Advances in Power System Modelling, Control and Stability Analysis (2nd Edition)
Power systems are becoming increasingly complex as well as flexible, able to integrate distributed renewable generation, EV, and additional loads. This expanded and updated second edition covers the technologies needed to operate modern power grids.
Initial chapters cover power system modelling, telegrapher equations, power flow analysis, discrete Fourier transformation and stochastic differential equations. Ensuing chapters deal with power system operation and control, power flow, realtime control and state estimation techniques for distribution systems as well as shipboard systems. The final chapters describe stability analysis of power systems and cover voltage stability, transient stability, time delays, and limit cycles. New content for the second edition includes four new chapters on recent modelling, control and stability analysis of power electronic converters and electric vehicles.
This new edition is an essential guide to technologies for operating modern flexible power systems for PhD students, earlycareer researchers and practitioners in the field.
 Book DOI: 10.1049/PBPO217E
 Chapter DOI: 10.1049/PBPO217E
 ISBN: 9781839535758
 eISBN: 9781839535765
 Page count: 764
 Format: PDF

Front Matter
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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 Reliable solutions of uncertain optimal power flow problems by affine arithmetic
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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 PF (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, 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. Finally, the concept of secondorder AA is introduced for mitigating the overconservatism of the conventional AAbased operators in solving uncertain programming problems.

3 DFTbased synchrophasor estimation processes for Phasor Measurement Units applications: algorithms definition and performance analysis
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This chapter has discussed the main elements related to the definition of DFTbased SE algorithms since they represent the most commonly adopted ones in real PMU devices. In particular, this chapter has focused on the analysis of spectral leakage as it represents the most relevant source of uncertainty when using the DFT to estimate the parameters of a sinusoidal signal. This aspect is of importance in SE processes since they usually adopt relatively short windows to reduce the PMU measurement reporting latencies and RTs.
This chapter has also presented stateoftheart SE algorithms belonging to the family of IpDFT estimators. This chapter has discussed both the classical IpDFT as well as its iterative counterpart (iIpDFT) capable of dealing with the compensation of the effects of the selfinteraction between the positive and the negative images of the spectrum. Such a technique has demonstrated to improve the classical IpDFT performances during both static and dynamic conditions described in the IEEE Std. C37.118 and to be immune to the instantaneous frequency variations of a power system. Furthermore, it has been demonstrated that the iIpDFT technique outperforms classical IpDFT methods, also by adopting shorter windows (up to two periods) that are usually worsening the estimation uncertainty of any SE algorithm.

4 Modeling 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 nondispatchable technologies, e.g., wind and solar parks, has increased the degree of randomness in power systems.
This chapter summarizes the stateoftheart of the modeling of stochastic perturbations in power systems by means of stochastic differential equations (SDEs). The chapter begins with a brief theoretical introduction to SDEs and provides 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 1,479 bus Irish power system model.

5 Detailed modeling of inverterbased resources
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WPs consist of a large variety of interconnected components including mechanical parts, power electronic devices, control and protection systems, etc. Accurate and generic models for different types of WTs are crucial for reliable design and planning of modern power systems that incorporate WPs. This chapter introduces an effective approach towards detailed modeling and simulation of WPs that employ variable speed WTs. Specifically, converters and their control schemes for DFIG and FSC WTs are thoroughly discussed, and their main parameters are explained. Moreover, software implementation, average value and detailed models, and controller design are addressed. EMTP® is used to verify the accuracy of the generic models under different test cases. Timedomain simulation results are analyzed and compared with the reallife measurements of postfault transients in the test scenarios. The results confirm that the developed models in conjunction with the EMT simulations can accurately predict the response of DFIG and FSC WTs under both steadystate and transient conditions.

6 Isomorphismbased simulation of modular multilevel converters
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This chapter shows how the isomorphismbased approach can be applied to perform efficient EMT simulations of MMCS. This approach has three main features, which have been validated by adopting it to simulate a benchmark system comprising two MMCs. First, the approach is compatible with any MMC operating condition (ranging from ordinary operation to faults in the AC and DC networks) and SM model. Indeed, depending on the degree of detail required, the user can flexibly choose the complex FP, the simple BVR representation, or even a brand new custom SC representation. Second, regardless of the SM model adopted, the CPU time required to simulate MMCS increases only almost linearly with the number of SMS in each MMC arm. Lastly, the proposed simulation approach paves the way for component level studies that require the adoption of the FP model of SMS . This kind of analysis would be possible with neither conventional simulation approaches nor the MMC representations outlined in Section 6.2. Indeed, in the former case, the adoption of the FP representation would lead to prohibitive simulation times. On the contrary, in the latter case, the use of accelerated MMC models prevents the analysis of switching transients within the SMS either because they adopt too simplified SM models or they do not retain their individual behavior.

7 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.

8 Static and recursive PMUbased state estimation processes for transmission and distribution power grids
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The chapter starts by providing the measurement and process model of WLS and KF SE algorithms and continues with the analytical formulation of the two families of state estimators, including their linear and nonlinear versions as a function of the type of available measurements. Finally, two case studies targeting IEEE transmission and distribution reference networks are given.

9 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.

10 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 13 and 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 10^{−6}  10^{−9}). 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.

11 Control of converter interfaced generation
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The chapter presented an overview of implementation options for grid connected converters. The evolution of the presented solutions also represent how the role of power electronic converters has been evolving over the years. The gridfollowing concept can be seen as the first stage of an ongoing development, in which the role of renewable power generation was initially considered marginal in the power system context. In the recent years, correspondingly with the growing penetration of wind and solar installations, the attention has been moved toward gridforming approaches because of their capability to play a key role in grid dynamics. Solutions like VSM can be seen as a sort of continuity with the past but reinterpreted with the programmability of power electronics. The control freedom provided by power electronics is now offering the possibility to consider also completely new approaches that are intended for a fully electronic grid. The VOC is one example in this direction which, by exploiting nonlinear dynamics, offers a new and more robust way to work in grids without significant physical inertia from rotating machines.

12 Combined voltagefrequency control with power electronicsbased devices
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A relevant question related to the regulation provided by power electronicsbased devices is how to utilize efficiently their active and/or reactive control loops and which control signals to dedicate to which control objectives. These devices are faster and, at least with respect to control capabilities, more versatile than conventional synchronous machines. Powerelectronic converter represents thus both a great challenge and an unprecedented opportunity for the dynamic performance of power systems. This chapter focuses on the opportunities and describes a set of control schemes that improve the overall power system dynamic response through a combined voltagefrequency regulation strategy. The schemes described in this chapter are intended for any power electronicsbased devices, including distributed energy resources (DERs) and energy storage systems (ESSs), as well as flexible AC transmission system (FACTS) devices. The performance of the control schemes in this chapter is illustrated through examples based on benchmark test systems as well as on a realistic model of the Irish transmission system.

13 Smart transformer control of the electrical grid
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This chapter introduces the smart transformer (ST) concept, a power electronicsbased transformer that, in addition to the voltage transformation, can offer enhancing services to the grid. The chapter begins with a brief introduction on the ST concept and gives an overview of the offered services. In the second section, the ST architecture and control for each transformation stage are described, analyzing different topology alternatives. Basic and more advanced services provided to the grid are described in the third section that includes the innovative concept to regulate the grids power consumption by means of controlled voltage or frequency variations. The chapter closes with an overview of innovative grid concepts, such as DC grids.

14 On the interactions between plugin electric vehicles and the power grid
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Growing concerns over climate changes have driven regulatory pressures to reduce urban pollution, emissions from CO_{2} and other particulate matters, and city noise, which have motivated intense activity in the search for alternative road transportation propulsion systems. In this context, plugin electric vehicles (PEVs), either as purely electric vehicles or as plugin hybrid vehicles, have become a significant fraction of the overall transportation fleet in many countries worldwide.
This everincreasing penetration level of PEVs is posing significant challenges to the existing power grids' infrastructures, as a significant load in terms of the required energy (e.g., during night charging of the vehicles connected for charging) or in terms of the required power (e.g., during fast charging events). The uncertainty of the charging events (in terms of time, space and required energy) further challenge the ability of the power grids to seamlessly handle such new electrical loads in addition to the already existing base load. At the same time, however, the ability of PEVs to operate in a vehicletogrid (V2G) mode may be also exploited to provide ancillary regulatory services to facilitate the general operation of the power grid. The optimal tradeoff between these two contrasting aspects of the charging problem is still a subject of intense study in the power research community.
Accordingly, this chapter reviews the most recent research and technological advances in the charging process of PEVs; it presents through simple simulations the importance of controlled charging, and the advantages of utilizing decentralized schemes for practical implementation; also, it identifies some particularly interesting new trends that can be observed at the intersection between the transportation and the power networks, and it outlines interesting future directions in the context of electric vehicles.

15 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.

16 Voltage security in modern power systems
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The control of voltages and reactive power has become more and more critical in the power system operation, in recent years, due to the presence of the electricity market and the strong penetration of the RES that push system operators and electrical utilities to operate the transmission networks as close as possible to their maximum capacity. To improve voltage control in transmission grids, many projects have been developed around the world. The Hierarchical Voltage Control System, which is based on network area and resources subdivision, although developed by vertically integrated utilities in the past, is widely recognised as a viable solution and was adopted in several countries around the world.
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: (i) defining the constraints of the optimisation problem in order to fulfil the actual operating condition of the SVC system and (ii) testing different objective functions. A primaldual 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.
Different objective functions are presented and discussed with and without the presence of SVC scheme and comparative tests were made on two different network models. The first one is a small test system (the New England test system modified to include the SVC), and the second one is a model of the real Italian transmission system; the main characteristics of these different OPF models are emphasised for these particular grids. In both of these cases, and for each objective function, the solution of the OPF problem is found within few iterations and no particular numerical problems are identified. In general, the described approaches show good convergence properties also for systems of high dimension.

17 Smallsignal stability and timedomain analysis of delayed power systems
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This chapter describes the impacts that time delays in feedback control loops have on the smallsignal as well as on the transient stability of power systems. We present a power system model comprising of delay differential algebraic equations (DDAEs) and describe general techniques to compute the spectrum and numerically integrate such model. The focus is on delays arising in measured signals, e.g., remote frequency measurements for power system stabilizers (PSSs) of synchronous machines (SMs). Several examples are discussed based on the IEEE 14bus benchmark system as well as on a realistic model of the Irish transmission system.

18 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 right hand side of differential equations. Moreover, the technique shows a lower computational burden than other techniques proposed in the literature, e.g., [21].
The chapter also discusses an unconventional formulation of the power system model 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 centre of inertia and involves a recast of the variables to avoid aperiodic drifting of machine angles. The proposed generalization of the centre of inertia indicates that a proper reformulation of synchronous machine equations allows applying techniques that are wellassessed in circuit analysis. Rethinking power system models based on rigorous formalism appears as a challenging field of research.

19 Stability assessment in advanced DC microgrids
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The DC technology appears as promising in enabling new advanced microgrids. The reason is to be sought in the distributed implementation of PV renewable energy sources and battery storage systems, which actually operate by a DC distribution. When a widespread use of power converters guarantees the DC microgrids functionality, an attentive evaluation is to be carried out on the DC stability matter. The present chapter wants to investigate the methods to assess the stability in isolated DC distribution systems. As the latter can be lost when high are the converters control bandwidths, analytical developments are proposed to correlate requested control performance and poles positioning. If the methodology is initially conceived for a classical DC radial distribution, a final study will transfer it on complex DC Zonal Electrical Distribution Systems.

20 Scanning methods for stability analysis of inverterbased resources
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Recently, attempts have been made to understand and analyze the dynamic interactions between the power grids and the inverterbased resources (IBR) such as wind and solar photovoltaic (PV) parks. These adverse incidents that mainly stem from the controller interactions can lead to unwanted oscillations in sub or supersynchronous frequency ranges, and thus jeopardize the reliable operation of power systems. To address such stability analysis issues, frequencydependent impedance scanning techniques based on smallsignal perturbations have been developed. This chapter deals with detailed explanation of different scanning methods that can be employed for predicting the stability issues and characterizing the sub or supersynchronous oscillations. Implementation, computational burden, accuracy, and stability criteria for different scanning methods are also discussed. Stability assessment based on the scanning methods is investigated in three practical benchmark systems that involve full size converter (FSC) and doublyfed induction generator (DFIG) wind parks. The positivesequence, the dq and the αβ scans are applied to stable and unstable cases in each benchmark and the results are verified by the electromagnetic transient (EMT) simulations.

Appendix A: Stochastic process fitting procedure
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The fitting procedure presented in Section 4.2.2 involves fitting the function in (4.23) to the Autocorrelation Function (ACF) of the data as well as identifying a Probability Density Function (PDF) that best captures the probability distribution of the data. The procedure is demonstrated below with an example measured wind speed data set. The data set includes 3 years of data sampled and averaged hourly. The measurement location is Mace Head, Galway, Ireland [12].

Appendix B: Modeling correlated stochastic processes
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No Abstract available.

Appendix C: Data of lines, loads and distributed energy resources
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No Abstract available.

Appendix D: Proofs and tools for DDAEs
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No Abstract available.

Appendix E: Numerical aspects of the probeinsertion technique
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No Abstract available.

Back Matter
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