such threats, evaluate their impact, and determine appropriate measures to prevent, mitigate, and recover from any threat or disruptive event so that the operations and profitability of the organizations are maintained and maximized.

Chapter 6 presents the framework of CPSs under cyberattacks from a game‐theoretic perspective, which makes use of statistical data to model the behavior of cyberattacks and study the dynamic game between the network defender and attacker at the system level. For current utility CPSs, cyber threats from supervisory control and data acquisition (SCADA) systems, and spear‐phishing attacks on the accounts of internal employees to gain access to dedicated communication networks are investigated in Chapters 6 and 7. In addition, these chapters focus on how to modify the basic modeling techniques presented in Chapter 2 to describe cyber vulnerabilities, such as seizing the SCADA system under control, disabling/destroying IT infrastructure components, and denial‐of‐service (DoS) attacks on the control center in smart grids. Based on historical data for IT security spending, the cost of launching distributed DoS attacks, and the occurrence probability of cyber event losses, the contest intensity between the attacker and defender can be accurately predicted for the next period to guide the design of an effective network protection plan, that is, a game‐theoretic protection plan and a Bayesian‐based cyberteam deployment.

Chapter 7 investigates sequential control problems (i.e., sequential cyberteam deployment) in modern CPSs by introducing an adversarial cost sequence with a variation constraint. Chapter 6 reviews the data‐driven vulnerability model, and Chapter 7 deals with the dataset of the arrival time of cyberattacks, which uncovers the statistical pattern of attackers. To solve such problems, a fundamental idea is to first obtain sampled parameters for the arrival model of cyberattacks from the posterior distribution of realistic cyberattack arrival records. The reinforcement learning model for estimating parameters is formulated as a partly parameterized Bayesian model. As a result, the sampled parameters are used instead of the true parameters. The paradigm of this framework can also be applied to other classical models, although specific models are used here for illustration purposes. Next, a Bayesian multi‐node bandit is built to cope with the problem, and an online learning algorithm (the Thompson‐Hedge algorithm) is forwarded to retain a converging regret function that is a function of the cyberteam deployment. By comparison with the existing algorithm, the convergence rate of the regret function in the proposed algorithm is found to be superior.

Each of Chapters 3–7 can be read independently when one is interested in a specific type of application or further research, making the book attractive to readers from different areas or positions. This book has the following distinct features:

• It is the first book to systematically focus on CPSs with respect to modeling and reliability analysis.

• It provides a comprehensive treatment of imperfect fault coverage (single level/multi‐level or modular), functional dependence, common cause failures (deterministic and probabilistic), competing failures (deterministic and probabilistic), and dynamic standby sparing.

• It includes abundant illustrative examples and case studies based on real‐world systems.

• It covers recent advances in combinatorial models and algorithms for CPS modeling and analysis.

• It has a rich set of references, providing helpful resources for readers to pursue further research and study of the topics.

The target audience of the book is undergraduate (senior level) and graduate students, engineers, and researchers in system science and related disciplines, including those in computers, telecommunications, transportation, and other industries. Readers should have some knowledge of basic probability theory, control theory, computer science, optimization, game theory, and stochastic processes. However, the book includes a chapter reviewing the fundamentals that readers need to know for understanding the content of the other chapters covering advanced topics in CPSs and case studies. The book can provide readers with knowledge and insight of CPS behaviors, as well as the skills of modeling and analyzing these behaviors to guide the resilient design of real‐world critical systems. Thus, the book includes necessary background information, making it self‐contained. For some detailed topics, selected for their importance and application potential, references are provided for those interested in further details.

We would like to express our sincere appreciation to the many researchers who have proposed some underlying concepts, frameworks, and methods used in this book, or who have co‐authored with us some topics of the book and provided their insights; to name a few, Professor Enrico Zio from Politecnico di Milano, Professor Yanfu Li from Tsinghua University, Dr. Gregory Levitin from the Israel Electric Corporation, Professor David W. Coit from Rutgers University, and Professor Junlin Xiong from the University of Science and Technology of China. Though there are many other researchers to mention, and we have tried to recognize their significant contributions in the bibliographical references of the book.

Finally, it was our huge pleasure to work with Juliet Booker, managing editor of Electrical & Computer Engineering, John Wiley & Sons Ltd., and her team, who have assisted in the publication of this book. We deeply appreciate their efforts and support.

December 15, 2020

Huadong Mo

Min Xie

Giovanni Sansavini

Acronyms and Abbreviations

ACE | Area Control Error

AERs | All‐Electric Ranges

AGC | Automatic Generation Control

AGC' | Actual Global Cost

B | Biomass Power

BESS | Battery Energy Storage System

CBM | Condition‐Based Maintenance

CC | Control Center

CCC | Centralized Control Centers

CCF | Cross‐Correlation Factor

CPS | Cyber‐Physical System

CSMA/AMP | Carrier Sense Multiple Access with Arbitration on Message Priority

CSMA/CD | Carrier Sense Multiple Access with Collision Detection

CTMC | Continuous‐Time Markov Chains

DC | Direct Current

DEG | Diesel Engine Generator

DERs | Distributed Energy Resources

DG | Distributed Generation

DGS | Distributed Generation Systems

DHMM | Discrete HMM

DoS | Denial of Service

DR | Demand Response

EENS | Expected ENS

EM | Expectation‐Maximization

ENS | Energy Not Supplied

ERP | Expected RP

ESS | Energy Storage System

EV | Electrical Vehicle

EWMA | Exponentially Weighted Moving Average

FDs | Feeders

FERC | Federal Energy Regulatory Commission

FESS | Flywheel Energy Storage System

FFT | Fast Fourier Transform

G | Natural Gas Plant

GA | Genetic Algorithms

GC | Global Cost

GM | Grey Differential Model

Ls | Loads

LAN | Local Area Networks

LC | Long‐Run Cost Rate

LFC | Load Frequency Control

LP | Linear Programming

HAN | Home Area Networks

HGSAA | Hybrid Genetic‐Simulated‐Annealing Algorithm

HMM | Hidden Markov Model

HPS | Hybrid Power System

MAC | Media Access Control

MADT | Maximum Allowable Delay Time

MCR | Maintenance Cost Rate

MCS | Monte Carlo Simulation

MDEM | Missing Data Expectation Maximization

MLE | Maximum Likelihood Estimation

MNB | Multi‐Node Bandit

MS | Main Supply

MSE | Mean Squared Errors

MTU | Maximum Transmission Unit

NAN | Neighbourhood Area Networks

NCS | Networked Control System

NHPP | Non‐Homogeneous Poisson Process

OPF | Optimal Power Flow

O&M | Operation & Maintenance

PBM | Performance‐Based Maintenance

PCLPs | PHEV Charging Load Profiles

PCM | Percentage of Corrective Maintenance

PI | Proportional–Integral

PID | Proportional–Integral–Derivative

PHEVs | Plug‐In Hybrid Electric Vehicles

PMU | Phasor Measurement Units

PO | Percentage Overshoot

PPM | Percentage of Preventive Maintenance

PSO | Particle Swarm Optimization

PV | Photovoltaic Power

RBD | Reliability Block Diagram

RERs | Renewable Energy Resources

RP | Redundant Power

RT | Rising Time

RTU | Remote Terminal Unit

RUL | Remaining Useful Lifetime

SA | Simulated Annealing

SCADA | Supervisory Control and Data Acquisition

SOC | State of Charge

ST | Settling Time

TCo | Total O&M Cost

TENS | Total ENS

W | Wind Power

WAMS | Wide‐Area Measurement Systems

WAN | Wide Area Networks

WAPS | Wide‐Area Power System

WCSS | Within‐Cluster Sum of Square

WLAN | Wireless Local Area Networks

WTG | Wind Turbine Generator

1
Introduction

In this chapter, descriptions of traditional physical and cyber systems are provided to identify existing challenges. Current research trends of cyber‐physical systems (CPSs) are then illustrated to address these challenges. The major applications of the proposed methods in CPSs are reviewed.

1.1 Challenges of Traditional Physical and Cyber Systems

Over the past three decades, studies have addressed numerous concerns regarding the capability of traditional static modeling methodologies, such as the fault tree method and the event tree method, to adequately and quantitatively analyze the impact of hardware and software interaction on the stochastic behavior of CPSs ¹². During the past decade, the dynamical Markov reliability model was proposed to solve similar problems in CPSs ³. Control block diagrams were presented for cooling loop systems. The reliability block diagram (RBD) was then established and used to describe the overall reliability status of individual components in a simplified form ⁴⁵. However, RBDs are incapable of describing the dynamic maintenance and repairable activities; thus, various dynamic modeling methods have been reviewed in ⁶⁷. The Markov methodology has the advantage of tracking the dynamic changes and time‐dependent features of CPSs, and simply integrates all failure states that occur after each working state into one failure state. The Markov methodology eliminates most of the failure states into a system failure state (absorbing node) by conducting a necessary fault injection test and achieving a sparse transfer matrix but may still result in a very large model due to many existing surviving states. Its modeling precision largely depends on the number of fault injection tests, and more cycles yield higher accuracy. To avoid the disadvantages of these two methodologies, some studies have proposed hybrid reliability models combining RBDs and Markov models for CPSs ⁸.

The control block diagram introduces blocks to represent each part of the control system, including the controllers, actuators, and control objectives. Control block diagrams are widely used in modern control systems because they can visually describe the relations among the important components, data flow, and control sign flow. In addition, compared with other mathematical models, they have the advantage of simply reflecting the actual correlations in a CPS. It is reasonable to build a reliability model based on the control block diagram of a CPS. In the model, the controller has many input signals, including commands and system state feedback. In general, commands are the system's expected outputs. Control signal flows are given in the control block diagram, and sensors play an important role in this feedback system. This control block diagram clearly indicates the internal dynamic relations of the system, covering most of the aspects that need to be studied.

For applications in CPSs, we are interested in real‐time performance. Therefore, from a control perspective, the ability to adjust the transient and steady‐state response of a feedback CPS is a beneficial outcome of the design of the CPS. One of the first steps in the design process is to specify the performance measures. In this chapter, we introduce common time‐domain specifications, such as percent overshoot, settling time, time to peak, time to rise, and steady‐state tracking error. We will use selected input signals, such as the step and ramp, to test the response of the CPS. The correlations between the system performance and the stability, reliability, and resilience strategies of CPSs are investigated. We will develop valuable relationships between the performance specifications and the component states for CPSs.

The ability of a feedback CPS to compensate for the consequences of the inherent faults redefines the concept of failures, i.e., the reliability of the CPS is dependent not only on the type of failure that may occur, but also on the evolving states of system output and control signals in each period ⁹¹⁰. Classical reliability evaluation methods, such as fault tree analysis, event tree analysis, and failure mode and effect analysis, are not appropriate for application to these evolving states due to the level of complexity and dynamics of CPSs. In ¹¹¹², structured analyses and design techniques based on Monte Carlo simulation (MCS) for reliability evaluation are presented. This approach explicitly formalizes the functional interactions between subsystems, identifies the characteristic values affecting the reliability of complex CPSs, and quantifies the reliability, availability, maintainability, and safety (RAMS) parameters related to the operational architecture. As the remaining ability of the system to maintain the expected control goal after faults occur is crucial, ordered sequences of multi‐failure methods have been applied to assess the reliability of all possible CPS architectures ¹³. A new methodology called a multi‐fault tree is proposed, and time‐ordered sequences of failures are addressed.

In contrast to the aforementioned studies, the reliability of a CPS as a function of the required performance from a control viewpoint is evaluated in ¹⁴. The CPS is regarded as a failure if the dynamic performance does not satisfy all the requirements. Difference equations are introduced to describe the stochastic model of the CPS, explicitly illustrating the influence of the transmission delays and packet dropouts on changing the model parameters. A linear discrete‐time dynamic approach for modeling the signal flow in, out, and among all subsystems promotes straightforward calculation of fundamental dynamic aspects, such as times and fault characteristics ¹⁵.

MCS has been shown to be a straightforward yet accurate approach for the study of such complex systems ¹⁶¹⁷¹⁸. The general approach in MCS for reliability assessment is to generate operational requirements that lead to the failure of the entire system. However, this approach requires knowledge of the system requirements‐to‐failure distribution in advance. In ¹⁹, an event‐based MCS method was proposed for multi‐component systems, in which the failure time for each component is generated and then used to verify the success or failure of the system subject to the required operational time. Because no attempt is made to generate the failure time for the entire system, which requires knowledge of the time‐to‐failure distribution of the entire system as well as the distribution approximation at the component level, it is quite different from previous methods and can reduce the possible error and computational effort in estimating the system reliability.

In ²⁰, this method was extended to estimate the reliability of CPSs and replaces the constraint on the number of replications used in ²¹ with two other constraints, namely, a precision interval and a percentage of simulations belonging to this interval. The networked degradations for each channel are generated and are then used to determine the success or failure of the CPS for a given combination of operational requirements. Therefore, the reliability of the CPS is estimated as a tabulated function of the operational requirements. Compared with the results in ²², the results obtained in ²³ guarantee the estimated reliability to satisfy a given precision.

1.2 Research Trends of CPSs

1.2.1 Stability of CPSs

In power systems, communication delays always occur in the transmission of frequency measurements from sensors to the control center (S‐C channel), and control signals from the control center to the plant side (C‐A channel) ²⁴²⁵. In local networks, time delays are usually ignored because the control is mainly applied locally, and communication delays are negligible compared with the subsystem time constants ²⁶. In recent years, with the rapid development of wide‐area measurement systems (WAMSs), a large number of phasor measurement units (PMUs) have been deployed to facilitate the real‐time control of wide‐area power systems (WAPSs) to improve the load frequency control (LFC) performance ²⁷²⁸²⁹. However, when data packets are transmitted across a WAMS, communication delays may become significant and cannot be ignored ³⁰³¹³².

The conventional control of WAPSs is centralized and employs dedicated communication channels over a closed communication network. However, new regulatory guidelines require coordination across multiple hierarchical levels of power systems for more effective market operations and, as a result, open communication infrastructures have been deployed to promote the control of these increasingly complex systems ³³³⁴. While open networks have economic, maintenance, and reliability advantages, they are subject to time delays that are inherently stochastic (e.g., multiple delays ³⁵ and probabilistic interval delays ³⁶), and thus cannot be calculated based on the procedures for dedicated networks. Numerical investigations show that time delays the open communication network have the potential to destabilize a WAPS ³⁷³⁸³⁹. For instance, the deregulation of the power industry has pushed many tie lines between control areas to operate close to their maximum capacity. This is especially true for tie lines serving heavy load centers, for example, in southern California ⁴⁰. Under these circumstances, operational stresses, such as large time delays, increase the possibility of inter‐area oscillation, reducing the effectiveness of control system damping, and potentially leading to loss of system synchronism ⁴¹.

Recent studies on the LFC of WAPSs with communication delays have mainly focused on: 1) the influence of time delays on the WAPS and effective approaches for delay compensation ⁴²⁴³; 2) control methods for providing robust performance against delays (e.g., event‐triggered control methods ⁴⁴, wide‐area phasor power oscillation damping controllers ⁴⁵, and robust controllers ⁴⁶⁴⁷); 3) exact methods for evaluating the delay margin for stability, that is, the maximal allowable delay (upper bound on the time delays), when WAPSs and open networks are integrated ⁴⁸⁴⁹⁵⁰. A WAPS is unstable if the real‐time delays exceed the delay margin.

Two classes of methods are available for computing the delay margin in a WAPS for constant and time‐varying delays. Frequency‐domain direct methods quantify the delay margin based on computing critical eigenvalues for constant delays with a known upper bound, for example, the Schur‐Cohn‐based method for commensurate delays ⁵¹, Rekasius substitution ⁵², and the elimination of exponential terms in the characteristic equation ⁵³. Indirect methods can deal with time‐varying and constant delays; they are derived from Lyapunov stability theory ⁵⁴⁵⁵⁵⁶, linear matrix inequality techniques ⁵⁷⁵⁸⁵⁹, H‐infinity robust synthesis ⁶⁰, and the dual‐locus diagram method ⁶¹.

These methods assume well‐defined time delay models, that is, constant ⁶², uniformly distributed ⁶³, multiple ⁶⁴, and probabilistic interval time delays ⁶⁵, and require prior knowledge of the lower bound, upper bound, and parameters of the delay distribution. However, delays in an open communication network vary with the number of active end‐users ⁶⁶ and media access control (MAC) protocols ⁶⁷; therefore, no prior knowledge of time delays is available, and they are intrinsically stochastic. These assumptions hinder the application to real systems and lead to excessively conservative results. To overcome these limitations, the LFC model of WAPSs should account for a real open communication network with various MAC‐level protocols. The TrueTime simulator has been used to extract realistic scenarios of networked control systems (NCSs), in which the characteristics of the random delays are unknown and are statistically inferred after collecting sufficient delay observations ⁶⁸⁶⁹⁷⁰. As a specific application of NCSs, insights obtained from the integration of traditional control systems and open communication networks can foster the integration of open communication networks and WAPSs. However, to the best of the author's knowledge, few studies have analyzed the stabilization of WAPSs via real open communication networks.

During the computation of the control signals, the prediction of the C‐A delay in the current period and the S‐C time delay in the successive period can greatly improve the controller performance. However, this possibility has not been investigated in previous works on delay margins or robust LFC strategies. Several methods for delay estimation are available, such as the Markovian model approach ⁷¹, backpropagation neural network prediction ⁷², adaptive wide‐area power oscillation damping ⁷³, dynamic Markov jump filters ⁷⁴, hidden Markov models (HMMs) ⁷⁵⁷⁶, and the exponentially weighted moving average (EWMA) method ⁷⁷. Time delays depend on the underlying network state, which is ever‐changing and concealed ⁷⁸⁷⁹. Indeed, network states cannot be observed directly, but random time delays can be measured using the time‐stamp technique ⁸⁰. Therefore, the state variables of the open communication network can be estimated using the measured time delays; the transitions among these states can be modeled by a discrete HMM (DHMM). As a result, random delays are observations of the DHMM ⁸¹⁸²⁸³.

Traditionally, in HMM‐based delay models, random delays are mapped to a discrete observation space through scalar quantization techniques, for example, uniform quantization and k‐means clustering quantization ^84858687. Because the evolution of random delays is described by a finite‐state Markov chain, the communication network action can be learned via the DHMM ⁸⁸⁸⁹⁹⁰. In addition, DHMMs have been used for other applications in power systems, for example, the dynamic detection of transmission line outages ⁹¹, the generation of distributed photovoltaic systems ⁹², and residential energy use ⁹³⁹⁴⁹⁵. Chapter 3 introduces the missing data expectation maximization (MDEM)‐based Baum‐Welch algorithm ⁹⁶⁹⁷⁹⁸ for the estimate of the parameters of the DHMM online, without previous knowledge of the state transition matrix. Following this step, the time delay can be predicted using the Viterbi algorithm ⁹⁹. The predictions are used as the inputs of the Smith predictor to compensate for time delays and improve the LFC strategy performance ¹⁰⁰¹⁰¹. Our work improves the conventional Smith predictor, which simply employs the summation of the latest measured C‐A time delay and the S‐C time delay to construct the prediction ¹⁰². The conventional Smith predictor cannot capture the evolution of the network states, which is the root cause of stochastic time delays.

The relationship between the delay margin and the controller gains can only help achieve a compromise between the LFC performance and the maximum allowable delay. However, the delay margin‐based method cannot compensate for random delays because it does not involve the real‐time prediction of delays in each period. Therefore, in Chapter 3, the Smith predictor estimates the real‐time delay, which is then integrated into the delay margin‐based method to enhance the frequency stabilization performance. The Smith predictor can also enhance the LFC performance of robust proportional‐integral‐derivative (PID) controllers, whose gains are tuned via robust evolutionary algorithms ¹⁰³. Improvements to control strategies currently implemented in real systems, e.g., delay margin‐based proportional‐integral (PI) controllers ¹⁰⁴¹⁰⁵ and PID controllers ¹⁰⁶, are presented to demonstrate the effectiveness of the proposed methodology for the control of WAPSs.

The power sector is experiencing a structural trend toward decentralization, stemming from the integration of large shares of renewable energy resources (RERs) ¹⁰⁷. This is fostered by distributed energy resources (DERs), which require the integration of power generation means located at or near the end‐user side ¹⁰⁸¹⁰⁹. However, the stochastic nature of RERs and the load demand induces system frequency fluctuations ¹¹⁰¹¹¹. An effective control strategy is needed to maintain the system frequency at its nominal value by balancing the power generation and demand in real time. To this end, automatic generation control (AGC) schemes have been developed for damping frequency oscillations in distributed generation systems (DGSs) ^112113114115. AGC is performed by computing control signals based on the system frequency and delivering balancing inputs to various energy storage systems (ESSs) to absorb (release) the surplus (deficit) power from (to) the grid ^116117118. However, the ubiquity of DERs across wide areas and the complex structure of DGSs hinder the development of dedicated communication infrastructures for DGSs with massive DERs ^119120121122.

Recently, AGC has been integrated with open communication networks because of its low cost, high speed, simple structure, and flexible access. Data exchanges among PMUs, generators, and the control center are provided by the open communication network in the form of time‐stamped data packets ^123124125126. Stable AGC depends heavily on the performance of the open communication network ^{127128129130131132133134135}. Cognitive radio networks, cellular networks, local area networks (LANs), wide area networks (WANs), and wireless local area networks (WLANs) are employed as open communication infrastructures in these NCSs ^136137138.

However, open communication networks are exposed to various types of degradation processes, such as network‐induced time delays ^139140141142, packet dropouts ¹⁴³¹⁴⁴, failures of the communication infrastructure ¹⁴⁵, uncertain communication links ¹⁴⁶, and cyberattacks ¹⁴⁷. As a result, the measurement signals (control signals) received by the control center (ESS or generators) degrade, effective AGC cannot be carried out, and the system frequency response worsens ^{148149150151152}. Studying the performance of open communication networks is critical for understanding the occurrence of time delays and packet dropouts. To this end, medium access and packet transmission must be analyzed. The MAC layer is the lower layer of the data link layer of the Open System Interconnection model, and it is responsible for moving data packets among network interface cards across communication channels. Several MAC protocols, for example, carrier‐sense multiple access with collision detection (CSMA/CD, Ethernet), CSMA with arbitration on message priority (controller area network), and IEEE 802.11b/g (WLAN), prevent the collision of packets sent from different nodes across the same channel ^153154155156.

Time delays are variable, challenging to predict, deteriorate AGC performance, and reduce the stability region ¹⁵⁷¹⁵⁸. Packet dropouts refer to lost messages, which occupy network bandwidth but cannot reach the destination. They affect the operations of DERs and the reduction of frequency fluctuations, particularly in uncertain network environments. Optimal feedback AGC regulators for DERs have been investigated in numerous works for perfect communication networks, and the impact of transmission delays and packet dropouts on the controller cannot be captured ¹⁵⁹. Robust PID controllers against constant or uniformly distributed time delays ^160161162163 are designed to cope with perturbations of the control parameters. However, constant or uniformly distributed time delays cannot be generally assumed in realistic communication networks.

In addition, recent studies focusing on primary and secondary control levels have been extended to the power management level by considering fuzzy controllers ¹⁶⁴¹⁶⁵, decentralized power management and sliding mode control strategies ¹⁶⁶, static synchronous compensators ¹⁶⁷, and two‐degrees‐of‐freedom feedback‐feedforward robust controllers ¹⁶⁸¹⁶⁹. The reactive power reference can be determined and controlled by a novel application of radial basis function neural networks ^170171172 to improve the power sharing and stability of microgrids with multi‐DERs. To provide high reliability and robustness against network failure or time delays, droop‐based control schemes are designed to specify the frequency of each DER unit by using complementary loops and fuzzy logic controllers ¹⁷³, robust H_∞ controllers ¹⁷⁴, and PI controllers ¹⁷⁵¹⁷⁶. On the other hand, novel approaches for mitigating the impact of random time delays quantify robust delay margins ¹⁷⁷. The delay margin‐based sparsity‐promoting wide‐area control strategy, which requires few system observations, can reduce communication requirements and yield nearly optimal performance compared with centralized control ¹⁷⁸. Nevertheless, packet dropout still has the potential to affect the performance of this strategy.

1.2.2 Reliability of CPSs

Distributed renewable energy sources are increasingly connected to power distribution networks as a remedy for environmental and economic concerns ^179180181. However, their power outputs are dependent on the available intermittent natural resources, such as solar irradiation, wind velocity, and biofuel production ^182183184. The rapid deployment and commercialization of storage devices and electric vehicles (EVs) has become an attractive technological solution to facilitate the use of renewable energy sources, manage demand loads, and decarbonize the residential sector ^185186187. The above technological issues call for managing real‐time energy imbalance in DGSs to meet electricity demand over a long‐term horizon. In order to address the challenges of distributed control of energy sources, communication networks are being installed for accurate control of the different power sources and the timely operational scheduling of distributed generator (DG) units, with the objective of providing reliable and sustainable energy in a timely fashion ^{188189190191192193}. However, most existing research works do not formally investigate the capability of communication networks in providing real‐time power management and promoting the optimal power dispatch ^194195196197. The effective integration of communication networks into DG systems is a key step in the realization of future smart grids ¹⁹⁸¹⁹⁹.

Most integrated system‐of‐systems models have been developed based on dedicated and closed communication networks, where the infrastructure is exclusively built for smart grid applications ^200201202. As the network is dedicated between the DG and the control center, the data exchange is assumed to be perfect and free of defects (e.g., induced time delays and packet dropouts ^{203204205206207}). However, experience has shown that dedicated communication networks are ill‐suited to future DG systems, which require a different, more complex but much cheaper network, as its dimension would be much larger ^208209210. Because of the low installation cost, high transmission speed, and flexible access, the open communication network has the highest potential for integration with future DGSs ^211212213. As end‐users have to share the limited bandwidth in the open communication networks, which could lead to local congestion, they can be unreliable and suffer from network‐induced delays and packet dropouts ^214215216.

Existing research works ^{217218219220221222223224225226} do not model explicitly and adequately the behaviors of transmission delays and packet dropouts. Most of the aforementioned models are limited to constant or less stochastic transmission delays, which are not true in reality ²²⁷²²⁸. The delays are described by discrete‐time models or are neglected by assuming that they are much smaller than the communication interval ^229230231232. Packet dropouts are usually modeled by a two‐state Markov chain and the associated quantitative loss rates; the detailed state evolution is masked and only input/output information is made available. The state transition matrix is known by assuming that the evolutions of packet dropouts can be fully observed ²³³²³⁴. Additionally, the models of uncertain renewable power sources do not consider time‐correlated properties ^{235236237238239240}. Consequently, the control schemes derived based on such assumptions can be very conservative and may not be readily applicable to real systems.

To bridge this gap, a generic and transparent mathematical model is necessary for the analysis of the impacts of integration of unreliable open communication networks (e.g., home area networks, neighborhood area networks, and WANs). The major challenges lie in the modeling and simulation of the interactions between degraded communication networks and DG systems, and the optimization of the real‐time energy management problem on such system platforms. Note that the specific requirements (i.e., high‐frequency data and data prediction) introduced by communication network integration need to be taken into account in this modeling, simulation, and optimization framework.

1.3 Opportunities for CPS Applications

1.3.1 Managing Reliability and Feasibility of CPSs

CPSs perform critical tasks in many industrial applications, for example, manufacturing systems ²⁴¹, transportation systems ²⁴²²⁴³, and power systems ²⁴⁴²⁴⁵. The components of control systems, that is, actuators and sensors, are subject to degradation when operating under severe working conditions ^{246247248249250}.

Sudden load variations in electric power systems are often balanced by promptly changing the output of natural gas power plants following the LFC strategy ²⁵¹. However, the degradation of gas turbine compressors, that is, the deviation of compressor flow capacity and isentropic efficiency ²⁵², and the degradation of PMUs, that is, measurement drifts and errors ²⁵³²⁵⁴, reduce the LFC performance by decreasing the available balancing power, and by producing inaccurate frequency readings. Deteriorated LFC performance may result in power system failures because the system frequency exceeds its maximum allowable drop or fails to attain the steady‐state frequency tolerance band in the required time in compliance with ISO 8528‐5 ²⁵⁵. As a result, power system failures are determined by the LFC performance, stemming from the partial information on the power system conditions, that is, the health indexes of gas turbines (flow capacity and isentropic efficiency), and measurement drift. Therefore, it is necessary to study these degradation processes to predict the real‐time LFC performance loss and ensure adequate LFC through proper maintenance activities.

Lifetime prognostic studies of gas turbines ²⁵⁶ have shown that the failure time of these systems varies between 24 000 h and 35 000 h. Such variability stems from different working conditions ²⁵⁷ and different starting points of the degradation paths ²⁵⁸. Therefore, the degradation model should reflect the unit‐to‐unit variability in the degradation process of gas turbines ²⁵⁹. Moreover, many condition‐based maintenance (CBM) models determine maintenance activities based on the estimated remaining useful lifetime (RUL) of gas turbines using health indexes (reduction in flow capacity and isentropic efficiency ²⁶⁰) derived from on‐line measurements, for example, rotor speed, inlet temperature, and pressure ²⁶¹²⁶². Maintenance activities are scheduled based on the two health indexes. However, in control systems, failures are determined by the system performance ²⁶³ and not by the health indexes of individual components, that is, they are system‐level failures rather than component‐level failures ^264265266. Control systems may still operate normally even if the component health indexes exceed failure thresholds ²⁶⁷²⁶⁸. In addition, the feedback control mechanisms hide the explicit mapping from the individual component degradation state to the control performance loss ²⁶⁹. It is easy to estimate the RUL of an individual component, but difficult for degraded control systems, because their failure time cannot be described by any particular distribution ²⁷⁰²⁷¹. Therefore, when applied to degraded control systems, general CBM models should connect the component degradation and the control performance loss, so that maintenance activities depend on the reduction of control performance ²⁷².

The Wiener process is often employed in degradation models because of its favorable mathematical properties; in particular, it can capture non‐monotonic degradation signals frequently encountered in practical applications, because consecutive independent increments are normally distributed ²⁷³. Therefore, this stochastic process has been widely applied to characterize the path of degradation in realistic scenarios, where fluctuations are observed in the degradation process, for example, brake‐pad wear for automobiles ²⁷⁴, bearing degradation ²⁷⁵²⁷⁶, gyroscopes in inertial navigation systems ^277278279, contact image sensors in copy machines ²⁸⁰, the resistance of carbon‐film resistors ²⁸¹, and the pitting corrosion process ²⁸². To overcome the aforementioned limitations, the Wiener degradation model with unit‐to‐unit variability is introduced to describe gas turbines exhibiting different lifetimes. The Wiener model considers the random starting time of the degradation process, which follows a non‐homogeneous Poisson process ²⁸³²⁸⁴. Furthermore, the drift parameter, which denotes the aging rate, is also variable and follows a normal distribution, where the mean denotes the average aging rate and the variance represents the variability in the aging rate ^{285286287288289}. These parameters can be estimated from the lifetime dataset via an expectation‐maximization (EM) algorithm ^{290291292293294295}.

Thus, the data‐driven degradation model with unit‐to‐unit variability is integrated into the control system model described by control block diagram, resulting in a real‐time simulation model. In such control models, the interplay among the reduction in the control signal due to component degradation, the transfer functions of the subsystems, and the feedback control loop, provides the mapping between the component degradation states and the system performance loss. This interaction is modeled via control‐block diagrams, which implement the feedback control mechanism and quantify the control signal by comparing the control performance to the setpoint. Therefore, such an integrated model does not require explicit mapping from the component degradation states to the system performance loss and is well‐suited to represent a degraded control system. As such, this simulation model realistically predicts the performance of the control system at different operating times and degradation stages.

1.3.2 Ensuring Cybersecurity of CPSs

Attacks on complex systems, for example, CPSs, are fundamentally different from traditional internal failures (e.g., degradation and design) and external failures (e.g., natural disasters) ^296297298299. Many attack models for complex systems embrace a partial perspective, which only focuses on component vulnerability, and neglects the dependence of system performance on it ^300301302. As a result, the insights provided by these models are not adequate for providing general recommendations in realistic applications. To address this limitation, recent studies investigate the influence of component vulnerability (attacks at the component level) on system performance ^303304305306.

Pioneering works ^{307308309310311312} develop optimal defense strategies to minimize the attachment vulnerability of parallel systems, assuming that attackers maximize either the damage probability or the expected damage over a time horizon. They also consider general features, that is, imperfect false target techniques and genuine targets ³¹³³¹⁴. These defense strategies reach a trade‐off between increasing the protection of existing components and providing redundancy by allocating additional components ^315316317318.

System performance is an essential feature in CPSs that can still operate if some components are unavailable and, therefore, are characterized by multiple performance levels ^{319320321322323324}. System performance degrades with increasing component destruction or unavailability; if the system performance level decreases, the required demand may be partially unsatisfied. Two risk measures can be used for multi‐state complex systems ^325326327: 1) the probability that the demand is not satisfied is considered for complex systems that fail if performance cannot meet demand, for example, automatic train protection and block systems ³²⁸³²⁹, and power system dynamic security systems ³³⁰; 2) the expected damage proportional to the unsupplied demand is considered for complex systems that can operate even if the demand is partially supplied, for example, mobile ad hoc networks ³³¹, NCSs ³³²³³³, supervisory control and data acquisition (SCADA) systems ³³⁴³³⁵, water distribution networks ³³⁶, and electric power grids ^337338339.

Several works consider both the vulnerability and performance of complex systems subject to attacks ^{340341342343344345346347348}. These works generally describe a case as a dynamic contest between an attacker and a defender to develop a component vulnerability model and a multi‐state system performance model. The number of destroyed components quantifies the demand loss and expected damage costs ³⁴⁹³⁵⁰. To make the above contest more realistic, attack time uncertainties and the attacker's preference on the attack time should be considered.

In the literature, two different approaches exist for determining the attack time, that is, the strategic selection and the selection based on probability distributions. In the former, the attacker strategically selects whether to attack at some point in time or at a later point in time, based on the outcome of the game, given that the attack occurs at a specific time ³⁵¹. Thus, complex attack and defense strategies can be derived from a two‐stage min‐max multi‐period game. Extensive attack or defense in one period limits the attack or defense that can be exerted in the next period, and vice versa. Thus, players strategically choose whether to exert effort now or in the future ^352353354. The defender may determine optimal resource allocation strategies for redundancy ³⁵⁵ and protection, that is, individual or overarching protection ^356357358359. On the other hand, the attacker may distribute the constrained resources optimally across sequential attacks ^360361362363.

In the second approach, the attacker prefers to conduct the attack at the time of the critical event ³⁶⁴. Indeed, attacks in Nice, Berlin, Manchester, and London occurred several days before and after Bastille Day, Christmas, a concert, and the Champions League 2017 Final, respectively. In these cases, the defenders have increased the protection level in the immediate aftermath; therefore, it is not worthwhile and cost‐effective for the attacker to deploy another attack in a short period. Because attacks occurred at critical times, they can greatly influence public opinion. As a result, the attacker aims to maximize the system loss by strategically selecting a set of elements to attack based on the two‐stage min‐max game ³⁶⁵. Because we can predict the distribution of the time at which the critical event occurs, the attack time can be inferred from a data‐driven probability distribution ³⁶⁶. The two approaches aim to maximize the outcomes of the game given that the attack occurs at a specific time under a similar system structure and variable resources.

The truncated normal distribution is used to describe the uncertainty of the most probable attack time, that is, the time of the critical events, and the accuracy of the defender's estimate of it ^367368369. The truncated normal distribution has been adopted to represent uncertainties in many realistic applications, for example, traffic peaks of online video websites ³⁷⁰, the peak season of power supplies ^371372373, the peak demand of water distribution systems ³⁷⁴, and the rush hour of public transportation ³⁷⁵. Accounting for the influence of this uncertainty increases the relevance of the insights gained for the optimal resource allocation strategy against attacks.

CPSs are a new class of engineered complex systems that provide tight interactions between cyber and physical components. The corruption of a small subset of their components has the potential to trigger system‐level failures leading to entire system performance disruptions ^376377378379. Previous studies on attack vulnerability and performance of complex systems can be extended to identify resource allocation strategies for cyber components and promote system performance during cyber‐attacks in CPSs ³⁸⁰³⁸¹. Cyber vulnerabilities are exploited by attackers to launch insidious attacks on the integrity, confidentiality, and availability of cyber data by injecting false data into measurement devices, eavesdropping estimation of system states, and deploying denial of service (DoS) attacks on communication networks ^382383384385. More sophisticated attack models specifically target weaknesses to cause maximal damage ³⁸⁶. In this respect, it is key to capture the uncertainties intrinsic to the behavior of the attacker and the defender.

With respect to applications in smart grids, upgrading traditional grids to smart grids has brought many benefits to the overall management of power and energy systems, including higher reliability, better efficiency, improved integration of RERs, more flexible choice for stakeholders, and lower operation costs ^387388389. However, the core technologies, for example, communication techniques and SCADA systems ^390391392393, which deliver the advantages of smart grids, also open the grids to vulnerabilities that already exist in the information and communications technology (ICT) world. These vulnerabilities pose threats to smart grids, such as DoS attacks, false data injection, replay attacks, privacy data theft, and sabotage of critical infrastructure ^394395396. In addition, the failures in a smart grid caused by cyberattacks can easily cascade to other highly dependent critical infrastructure sectors, such as transportation systems, wastewater systems, health care systems, and banking systems, resulting in extensive physical damage and social and economic disruption ³⁹⁷³⁹⁸.

While government, the private sector, and academia are recognizing the cyber vulnerability of smart grids, the likelihood and impact of a cyberattack are difficult to quantify. Furthermore, for a smart grid, there may be mandatory standards and operational requirements from grid stakeholders. Current risk management strategies are generally qualitative or heuristic ³⁹⁹. In these strategies, some assumptions, for example, constant reward with respect to successful anti‐cyberattack ⁴⁰⁰⁴⁰¹, may be unrealistic for most smart grids.

Chapter 7 presents a probabilistic risk analysis framework to enhance smart grid cyber security. In particular, the dynamic and stochastic characteristics of smart grids, such as uncertain demands, are taken into account to investigate the effect of defending strategies on the real operation cost. The optimal power flow (OPF) model ⁴⁰² is applied to an 11‐node radial smart grid originating from the Elia grid in Belgium. Compared with the existing studies that focus on the inherent risk ⁴⁰³⁴⁰⁴, such as the natural degradation and uncertain RERs for better maintenance actions and power dispatch, Chapter 7 addresses the impact of the external threat (cyberattacks) on the operation cost for effective deployment of cyber defense teams. In previous works, the cost of each attack on a node was assumed to be a constant ⁴⁰⁵. Nevertheless, by investigating some practical scenarios, it has been found that the costs are more likely to be determined by some adversarial factors. Therefore, an adversarial cost sequence associated with each node is assumed, and a widely used variation constraint is introduced for each cost sequence. To cope with the objective of sequential decision strategies, the problem is formulated using the reinforcement learning framework ^406407408. In particular, the Bayesian prior method ⁴⁰⁹ is employed for the model parameters, and the problem is formulated as a Bayesian adversarial multi‐node bandit model. In addition, a Bayesian minimax type regret function is constructed, which is subject to the learning context.

2
Fundamentals of CPSs

In this chapter, fundamental Cyber-Physical System (CPS) models, evaluation processes, verification procedures and optimization methods are introduced.

As the CPS lacks an explicit formulation due to the feedback control mechanisms ⁴¹⁰⁴¹¹, the closed‐form solution of the optimal stability, reliability and resilience strategies is unavailable. Therefore, meta‐heuristics algorithms are employed to assist the search of the optimal strategy. For maintenance of CPSs, we use the hybrid Genetic‐Simulated‐Annealing algorithm (HGSAA), which has been applied in many maintenance optimization works ^412413414. It combines the Genetic Algorithms (GA) and the Simulated Annealing (SA) to improve the quality of solutions and reduce computation efforts ⁴¹⁵. For the optimal control strategy of CPSs (single objective), A particle swarm optimization (PSO) method based on Monte Carlo Simulation (MCS) is introduced in this book to solve the Proportional–Integral–Derivative (PID) controller optimization problem ⁴¹⁶⁴¹⁷. The method can achieve higher search efficiency values since it combines local search with global search. The technique has been widely employed to find optimal solutions for realistic applications, especially with regard to the optimization of control systems ^418419420. A solution including the PID control strategy is encoded as a finite‐length string called “a particle” in PSO. The fitness value of each particle is determined by a fitness function. However, the optimization of PID usually involves simultaneous optimizing two objectives, i.e., control performance and control effort, which are generally non‐commensurable and conflicting with each other ⁴²¹⁴²². Better control performance means more control efforts and vice versa. Therefore, the multi‐objective PSO algorithm is applied to achieve a best trade‐off between the two objectives ^423424425.

2.1 Models for Exploring CPSs

2.1.1 Control‐Block‐Diagram for CPSs

Consider the CPS with degraded components shown in Figure 2.1, which is a typical case of CPSs consisting of a forward channel and a feedback channel.

Figure 2.1 The control block diagram of CPSs with degraded components.

2.1.1.1 Control Signal in CPSs

At the total run time T_o, for an operational task with total K sampling periods, the time‐varying model of the forward channel in the k^th period (k ∈ {1, …, K}) is first derived.

As the sensor measures the system output during every sampling period T_P, the control signal u_k remains the same during the interval [(k − 1)T_P, T_P], which is the inherent discrete‐time property. Thus, the control signal u_k can be represented by the following sum of input steps:

(2.1)

where .

Arriving at the relationship between control signal u_k and output y_k is not a trivial task. In the time domain, it cannot be represented by the simple multiplication of time‐varying model of the actuators, their respective degradations, and the process, but their convolution. Inspired by a similar problem solved by control theory, by applying the Laplace transform to respective time‐varying models of the actuators, degradations, and the process, their respective complex domain models can be obtained. Once this is done, the overall relationship can be described through simple multiplication of the complex domain models.

Firstly, applying the Laplace transform to u_k, its complex domain representation is:

(2.2)

where with .

2.1.1.2 Degraded Actuator and Sensor

Since the system has N identical actuators, for any n ∈ {1, 2, …, N}, the degradation of the n^th actuator can be modeled by a Wiener‐process subject to unit‐to‐unit variability ⁴²⁶⁴²⁷:

(2.3)

where da_n(T_o) denotes the loss in the effectiveness of the n^th actuator at total run time T_o. da_n(0) is a known initial degradation state. Without loss of generality, it is assumed that da_n(0) = 0 in this book. For any T_o > 0, σB(T_o) ∼ N(0, σ²T_o) describes the stochastic dynamics of the degradation process.

It is found that (2.3) is able to describe the time‐varying variability arising from the dynamic features of B(T_o). Nevertheless, since each actuator operates under different working conditions, different actuators follow different degradation paths at different rates. Compared with previous works where the degradation processes are identical, it is more realistic to consider the unit‐to‐unit variability in the degradation process. Thus, θ is regarded as a random parameter denoting the unit‐to‐unit variability with , and σ is regarded as a constant capturing the nature of the degradation process common to all actuators in the same class.

Assumption 2.1: For any kT_P ≪ T_o, da_n(T_o + kT_P) equals to da_n(T_o). Thus, da_n(T_o + kT_P) is a constant value during an operational task starting at total run time T_o. The values of θ and B(T_o) are independent of each other.

Since the operational time of one control process is much smaller than the total run time of entire CPSs in real applications, it is general to consider that the change in the degradation of one component is negligible in such a short time ⁴²⁸⁴²⁹. Therefore, above assumption is reasonable and not restrictive.

Since the degradation of each actuator is constant during an operational time interval [T_o, …,T_o + KT_P], the Laplace transform cannot be applied to a constant value. Even so, the relationship between control signal and system output can still be represented by simply multiplying the time‐varying models of the actuators and the process and the degradation constant as:

(2.4)

where and da(T_o) = [da₁(T_o), …, da_N(T_o)]^tr, where tr stands for the transpose of matrix.

In industrial applications, the perfect sensor measurement of system output state is often impossible. It is the common case for the controller that the received sensor measurements are not only subject to measurement error but also to sensor gain degradation due to the noise and severe working conditions ⁴³⁰⁴³¹. Therefore, the received measurement in the k^th period is a superposition of the measurement error w_k and sensor gain degradation ds(T_o + kT_P) with

(2.5)

where the ds(T_o) accounts for the stochastic sensor gain degradation at total time T_o, ds(0) is the initial state with ds(0) = 0, and λ is the drift coefficient with λ < 0. Compared with the aforementioned models in which ds(T_o) is restricted to be a constant or follow a uniform or Bernoulli distribution, (2.5) is more suitable to describe the time‐varying properties of the sensor gain degradation.

Assumption 2.2: For any k and kT_P ≪ T_o, ds(T_o + kT_P) equals to ds(T_o) during an operational task starting at time T_o. ds(T_o), and w_k and da(T_o) are independent of each other.

Thus, the measurements received by the controller can be described as

(2.6)

2.1.1.3 Time‐Varying Model of CPSs

Therefore, substituting (2.2) into (2.4), and using the linearity property of Laplace transform and applying the inverse Laplace transform to (2.4), the output y_k can be obtained as

(2.7)

where and .

After completing the forward and inverse Laplace transform, the time‐varying model of the forward channel is built. Consider now the time‐varying model of the feedback channel.

Right after the controller receives the measurement of the system output y_k in (2.6), it will compute the control signal for each actuator at the (k + 1)^th period, separately. For simplification, it is assumed that all actuators share the control signal evenly.

Thus, the control signal for actuator n based on the following PID strategy is determined as:

(2.8)

where e_k + 1 = r_k + 1 − y_k.

From (2.8), it can be concluded that the control signal computed is determined actually by historical errors. Properly designing the parameters of the control strategy can definitely help the system eliminate the influence caused by degraded components and make the system output converge to the expected target with required qualities.

Therefore, the stochastic model of CPSs is closed through the forward channel model expressed by (2.1) and (2.7), and the feedback channel model expressed by (2.6) and (2.8). The performance analysis and reliability improvement can be conducted based on the explicit system structure by optimally choosing the parameters of the system controller.

2.1.2 Implementation in TrueTime Simulator

2.1.2.1 Introduction of TrueTime Simulator

The architecture for the Automatic Generation Control (AGC) of Distributed Energy Resources (DERs) via Ethernet and hybrid network, is illustrated by the Figure 2.2. To implement the microgrid with two different communication network architectures in the Matlab/Simulink environment, the TrueTime simulator is deployed for simulating different types of communication protocols ⁴³².

Figure 2.2 Simulink realization of the microgrid. (a) Ethernet, (b) Hybrid network.

TrueTime is a Matlab/Simulink‐based simulator for real‐time CPSs, which facilitates co‐simulation of controller task execution in real‐time kernels, network transmissions, and continuous plant dynamics. It can simulate most realistic communication networks, e.g., Ethernet, CAN, TDMA, FDMA, Round Robin, Switched Ethernet, FlexRay, PROFINET, 802.11b WLAN and 802.13.4 ZigBee. Multiple activity levels of the interfering traffic are simulated via the Interference node, which sends random interfering packets over the network. Packet dropout is described by Bernoulli‐distributed variables ⁴³³. More details of TrueTime simulator and its realization in CPSs are given in this Section.

Above procedure is detailed in the “TrueTime 2.0‐Reference Manual”, which provides representative examples of CPSs ⁴³⁴. Figure 2.2 (a) and Figure 2.2 (b) present the Matlab/Simulink implementations of the microgrid with Ethernet and of the microgrid with Hybrid network, respectively. The network blocks from the TrueTime simulator are directly linked with physical components, i.e., the PID controller, Phasor Measurement Units (PMU), DERs, to enable data exchanges.

The data exchange among the control center, DERs, interference node and PMU are wired in the Ethernet architecture. In the hybrid architecture, the data exchange between the routers and the Remote Terminal Unit (RTU), such as Battery Energy Storage Systems (BESS), Flywheel Energy Storage Systems (FESS), Diesel Engine Generator (DEG) and PMU, is wireless and provided by the 802.11b/g. The data exchange among routers is wired and provide by the Ethernet. Low product prices make 802.11b/g more convenient and cost‐effective compared to the Ethernet. As such, 802.11b/g has become an efficient approach to provide flexible data communication between routers and RTU, and is employed for monitoring and controlling DERs, offshore wind farms and smart home energy management systems ⁴³⁵.

The interference node simulates the interference user in the open network, who sends disturbing traffic over the channels and cause congestion. Generally, the length of data traffic is constant (i.e., 80 bytes in this chapter, same as the length of PMU measurement ⁴³⁶⁴³⁷), and the interference node sends it to the network at every time period T_i if

(2.9)

where UN_i is a uniformly distributed random number sampled at time period T_i in the interval [0, 1], and BWShare is the expected ratio of the network bandwidth used by the interference node ⁴³⁸.

Remark 2.1: The time interval T_i determines the activity level of the interference user. For example, T_i = 0.01 s means that the interference user tries to send out disturbing traffic every 0.01 s.

Remark 2.2: BWShare indicates the expected percentage of bandwidth used by the interference user. If the node sends out a message, it occupies the entire channel bandwidth, and other nodes cannot send messages until this channel is free, to avoid packet collisions.

For example, BWShare=0.1 means that 10% of bandwidth is used by the interference user, and from a statistical viewpoint, the interference user sends out disturbing traffic with probability 0.1.

2.1.2.2 Architectures of CPSs in TrueTime

The architectures of the real‐time AGC of DERs via the Ethernet and hybrid network all consists of:

• PMU node (time‐driven): the PMU takes measurements of the system frequency at every sampling interval T_s = 0.01 s and sends Δf(kT_s) to the control center over the network. The size of data packets is 80 bytes and the phase delay caused by the filter of the PMU is 0.006 s ⁴³⁹⁴⁴⁰.

• Control center node (event‐driven): when a data packet from the PMU reaches the control center, the controller takes 0.002 s to compute the control signal u(t_n) and sends it to the DERs. The length of control signal is 500 bytes.

• DERs nodes (event‐driven): the DEG, BESS and FESS adjust their operations based on the control signal received.

• Interference node (time‐driven): it sends disturbing traffic over the network with period T_i and causes congestion, to generate different scenarios of time delay and packet dropout.

The above two architectures for the open communication network are simulated using the TrueTime simulator ⁴⁴¹, which is a Matlab toolbox that generates scenarios of realistic network with different MAC protocols ^442443444.

Example 2.1: Figure 2.3 presents the dynamics of sending and receiving time for data packets at the interference node, PMU, control center and DERs under the Ethernet architecture, for BWShare = 0.3, T_i = 0.005s and L_d = 5%. The continuous lines show steps when a message is received by the corresponding node. The x coordinate of the steps identifies the receiving time of data packet, and the y coordinate identifies the time at which that corresponding packet was sent. Dotted vertical lines indicate that one of the nodes, i.e., interference node, PMU and control center, respectively in Figure 2.3, sends out a data packet. The sending time marked by the dotted lines is consistent with the y value of step.

Figure 2.3 Sending time and receiving time for the (a) interference node, (b) control center, (c) DGE, (d) BESS and (e) FESS in the Ethernet architecture, where BWShare = 0.3, T_i = 0.005s and L_d = 5%.

In Figure 2.3 (a), the step (55, 46) in the control center shows that the PMU sends out a system frequency measurement at 46 ms, which is consistent with the dotted line showing a sending pulse from the PMU at 46 ms, and the control center receives it at 55 ms. Furthermore, Figure 2.3 allows identifying the dropped packets. In Figure 2.3 (d), the control center sends out a packet to FESS at 29 ms (dotted lines marked with cross), but the FESS cannot receive it (no corresponding step with y = 29 ms), indicating that a packet dropout occurs in this transmission. Above two scenarios are highlighted in the Figure 2.3 to provide better descriptions.

2.2 Evaluation and Verification of CPSs

2.2.1 CPS Performance Evaluation

2.2.1.1 CPS Performance Index

Having built the hybrid model of the closed‐loop feedback CPS with degraded components, an event‐based MCS method is introduced to quantitatively assess the reliability of such system. The quantitative analysis for system performance considers the domain performance requirements ⁴⁴⁵.

The qualities of a control process of CPSs consist of the rising/declining time, the percentage overshoot and the settling time, which are all relevant in assessing real‐time operational performances. Descriptions of the qualities are presented in Table 2.1. Rising/declining time is useful while ensuring that the control process responds rapidly to the command signal; percentage overshoot ensures that the control process exhibits sufficiently small oscillations during response; and the settling time ensures that the control process quickly reaches the expected output value and keeps stable.

Table 2.1 Domain Requirements and Descriptions.

If any of the quality parameters exceeds the corresponding maximal operational requirements within the total run time T_o, it can be concluded that the CPS is unable to satisfy the stipulated requirements. Hence, it has failed at total run time T_o, because of component degradation.

When considering the quality‐Rising time, it can be represented as

(2.10)

When considering the quality‐Declining time, it can be represented as

(2.11)

where RT, DT, PO and ST are the qualities of the control process at total run time T_o. x = 0 means that the CPS has failed at total run time T_o. RT_max, DT_max, PO_max and ST_max are the maximal operational requirements.

Besides the operational requirements, the nonfunctional requirement‐reliability is especially important. Reliability directly determines the ability of the CPS to maintain the expected qualities despite the presence of degraded components.

2.2.1.2 Reliability Evaluation of CPSs

It is noticed that the degradation process associated with each component is probabilistic and it is difficult to obtain the explicit reliability function of such a system. The event based MCS method is introduced in this book to estimate the ability of the system with a PID control strategy in the design phase ⁴⁴⁶⁴⁴⁷. The reliability of the CPS is then estimated as a tabulated function of the total run time subject to operational requirements. The detailed procedures for applying MCS method to hybrid model are described below.

INPUT: the PID control strategy K_p, T_i and T_d; the mathematical model for the CPS; the degradation paths of components; the total run time T_o; the operational requirements.

OUTPUT: the estimate R^*(T_o), of system reliability at total run time T_o.

STEP 1: Set current number of MCS replications h equal to 1.

STEP 2: Define precision interval L, the percentage of simulations belonging to this precision interval p, and the number of simulations N_T in each MCS run; set N_F, N_E and current number of simulations i in this MCS run to 0.

STEP 3: Conduct a simulation using the hybrid model described in section 2.1.1.3. If the simulation fails as judged by (2.10) or (2.11), then set N_F = N_F + 1, the current reliability and note the qualities of process RT_i/DT_i, PO_i and ST_i.

STEP 4: If , N_E = N_E + 1.

STEP 5: If or i < N_T, let i = i + 1 and go to STEP 3; otherwise, and go to STEP 6.

STEP 6: If h < H, let h = h + 1 and go to STEP 2; otherwise, and halt.

The estimated reliability R^*(T_o) has the following several important statistical properties which have been proved in literature ^448449450. These properties are rather useful in verifying whether the estimated reliability of system obtained from the event based MCS is a good estimator and defining the relationship among H, ε and α. The unbiased estimator with required precision derived from a small number of simulations is important for reliability assessment. Several properties about the estimated reliability are introduced as follows.

Property 2.1: The expectation of the estimated reliability R^*(T_o) of the CPS with degraded components obtained from MCS method is an unbiased estimator of the exact reliability R(T_o) at total run time T_o.

Proof: Let be the expected reliability of the CPS with degraded components at total run time T_o, where h = 1, 2, …, H. Then,

(2.12)

Thus,

(2.13)

Property 2.2: R^*(T_o) is an unbiased, consistent estimator of R(T_o):

(2.14)

Property 2.3: If the absolute error ε and the confidence level (1 − α)% of the MCS method are required, the total number of MCS replication H satisfies below approximate relation

(2.15)

Proof: Let R^*(T_o) be the estimated reliability and R(T_o) be the actual reliability. Consider the absolute error ε and the confidence level (1 − α)%, it exists

Using Z_α/2 as the z‐value of the (1 − α/2) percentile of the standard normal distribution, set the initial estimate of the replications required for MCS as the smallest integer H such that

where S₀ is the sample standard deviation of R^*(T_o) obtained from a sample with H₀ MCS replications. In this book, H₀ is selected as 1000. Setting the number of MCS replications at 1500 as well as the sample size of simulations in each MCS run at 5000 is adequate for an absolute error 0.005 at the confidence level 95%. It should be noted that if H₀ is small, it would be more appropriate to use t‐distribution instead of Z_α/2.

A high reliability indicates that the designed CPS should be able to satisfy the operational requirements in most cases while considering the degradation of the components. Otherwise, the system should be redesigned by optimizing the PID control strategy, in order to improve the ability of the feedback system compensating the loss effectiveness caused by degraded components, and finally provide a required reliability.

2.2.2 CPS Model Verification

The validation of data‐driven degradation model of the component in the CPS is usually via comparison and this chapter takes the model of generator degradation as the example to show how to conduct the cross‐verification with existing estimation model.

Example 2.2: The power system with a natural gas power plant consisting of four gas turbine units is shown in Figure 2.1. The values of the power system parameters are given in ⁴⁵¹⁴⁵², i.e., b = 1, c = 0.05, X = 0.6, Y = 1, T_c = 0.3, T_f = 0.23, T_t = 0.2, K_p = 120, T_p = 20, B = − 0.425 and R = 2.2. The discrete PID controller for the LFC is optimally tuned via the two‐degree‐of‐freedom internal model control design method and the PID approximation procedure ⁴⁵³.

This tuning method provides the system with disturbance rejection ability, and results in K_P = 0.13, K_I = 0.42 and K_D = 1.32, which ensure optimum LFC performance (PO_O =  − 0.015 pu and ST_O = 2.5 s), against the sudden load increase and the requirements in ⁴⁵⁴, i.e., failure thresholds L_PO =  − 0.016 pu and L_ST = 30 s. The steady output of the gas turbines is SO_O = 2.5 · 10⁻³ pu.

The degradation of the gas turbines and of the PMU reduces the LFC performance and ultimately results in the failure of the power system ^455456457. For simplicity, time T, λ and σ_B are expressed in units of 10000 h in this example, unless other units are specified. The degradation rates of the compressors flow capacity and efficiency are estimated by applying the EM algorithm to the dataset in ⁴⁵⁸:

• the flow capacity degradation of gas turbine is described by: λ∼N(1.8568, 0.0473²), σ_B = 0.1, a = 1;

• the efficiency degradation of gas turbine is described by: λ∼N(1.0252, 0.0427²), σ_B = 0.055, a = 1.

The model parameters for the measurement drift and the measurement error of the PMU (T_s = 0.1 s) are given in ⁴⁵⁹:

• the measurement drift degradation of sensor is described by: λ∼N(1.1575, 0.0605²), σ_B = 0.04, and a = 1 ⁴⁶⁰;

• the measurement error is described by: w[mHz]∼N(0, 2²).

In addition, Table 2.2 presents the parameters of the distribution of the aging rate, i.e., the drift parameter λ, obtained by the EM algorithm and the 95% confidence interval.

Table 2.2 The parameters and the 95% confidence interval of the distribution of the drift parameter λ.

Figure 2.4 (a) and Figure 2.4 (b) present the comparisons among the degradation models used in ⁴⁶¹⁴⁶², and the degradation model of (2.3) for two gas turbines.

Figure 2.4 Comparisons among the Wiener degradation model of (2.3) and linear and quadratic degradation models used in ⁴⁶³⁴⁶⁴ for two gas turbines.

The Wiener degradation model accounting for unit‐to‐unit variability better represents the degradation path of the turbine health index. The mean squared errors (MSE) of the Wiener process are 0.7498 and 0.7145; the MSE of the linear and quadratic regression‐based methods are, respectively, 0.8512 and 0.9665, and 0.8198 and 0.8757. Additionally, the Wiener degradation path can track the variable trend of the flow capacity drop. As such, the drift parameter λ introduces a stochastic component capturing the unit‐to‐unit variability.

2.3 CPS Performance Improvement

This section introduces the reliability enhancement of CPSs via using heuristic algorithms, such as GA and PSO, and the control performance improvement of CPSs via solving a multi‐objective optimization problem. This section provides the basics of maintaining the performance indexes of CPSs.

2.3.1 PSO‐Based Reliability Enhancement

In the present section, the PSO method is selected as an example, where a solution consisting of three parameters of the PID control strategy ‐ K_p, T_i and T_d ‐ is encoded as “a particle” in PSO. The fitness value of each particle is determined by the same fitness function. It should be aware of the fact that the reliability of the system may stay at 1 if the total run time T_o is in an early stage of the system's life time, which indicates that the degradation of component is not so severe.

In such a case, the PID control strategy to be adopted not only needs to make the system reliability stay at 1, but also maintain the qualities of the control process at high levels. Therefore, the fitness function should consider two cases: system reliability at 1 and not. If system reliability is 1, the fitness function is a function of qualities of the control process; otherwise, the fitness function is a function of system reliability plus a penalty for system reliability not being at 1.

Thus, the fitness value of a particle can be evaluated by the following equation:

(2.16)

where

and

During each PSO iteration, information about the best position of each particle and the best particle in the entire swarm can be obtained after computing the fitness values of all particles based on (2.16). And then in the next iteration, the movements of the particles are guided by their previous best‐known positions within the search space, called pbest, as well as the best known position of the entire swarm, called gbest.

For a particle j with D dimensions, its local best known position can be represented by pbest_j = (p_j1, p_j2, …, p_jD) and the global best known position of the entire swarm is gbest = p_g. Therefore, the velocity , and the position, associated with the dimension d of the particle j after iteration m are updated according to:

(2.17)

(2.18)

where m ∈ {1, …., M}; j ∈ {1, …., J}; d ∈ {1, …., D}; w is the inertia weight; c₁ and c₂ are the cognition learning factor and the social learning factor; c₁ + c₂ generally equals to 4 and ρ₁ and ρ₂ are random factors restricted to [0, 1].

The followings are some important details of the proposed MCS‐PSO method:

STEP 1: Randomly initialize the particles representing the PID control strategies, including the velocity and the position.

STEP 2: Apply the MCS proposed in section 2.2.2.2 to obtain the R^*(T_o); then compute the fitness value for each particle based on (2.16).

STEP 3: Update the values of pbest and gbest and determine the velocity and position values for each particle in the next iteration on the basis of (2.17) and (2.18).

STEP 4: If the stopping condition ‐ the maximum number of iterations M is reached, go to STEP 5; otherwise, go to STEP 2.

STEP 5: Decode the particle with the minimum fitness value and take the result as the optimally designed PID control strategy which can ensure the maximum possible reliability or the best quality parameters associated with the control process with reliability at 1 at total run time T_o.

2.3.2 Optimal PID‐AGC

The AGC performance of the integrated system depends on the discrete‐time PID controller. Therefore, the PID controller is optimized to mitigate the communication network disturbances and offer optimum AGC performance by reducing system frequency fluctuations. As a result, system reliability is also optimized. In previous works, the objective function for the optimization of the PID controller is an integral performance index over the total operating time T, which quantifies frequency and control signal deviations ⁴⁶⁵⁴⁶⁶:

(2.19)

where η₁ indicates the relative importance of the two terms and η₂ is the normalizing constant to scale both terms in a uniform range and is set to 0.002. The first term is directly related to the reliability of the integrated system and the second term measures the disturbance rejection ability of the controller, which is the total control effort to be minimized. The total process time T is set to 30 s ⁴⁶⁷.

Motivated by ^{468469470471472473}, the objective function in (2.19) is separated into two objectives and multi‐objective optimization is applied. In fact, the weighting factors in the objective function in (2.19) can be changed dynamically using multi‐objective PSO algorithm:

(2.20)

where and . is defined in (2.20). The variable are the controller parameters, i.e., K_P, T_I and T_D, and the 3‐dimensional search space is pre‐specified as G = [0, 10]³ to widely span the optimization range of the controller design ⁴⁷⁴.

The fitness value of the stochastic population‐based algorithms is the expectation of the stochastic objective function obtained from MCS:

(2.21)

where J_i is defined in (2.20) and N = 200 samples.

In this section, we use the PSO rather than other heuristic algorithms, i.e., GA ⁴⁷⁵, to solve the optimization problem for , because PSO can avoid being trapped into local minima ^476477478. The particle movement in the PSO can be interpreted as a form of path re‐linking via the exploitation of the local best‐known positions, i.e., pbest ⁴⁷⁹.

In this sense, both the PSO and GA can be regarded as generating new solutions in the neighborhood of two parents, i.e., using crossover in the GA and using attractions to two pbest positions in the PSO. This multi‐parent effect is a key advantage over single‐point techniques such as the simulated annealing and tabu search.

As compared to GA, the PSO utilizes three available information sets to facilitate the search process. These are the local best‐known positions (pbest), the global best‐known position (gbest), and the current positions ^480481482. This allows greater diversity and exploration over a single population (which with elitism would only be a population of pbests).

In addition, the momentum effects on particle movement allows fast convergence (e.g., when a particle is moving in the direction of a gradient) and increased variety/diversity in search trajectories. The performance of PSO has been shown superior to GA in parameters optimization of PID controller ⁴⁸³⁴⁸⁴. The MOPSO is preferred to the multi‐objective GA because no evolution operators, i.e., the crossover and mutation, are required, and the information of the current optimum particles effectively identifies candidate solutions throughout the problem space.

The detailed introduction to PSO and MOPSO are provided in ^485486487 and ^{488489490491492493}, respectively. The core part of the MOPSO algorithm is detailed in the following.

Notation to be used in MOPSO Algorithm:

A: solution set which provides optimal values of the PID controller parameters K_P, K_I and K_D

NP: number of particles

x_i and v_i: current position and velocity of particle i

MP: number of dimensions (MP = 3)

I and NI: current and maximum number of iterations

ω: inertia coefficient (ω = 0.5 × 0.99^I which tunes the impact of the previous history velocities on the current velocity)

c₁ and c₂: cognitive and social factors (which can accelerate the search towards the local and global best directions, and equal to c₁ = 1 and c₂ = 2)

r₁ and r₂: random numbers drawn from the uniform distribution [0, 1].

Procedures of MOPSO Algorithm:

STEP 1: Randomly initialize the position x_i of the particle i (i = 1, 2, …, NP) sampling from a uniform distribution within the solution space, i.e., G = [0, 10]³. The coordinates of the particle position represent the PID controller parameters, i.e., K_P, K_I and K_D. Initialize v_i, pbest_i and gbest_i to zero.

STEP 2: Compute the fitness value for each particle based on (2.20).

STEP 3: Update the values of pbest and gbest based on calculated fitness values; determine the velocity and position values for each particle in the next iteration using following equations:

(2.22)

(2.23)

where P_i = [pbest_i1, pbest_i2, …, pbest_iMP]^T, G_i = [gbest_i1, gbest_i2, …, gbest_iMP]^T and d = 1, 2, …, MP. The current best solution is put into the set A.

STEP 4: If the maximum number of iterations NI is reached, go to STEP 5; otherwise, go to STEP 2.

STEP 5: Decode the particle with the minimum fitness value and choose the result as the optimally designed PID controller.

As indicated by ^{494495496497498499}, the multi‐objective optimization generates the Pareto optimal set of the non‐dominated solutions from which the global optimum solution is selected based on the specific application. In this study, fuzzy set theory is applied to model the trade‐off between the two objectives ^{500501502503504505}. First, a linear membership function is defined for each objective function :

(2.24)

According to (2.24), the minimum of the objective function J_i is associated to the minimum of the membership function, therefore it has maximum degree of achievement of the fuzzy objective. Then, for every non‐dominated solution k, the aggregate membership function for the PID controller optimization is computed as:

(2.25)

The solution which minimizes (2.25) is regarded as the best compromise between the two objectives and named “Best Tradeoff I”. For comparison, the distance of each non‐dominated solution k from the origin in the fuzzified coordinate is also computed ^{506507508509510511}:

(2.26)

And the solution with minimum MF2^k is chosen as the best compromise between the two objectives and named “Best Tradeoff II”. The convergence of optimization problem solved by the single PSO has been investigated in detail in works ⁵¹²⁵¹³. These works investigated similar optimization problem of PID controller, where both the objective value and the controller parameters become almost constant toward the end of 50 iterations (the number of populations is 30). As for the convergence of optimization problem of PID controller solved by the MOPSO, works ^{514515516517518519} showed fast convergence to the Pareto front in 50 iterations (the number of populations is 50).

Note

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