Advances in Power System Modelling, Control and Stability Analysis
Advances in Power System Modelling, Control and Stability Analysis captures the variety of new methodologies and technologies that are changing the way modern electric power systems are modelled, simulated and operated. The book is divided into three parts. Part 1 presents research works on power system modelling and includes applications of telegrapher equations, power flow analysis with inclusion of uncertainty, discrete Fourier transformation and stochastic differential equations. Part 2 focuses on power system operation and control and presents insights on optimal power flow, realtime control and state estimation techniques. Finally, Part 3 describes advances in the stability analysis of power systems and covers voltage stability, transient stability, time delays, and limit cycles. A rich mix of theoretical aspects with practical considerations, as well as benchmarks test systems and realworld applications makes this book essential reading for researchers and students in academia and industry in electric power systems modelling and control
Inspec keywords: power system simulation; power system state estimation; load flow; power system control; discrete Fourier transforms; stochastic processes; differential equations; power system stability
Other keywords: discrete Fourier transformation; transient stability; time delays; telegrapher equations; power system control; realtime control techniques; benchmarks test systems; state estimation techniques; uncertainty inclusion; power system stability analysis; power flow analysis; power system modelling; stochastic differential equations; voltage stability
Subjects: General and management topics; Probability and statistics; Mathematical analysis; Mathematical analysis; Integral transforms; Integral transforms; General electrical engineering topics; Probability and statistics; Control of electric power systems; Power system control
 Book DOI: 10.1049/PBPO086E
 Chapter DOI: 10.1049/PBPO086E
 ISBN: 9781785610011
 eISBN: 9781785610028
 Page count: 496
 Format: PDF

Front Matter
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Part 1: Modelling
1 Telegrapher's equations for fieldtotransmission line interaction
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In this chapter, we discuss the transmission line theory and its application to the problem of external electromagnetic field coupling to transmission lines, with particular reference to lightninginduced overvoltages on overhead power lines. After a short discussion on the underlying assumptions of the transmission line theory, we provide the derivation of fieldtotransmission line coupling equations for the case of a singlewire line above a perfectly conducting ground. We also describe three seemingly different but completely equivalent approaches that have been proposed in the literature to describe the coupling of electromagnetic fields to transmission lines. The derived equations are extended to deal with the presence of losses and multiple conductors. The timedomain representation of the fieldtotransmission line coupling equations, which allows for a straightforward treatment of nonlinear phenomena as well as the variation in the line topology, is also described. Solution methods in the time domain are presented. The description of the main modelling features of an advanced computer code for the calculation of lightning originated voltages, i.e., the LIOVEMTPRV code, is given. The application of the illustrated theory and relevant computer codes to the case of a typical mediumvoltage multiconductor distribution feeder, which includes transformers and surge protection devices, is presented. The lightning performance assessment of traditional and compact overhead lines is dealt with as well.
2 An affine arithmeticbased methodology for uncertain power flow and optimal power flow analyses
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An affine arithmetic (AA)based computing paradigm aimed at achieving more efficient computational processes and better enclosures of uncertain power flow (PF) and optimal power flow (OPF) solution sets is presented in this chapter. The main idea is to formulate a generic mathematical programming problem under uncertainty by means of deterministic problems, based on equivalent AA minimization, equality and inequality operators. Compared to existing solution paradigms, the described formulation presents a different approach to handle uncertainty, yielding adequate and meaningful PF and OPF solution enclosures. Detailed numerical results are presented and discussed using a variety of test systems, demonstrating the effectiveness of the explained AAbased methodology and comparing it to previously proposed techniques for uncertain PF and OPF analyses.
3 DFTbased synchrophasor estimation processes for Phasor Measurement Units applications: algorithms definition and performance analysis
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In this respect, this chapter first analyses the DFT with a particular focus on the origin of the wellknown aliasing and spectral leakage effects. Then it formulates and validates in a simulation environment a novel SE algorithm, hereafter referred as iterativeInterpolated DFT (iIpDFT), which considerably improves the accuracies of classical DFTand IpDFTbased techniques and is capable of keeping the same static and dynamic performances independently of the adopted window length that can be reduced down to two cycles of signal at the nominal frequency of the power system.

Part II: Control
4 Modelling power systems with stochastic processes
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Any physical system and, thus, also power systems, contains randomness and uncertainty. For example, load power consumption is not fully deterministic. Moreover, in recent years, the massive installation of nondespatchable technologies, e.g., wind parks, has increased the degree of randomness in power systems. This chapter summarizes the state of the art of the modelling of stochastic perturbations in power systems by means of continuous stochastic differential equations (SDEs). The chapter begins with a brief theoretical introduction to SDEs and provides two designing methods of SDEs to represent perturbations with given statistical properties. Perturbation models derived from the application of the presented methods are illustrated through numerical examples. The chapter also describes a general procedure to define stochastic dynamic models for power system components. Practical issues related to the numerical integration of the resulting power system model are discussed. Finally, the dynamic behaviour of power systems subjected to stochastic phenomena is illustrated through simulations of the IEEE 145bus 50machine system.
5 Optimization methods for preventive/corrective control in transmission systems
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This chapter discusses a methodology for the assessment of corrective control actions to be implemented in power system after the occurrence of a severe contingency. This precalculation can be intended as one of the contingency screening functions to be developed within each Supervisory Control and Data Acquisition (SCADA)/Energy Management Systems (EMS) control cycle and in the framework of online Dynamic Security Assessment (DSA). It is assumed that, based on latest system state information and calculations carried out for a selected number of severe fault events, precalculated control actions can be applied on actuators during the power system transient that follows such contingencies. The methodology can be adopted, for example, for the arming of Remedial Actions Schemes based on load shedding, generation shedding, Flexible AC Transmission Systems (FACTS) or any other fast actuator. The goal is preserving power system stability and its integrity even after that a specific severe contingency had occurred. The proposed methodology is based on the conversion of a dynamic optimization problem in the continuous time domain, into a static optimization problem in the discrete time domain. In order to show the key working principle of the proposed approach, a dynamic optimization problem is first formulated in his general form and then solved with the classical method and through discretization. Finally, the methodology is explicitly adapted to the set of equations and constraints that represent the dynamic behaviour of power systems. The same general methodology is then also applied to the solution of a preventive control optimization problem.
6 Static and recursive PMUbased state estimation processes for transmission and distribution power grids
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In the operation of power systems, the knowledge of the system state is required by several fundamental functions, such as security assessment, voltage control and stability analysis. By making reference to the static state of the system represented by the voltage phasors at all the network buses, it is possible to infer the system operating conditions. Until the late 1970s, conventional load flow calculations provided the system state by directly using the raw measurements of voltage magnitudes and power injections. The loss of one measurement made the calculation impossible and the presence of measurement errors affected dramatically the computed state. To overcome these limitations, load flow theory has been combined with statistical estimation constituting the socalled state estimation (SE). The latter consists in the solution of an optimization problem that processes the measurements together with the network model to determine the optimal estimate of the system state. The outputs of load flow and SE are composed of the same quantities, typically the voltage magnitude and phase at all the network buses, but SE uses all the types of measurements (e.g., voltage and current magnitudes, nodal power injections and flows, synchrophasors) and evaluates their consistency using the network model. The measurement redundancy is key to tolerate measurement losses, identify measurement and network parameter errors, and filter out the measurement noise. The foregoing properties of SE allow the system operator to obtain an accurate and reliable estimate of the system state that consequently improves the performance of the functions relying on it.
7 Realtime applications for electric power generation and voltage control
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This chapter deals with realtime applications of digital devices aimed at voltage control of transmission and distribution systems. The focus is on the implementation of the aforementioned devices as digital control systems using microprocessors and generalpurpose computing systems, fitted out with operating systems for general use but with realtime characteristics. With this aim, high level simulation tools designed to model the plant process, and automatically generate a control code appear to be a promising approach to quickly prototype control systems. This chapter discusses in details the application and simulation results of such realtime operating systems (RTOSs) and hardware (HW)software (SW) platforms for voltage control applications in some realworld distribution networks with inclusion of distributed generation.
8 Optimal control processes in active distribution networks
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In this chapter, we consider a centralized realtime control architecture for voltage regulation and lines congestion management in ADNs that is based on a linearized approach that links control variables (e.g., power injections and transformers tap positions) and controlled quantities (e.g., voltages and current flows) by means of sensitivity coefficients. We validate the proposed analytic method by making reference to typical IEEE 13and 34bus distribution test feeders. The numerical validation of the computation of the coefficients is performed using the IEEE 13bus test feeder and it shows that the errors between the traditional approaches, i.e., based on the inverse of the Jacobian matrix, and the analytic method are extremely low (in the order of magnitude of 106109).The IEEE 34bus test feeder is used to show application examples related to a possible integration of the proposed method for the problem of optimal voltage control and lines congestion management in unbalanced distribution systems. The simulation results show that the proposed algorithm is able to improve the voltage and current profiles in the network, and also that when each of the three phases of the DERs can be controlled independently of the others, the resulting optimal voltage and current profiles are better than the ones corresponding to the balanced control of the threephase output of the set points of the DERs.

Part III: Stability Analysis
9 Timedomain simulation for transient stability analysis
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It has been widely recognized that timedomain simulation is the most accurate method to describe power system transient behaviour since it can represent `as they are' controls, nonlinearities, saturation, strong dissipative effects and the `silent sentinels', i.e., the protection system. To counterbalance this interesting feature of the approach, there is the formidable computational burden associated to the simulation of real systems when realtime framework is required for dynamic security assessment, control, etc. However, the structure of the problem presents some interesting characteristics which allowed the use of parallel/distributed computing. This chapter synthesizes the results obtained by the authors in this field.
10 Voltage security in modern power systems
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This chapter deals with the computation of an optimal voltage profile using different optimisation strategies. For this purpose, the mathematical model of the optimisation problem is defined and described considering two issues: defining the constraints of the optimisation problem in order to fulfil the actual operating condition of the SVC system and testing different objective functions. A primal dual interior point method is proposed to solve the OPF problem and the structure of the matrices used by the method is described in detail. In particular, in the OPF models, a quadratic formulation of the PF equations is adopted. In this way, no trigonometrical equations are adopted. The main advantage of this formulation is the robustness of the algorithm.
11 Smallsignal stability and timedomain analysis of delayed power systems
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This chapter describes the impacts that time delays in feedback have on small signal as well as transient stability of power systems. We present a power system model comprising of delay differentialalgebraic equations (DDAEs) and describe general techniques to compute the spectrum of DDAE and to integrate such equations in time domain. The focus is on delays arising in measured signals, e.g., remote frequency measurements for power system stabilizers (PSS) of synchronous machines. Several examples based on a benchmark system, the IEEE 14bus test system, as well as a realworld system are discussed and analysed.
12 Shootingbased stability analysis of power system oscillations
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This chapter shows an application of TDSM and PIT for the determination of limit cycles in power systems. The main advantage of the proposed technique is the ability to determine both stable and unstable periodic orbits in a unique framework. The proposed technique can also cope with hard limits and/or discontinuities, such as switched capacitor banks, on the righthand side of differential equations. Moreover, the technique shows a lower computational burden than other techniques proposed in the literature, e.g., Reference 21. This chapter also discusses an unconventional formulation of the PSM to cope with the requirements of the TDSM. The main feature of the proposed model concerns the representation of the speed reference of synchronous machines. This model is basically a generalized COI and involves a recast of the variables to avoid aperiodic drifting of machine angles. The proposed generalization of the COI indicates that a proper reformulation of synchronous machine equations allows applying techniques that are well assessed in circuit analysis. Rethinking PSMs based on rigorous formalism appears as a challenging field of research.

Part IV: Appendices
Appendix A: Outlines of stochastic calculus
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This appendix outlines relevant concepts of stochastic calculus that are required to the theoretical justifications of the designing methods of stochastic differential equations (SDEs).
Appendix B: Data of lines, loads and distributed energy resources
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The per unit (pu) of length resistance, reactance and susceptance of the used line configuration are given in Tables B.1a and B.1b, respectively, and are represented by threephase matrices. They correspond to the configuration #300 of Reference 1. Table B.2 gives the line data in terms of connected buses and line length.
Appendix C: Proofs and tools for DDAEs
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In this paper the algebraic equations are computed at (t  τ). Since algebraic constraints have to be always satisfied, the some steadystate condition must hold. Finally proofs and tools for DDAE are considered.
Appendix D: Numerical aspects of the probeinsertion technique
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This Appendix discusses the numerical aspects of the probeinsertion technique.

Back Matter
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