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This letter proposes a dual model for grid-forming (GFM) controlled converters. The model is inspired from the observation that the structures of the active and reactive power equations of lossy synchronous machine models are almost symmetrical in terms of armature resistance and transient reactance. The proposed device is able to compensate grid power unbalance without requiring a frequency signal. In fact, the active power control is based on the rate of change of the voltage magnitude. On the other hand, synchronization and frequency control is obtained through the reactive power support. The letter shows that the proposed dual-GFM control is robust and capable of recovering a normal operating condition following large contingencies, such as load outages and three-phase faults.

This paper introduces a novel concept of local synchronization of power systems devices based on the difference between the complex frequency of the voltage and current injected at terminals. Formal definitions are provided to account for bounded and asymptotic local synchronization. The definitions are suitable for modern power systems as they remove classical assumptions limiting the application of the concept of synchronization to synchronous machines and omitting voltage dynamics. The paper also provides a systematic analytical description of the synchronization mechanisms of common power system devices. Finally, a variety of examples is included to illustrate the theoretical value and practical application of the proposed definitions to power systems modeling and stability analysis.

This letter proposes a general dynamic formulation of slack bus. With this aim, the angle constraint imposed by the slack bus is redefined as a set of differential equations and an energy source. The existence and role of the transient component of this source is also discussed in the letter. Based on this framework, the letter shows that the swing equations of synchronous machines can be interpreted as distributed, dynamic, multi-variable, local slack buses. Other relevant cases, including primary and secondary frequency regulation, passive loads as well as grid following and grid forming converters are discussed.

This letter analyses the sources of asymmetry of frequency probability distributions (PDs) and their impact on the dynamic behaviour of power systems. The letter also discusses on how secondary control can reduce this asymmetry. We also propose an asymmetry index based on the difference between the left and right-hand side standard deviations of the frequency PDs. The IEEE 9-bus system and real-world data obtained from the Irish transmission system serve to show that losses, saturation's and wind generation lead to asymmetric PDs. A relevant result is that the droop-based frequency support provided by wind generation using a tight deadband of 15 mHz leads to significantly increase the asymmetry of the frequency PDs.

This paper discusses the stability assessment of low-inertia power systems through a real-world large-scale low-inertia system, namely, the All-Island power system (AIPS) of Ireland and Northern Ireland. This system currently accommodates world-record levels of system non-synchronous penetration namely 75% (planning to increase to 80% next year). The paper discusses one-month results obtained with the state-of-the-art stability tool called look-ahead security assessment (LSAT). This tool carries out rotor-angle, frequency and voltage stability analyses and is implemented in the control centres of the transmission system operators (TSOs). The paper shows that, at the time of writing, the main binding stability constraint of the AIPS is related to the limits on the rate of change of frequency (RoCoF).

This paper discusses emerging operational challenges associated with the integration of solar photovoltaic (PV) in the All-Island power system (AIPS) of Ireland and Northern Ireland. These include the impact of solar PV on: (i) dispatch down levels; (ii) long-term frequency deviations; (iii) voltage magnitude variations; and (iv) operational demand variations. A case study based on actual data from the AIPS is used to analyze the above challenges. It is shown that despite its (still) relatively low penetration compared to wind power penetration, solar PV is challenging the real-time operation of the AIPS, e.g., maintaining frequency within operational limits. EirGrid and SONI, the transmission system operators (TSOs) of the AIPS, are working toward addressing all the above challenges.

The paper revisits the concepts of instantaneous active and reactive powers and provides a novel definition for basic circuit elements based on quantities utilized in classical mechanics, such as absolute and relative velocity, momentum density, angular momentum and apparent forces. The discussion leverages from recent publications by the authors that interpret the voltage and current as velocities in generalized Lagrangian coordinates. The main result of the paper is a general and compact expression for the instantaneous active and reactive power of inductances, capacitances and resistances as a multivector proportional to the generalized kinetic energy and the geometric frequency multivector. Several numerical examples considering stationary and transient sinusoidal and non-sinusoidal conditions are discussed in the case study., Comment: 14 pages, 18 figures

The concept of complex frequency has been recently introduced on the IEEE Transactions on Power Systems to study bus voltage variations in magnitude and frequency and their link with complex power injections of a power system. In this paper, the complex frequency is applied to time-varying series connections, namely, RLC dynamic branches, regulating transformers and AC/DC converters. The proposed modeling approach allows deriving an explicit expression for the complex frequency of the voltage of a certain bus as a linear combination of three elements: net current injected by the devices connected to the bus, adjacent voltages, and time-varying series branches. The proposed formulation unifies the link between voltage and frequency dynamics in AC, DC, as well as hybrid AC/DC power systems. A variety of static and dynamic examples are presented to show the potential of the proposed formulation. Relevant applications of the proposed modeling approach are outlined.

The paper discusses the relationships between electrical quantities, namely voltages and frequency, and affine differential geometry ones, namely affine arc length and curvature. Moreover, it establishes a link between frequency and time derivatives of voltage, through the utilization of affine differential geometry invariants. Based on this link, a new instantaneous frequency estimation formula is proposed, which is particularly suited for unbalanced systems. An application of the proposed formula to single-phase systems is also provided. Several numerical examples based on balanced, unbalanced, as well as single-phase systems illustrate the findings of the paper.

This letter proposes a simple and inexpensive technique to improve the frequency control of distributed energy resources. The proposed control consists in modifying the conventional estimated bus frequency signal with an additional feedback signal that utilizes the rate of change of the voltage magnitude measured at the same bus. The case study showcases the benefits of the proposed control and compares its performance with standard frequency control schemes through time-domain simulations.

This paper discusses the conditions that a device needs to satisfy to replicate the behavior of a conventional synchronous machine (SM) connected to a power network. The conditions pertain to the device's stored energy, time scale of response, oscillation damping, and behavior during short-circuits. Relevant remarks for devices that do/don't satisfy these conditions are discussed through an illustrative numerical example as well as through simulation results based on a modified version of the well-known WSCC 9-bus test system.

The All-Island power system (AIPS) of Ireland and Northern Ireland currently accommodates up to 75% of system non-synchronous penetration (SNSP) (e.g., wind). These unprecedented levels of renewable penetration challenge the operation of the power system. The AIPS is not always able to accommodate all of the available renewable generation due to binding operational and technical constraints. In this context, this paper analyses wind energy curtailment in the AIPS using actual data. It is found that there is a positive correlation between the installed wind capacity and curtailment levels, and that the trend is that these levels increase. The paper also shows that the main driver for curtailment in AIPS during 2020-2021 was the operational constraint that imposes a minimum number of conventional units online (MUON) (80% of the time), with the SNSP limit accounting for less than 20%. Other system-wide limits, such as rate of change of frequency (RoCoF) and inertia are found to have a negligible impact on wind curtailment.

This paper analyses the issue of frequency quality in low-inertia power systems. The analysis is based on a real-world large-scale low-inertia power system namely, the All-Island transmission system (AITS) of Ireland and Northern Ireland currently accommodating up to 75% of non-synchronous generation. The paper is motivated by a recent trend of some frequency quality parameters such as the standard frequency deviation and the slow frequency restoration. The paper first discusses the frequency control services currently in place to ensure frequency quality in the AITS. An analysis of the frequency quality parameters of the AITS is then presented based on actual data. The paper also discusses, through an illustrative example, the effectiveness of automatic generation control as a potential approach to keep frequency within the operational range.

This paper proposes a taxonomy of power converter control schemes based on the recently proposed concept of complex frequency. This quantity captures local frequency variations due to the change of both the phase angle and amplitude of bus voltages and current injections. The paper derives the analytical expressions of the link between complex power variations and complex frequency of each converter controller as well as the identification of critical control parameters. The main contribution of this work is to provide a general framework that allows classifying converters synchronization mechanisms and controllers. This framework also allows comparing converters with synchronous machines. To validate the theoretical results, extensive simulations are performed using a modified version of the WSCC 9-bus system. Examples of how the theoretical formulations of the paper can be used to improve power converter control in power system applications are showcased., Comment: 10 pages, 22 figures

This letter presents a technique to calculate the variance of algebraic variables of power system models represented as a set of stochastic differential-algebraic equations. The technique utilizes the solution of a Lyapunov equation and requires the calculation of the state matrix of the system. The IEEE 14-bus system serves to demonstrate the accuracy of the proposed technique over a wide range of variances of stochastic processes. The accuracy is evaluated by comparing the results with those obtained with Monte Carlo time domain simulations. Finally, a case study based on a 1479-bus dynamic model of the all-island Irish transmission system shows the computational efficiency of the proposed approach compared to the Monte Carlo method.

This work presents a technique to estimate on-line the inertia of a power system based on ambient measurements. The proposed technique utilizes the covariance matrix of these measurements and solves an optimization problem that fits such measurements to the synchronous machine classical model. We show that the proposed technique is adequate to accurately estimate the actual inertia of synchronous machines and also the virtual inertia provided by the controllers of converter-interfaced generators that emulate the behavior of synchronous machines. We also show that the proposed approach is able to estimate the equivalent damping of the classical synchronous machine model. This feature is exploited to estimate the droop of grid-following converters, which has a similar effect of the swing equation equivalent damping. The technique is comprehensively tested on a modified version of the IEEE 39-bus system as well as on a dynamic 1479-bus model of the all-island Irish transmission system.

This paper proposes a general framework to interpret the concept of Instantaneous Frequency (IF) in three-phase systems. The paper first recalls the conventional frequency-domain analysis based on the Fourier transform as well as the definition of IF which is based on the concept of analytic signals. The link between analytic signals and Clarke transform of three-phase voltages of an ac power system is also shown. Then the well-known five paradoxes of the IF are stated. In the second part of the paper, an approach based on a geometric interpretation of the frequency is proposed. This approach serves to revisit the five IF paradoxes and explain them through a common framework. The case study illustrates the features of the proposed framework based on a variety of examples and on a detailed model of the IEEE 39-bus system., Comment: 12

The paper proposes a generalization of the Park transform based on the Frenet frame, which is a special set of coordinates defined in differential geometry for space curves. The proposed geometric transform is first discussed for three dimensions, which correspond to the common three-phase circuits. Then, the expression of the time derivative of the proposed transform is discussed and the Frenet-Serret formulas and the Darboux vector are introduced. The change of reference frame and its differentiation based on Cartan's moving frames and attitude matrices are also described. Finally, the extension to circuits with more than three phases is presented. The features of the Frenet frame are illustrated through a variety of examples, including a case study based on the IEEE 39-bus system., Comment: 11 pages, 2 figures, accepted for publication on the IEEE Transactions on Circuits and Systems I: Regular Papers

The paper provides a novel framework to study the accuracy and stability of numerical integration schemes when employed for the time domain simulation of power systems. A matrix pencil-based approach is adopted to evaluate the error between the dynamic modes of the power system and the modes of the approximated discrete-time system arising from the application of the numerical method. The proposed approach can provide meaningful insights on how different methods compare to each other when applied to a power system, while being general enough to be systematically utilized for, in principle, any numerical method. The framework is illustrated for a handful of well-known explicit and implicit methods, while simulation results are presented based on the WSCC 9-bus system, as well as on a 1, 479-bus dynamic model of the All-Island Irish Transmission System.

The paper discusses the relationships between electrical quantities, such as voltages, currents, and frequency, and geometrical ones, namely curvature and torsion. The proposed approach is based on the Frenet frame utilized in differential geometry and provides a general framework for the definition of the time derivative of electrical quantities in stationary as well as transient conditions. As a byproduct, the proposed approach unifies and generalizes the time- and phasor-domain frameworks. Other noteworthy results are a new interpretation of the link between frequency and the time derivatives of voltage and current; and a definition of the rate of change of frequency that includes the novel concept of "torsional frequency." Several numerical examples based on balanced, unbalanced, harmonically-distorted and transient voltages illustrate the findings of the paper.

The paper introduces the concept of complex frequency. The imaginary part of the complex frequency is the variation with respect of a synchronous reference of the local bus frequency as commonly defined in power system studies. The real part is defined based on the variation of the voltage magnitude. The latter term is crucial for the correct interpretation and analysis of the variation of the frequency at each bus of the network. The paper also develops a set of differential equations that describe the link between complex powers and complex frequencies at network buses in transient conditions. No simplifications are assumed except for constant elements of the network admittance matrix. A variety of analytical and numerical examples show the applications and potentials of the proposed concept.

The letter provides a geometrical interpretation of frequency in electric circuits. According to this interpretation, the frequency is defined as a multivector with symmetric and antisymmetric components. The conventional definition of frequency is shown to be a special case of the proposed theoretical framework. Several examples serve to show the features, generality as well as practical aspects of the proposed approach.

Virtual inertia controllers (VICs) for wind turbine generators (WTGs) have been recently developed to compensate for the reduction of inertia in power systems. However, VICs can induce low-frequency torsional oscillations of the drive train of WTGs. This paper addresses this issue and develops a new nonlinear VIC based on objective holographic feedbacks theory. This approach allows transforming the objectives that require improvement into a completely controllable system of Brunovsky's type. Simulation results under various scenarios demonstrate that the proposed method outperforms existing VICs in terms of suppression of WTG low-frequency drive-train torsional oscillations, enhancement of system frequency nadir as well as fast and smooth recovery of WTG rotor speed to the original MPP before the disturbance. The proposed method is also able to coordinate multiple WTGs., Comment: 8 pages, 14 figures

This paper investigates the chattering and deadlock behaviors of the proportional-integral (PI) controller with an anti-windup (AW) limiter recommended by the IEEE Standard 421.5-2016. Depending on the simulation method, the controller may enter a chattering or deadlock state in some combinations of parameters and inputs. Chattering and deadlock are analyzed in the context of three numerical integration approaches: explicit partitioned method (EPM), execution-list based method (ELM), and implicit trapezoidal method (ITM). This paper derives the chattering stop condition for EPM and ELP, and analyzes the impacts of step size and convergence tolerance for simultaneous method. The deduced chattering stop conditions and deadlock behavior is verified with numerical simulations.

This paper provides a systematic method to build wind speed models based on stochastic differential equations (SDEs). The resulting models produce stochastic processes with a given probability distribution and exponential decaying autocorrelation function. The only information needed to build the models is the probability density function of the wind speed and its autocorrelation coefficient. Unlike other methods previously proposed in the literature, the proposed method leads to models able to reproduce an exact exponential autocorrelation even if the probability distribution is not Gaussian. A sufficient condition for the property above is provided. The paper includes the explicit formulation of SDE-based wind speed models obtained from several probability distributions used in the literature to describe different wind speed behaviors., Comment: 17 pages, 25 references

This paper presents a generalized energy storage system model for voltage and angle stability analysis. The proposed solution allows modeling most common energy storage technologies through a given set of linear differential algebraic equations (DAEs). In particular, the paper considers, but is not limited to, compressed air, superconducting magnetic, electrochemical capacitor and battery energy storage devices. While able to cope with a variety of different technologies, the proposed generalized model proves to be accurate for angle and voltage stability analysis, as it includes a balanced, fundamental-frequency model of the voltage source converter (VSC) and the dynamics of the dc link. Regulators with inclusion of hard limits are also taken into account. The transient behavior of the generalized model is compared with detailed fundamental-frequency balanced models as well as commonly-used simplified models of energy storage devices. A comprehensive case study based on the WSCC 9-bus test system is presented and discussed.