The paper proposes a novel general definition of coherency among power system devices of any type. The proposed approach is thus not limited to synchronous machines. With this aim, the paper shows that coherency can be formally based on the difference in the complex frequency of the current injections of any two devices electrically connected to the same grid. The proposed definition is model-agnostic, making it general and suitable for modern power systems composed of a heterogeneous mix of technologies. The paper also provides a systematic analytical procedure to study the properties that specific device models must satisfy to be coherent. Time-domain simulations are conducted in three case studies whose results illustrate the ability of our definition to evaluate coherency among any type of device.

This paper proposes a novel control framework designed for Inverter-Based Resources (IBRs), denoted as Generalized Swing Control (GSC). The proposed GSC framework generalizes the definition of Grid-Forming (GFM) control schemes and exploits the coupling between active and reactive power dynamics. To validate the proposed scheme, we conduct extensive time-domain simulations and small-signal analysis using a modified version of the WSCC 9-bus system and a 1479-bus dynamic model of the all-island Irish transmission system. The case studies focus on evaluating the dynamic performance of the proposed framework under different configurations, including Virtual Synchronous Machine (VSM), coupled-VSM and dual-VSM schemes. To address the nonlinear nature of power system dynamics, sensitivity analysis based on Monte Carlo methods are employed to improve parameter tuning and assess the stability of GSC configurations in the studied systems.

This paper analyzes how power injections affect frequency quality in power systems. We first derive a general expression linking active and reactive power injections at buses to the system's frequency. This formulation explicitly considers both real and imaginary frequency components, providing a complete description of frequency behavior in power systems during transients. Next, we extend our analysis to incorporate stochastic variations of power injections. Using the frequency divider concept and power-based frequency estimation, we develop analytical relationships linking stochastic load fluctuations to frequency deviations. We discuss under which conditions the Central Limit Theorem cannot be applied to capture the frequency distribution, thereby clarifying how its hypotheses are not satisfied in power system applications. Then, we establish clear criteria for the appropriate use of statistical methods in frequency analysis. Finally, we validate our theoretical results through simulations on modified IEEE 14-bus and all-island Irish transmission test systems, highlighting the accuracy, practical utility, and limitations of our proposed formulation.

The utilization of voltage source converters (VSCs) in modern power grids has been increasing dramatically along with the rapid developments of renewable power generations and VSC-based high-voltage DC (VSC-HVDC) transmission systems, leading to overall reduced synchronous generator (SG) inertia and elevating the risk of large-scale power blackouts. Enabling VSCs to mimic SG inertial response has become a critical approach for the stable operation of VSC-dominated power grids. This paper presents a comprehensive review of VSC inertia emulation control (INEC) schemes. The grid low-inertia challenges and technological demands for INEC are first analyzed. Then, the existing VSC INEC algorithms are divided into four categories based on synchronization methods and control objectives, with their control principles elaborated. Frequency- and time-domain analyses reveal that while grid-following-based INECs offer simple implementation, they are prone to instability in weak grids. Conversely, grid-forming-based INECs exhibit superior performances in weak grids, but may be subject to power oscillation problems in stiff grids. Moreover, typical energy sources for INEC, including rotor kinetic energy of wind turbine generators, energy storage systems and VSC-HVDC interconnections, are comprehensively investigated. Techno-economic analyses identify supercapacitors as an attractive option due to their rapid response speed, high power density and long lifetime, despite comparatively high capital costs. Finally, several emerging issues and suggested future works related to VSC INEC are discussed. Importantly, it concludes that overcoming current implementation barriers of INEC technologies requires optimally selecting and sizing energy sources, optimizing dynamic performances, and establishing uniform technical specifications and mature market mechanisms to promote INEC transition from a theoretical concept to one of standardized ancillary services.

This industry-oriented paper originates from the observation that current frequency quality metrics utilized by transmission system operators (TSOs) fail to fully capture the dynamic behavior of the grid frequency. Motivated by this gap, the paper proposes novel frequency quality metrics based on second-order dynamics and stochastic autocorrelation. Using real-world data from the Irish, Great Britain and Nordic systems and running dynamic stochastic simulations, the paper shows that the proposed metrics bring new and counterintuitive insights in terms of how good or poor the frequency quality of power grids is beyond current well-known metrics. In particular, the paper shows that a power system may show good frequency quality using standard metrics and poor frequency quality using the proposed metrics. Overall, the paper contributes to improve the understanding of frequency quality.

This paper proposes a general framework to evaluate power system strength. The formulation features twelve indicators, grouped in three dynamical orders, that quantify the resistance of bus voltage phasors and their first and second order rates of change to sudden current injection changes. To quantify such changes the paper introduces a novel finite differentiation technique, that we named Delta operator, able to properly capture "jumps" of algebraic variables and utilizes the recently developed concept of complex frequency. The paper also shows how the proposed framework can be systematically applied to any system device, and provides a variety of examples based on synchronous machines, converters and loads models are given. Numerical results in a benchmark system validate the exactness of the formulation.

For grid-forming (GFM) converters controlled in the positive and negative sequence synchronous reference frames (pn frames), this paper develops the pn-frame small-signal impedance models under four symmetrical component extraction (SCE) and negative-sequence control (NSC) methods, namely, delay signal cancellation (DSC), dual second order generalized integrator (DSOGI), decoupled double synchronous reference frame (DDSRF), and notch-filter (NF)-based SCE method. Then, using these pn-frame models, we conduct the small-signal stability analysis of GFM converter systems and study the impact of SCE and NSC as well as of voltage and current control loop (VCL and CCL) bandwidth on system stability. The impedance analysis results indicate that a higher product of the VCL and CCL bandwidths is beneficial to the sub-synchronous oscillation (SSO) stability of the GFM converter system. Furthermore, while keeping the product unchanged, increasing the VCL bandwidth has a positive effect on the SSO stability equivalent to that of increasing the CCL bandwidth. Finally, the use of DSC-SCE and NSC reduces the SSO stability and raises the upper bound of the VCL and CCL bandwidth product, whereas the use of NFSCE exhibits SSO stability comparable to that without SCE and NSC. In contrast, comparing with DSC- and NF-SCE, the use of DSOGI/DDSRF-SCE makes the stable range of the VCL and CCL bandwidths much smaller (i.e. makes the upper bound of their product much higher), and introduces an additional high frequency instability issue in the GFM converter system.

This paper proposes a reliable and nonintrusive approach for calculating economical operating states and at the same time, maintaining the transient stability of the system. It focuses on providing a preventive control strategy to address transient rotor angle stability and short-term voltage stability issues resulting from severe disturbances. The objective function, steady-state and multiple types transient stability constraints are equivalently and simultaneously reformulated in polynomial expressions of the control variables such that the original differential-algebraic equations in the transient stability-constrained optimal power flow model are eliminated. This reformulation also allows imposing both rotor angle and voltage magnitude bounds that achieve transient stability and reduce the complexity in the optimization stage. A transient constraint reduction strategy is also proposed to address the large number of constraints introduced by the polynomial chaos expansion when transforming multiple types of transient constraints. This strategy enables the elimination of most transient constraints, thereby further simplifying the overall optimization model. The effectiveness of the proposed method is numerically illustrated and validated through the WECC 3-machine 9- bus system and the IEEE 69-machine 300- bus system.

This article proposes a hierarchical site selection framework of grid-forming energy storage (GFM-ESS) devices. This framework enhances stability of both the device itself and the power system across multi-timescales with respect to static, dynamic, and transient stability layouts. Specifically, the first layout optimizes static voltage stability margin and system losses. The second layout evaluates small-signal stability based on the generalized short-circuit ratio (gSCR). Finally, the third layout maximizes the GFM-ESS supporting performance in response to the system dynamics. A case study based on Western China serves to validate the effectiveness of the proposed GFM-ESS site selection strategy and to prove its scalability for large real-world power systems.

This industry-oriented paper introduces the concept of ‘frequency control strength’ as a novel approach to understand how different real-world power systems compare to each other in terms of effectiveness and performance of system-wide frequency control. It presents a comprehensive comparison, based on measurement data, of the frequency control strength of four real-world, renewable-based, synchronous island power systems, namely Great Britain (GB), the All-Island power system (AIPS) of Ireland, and Australia (AUS) mainland and Tasmania (TAS). The strength is evaluated by means of different frequency quality metrics. The common understanding is that the bigger the capacity of a power system, the bigger its robustness with respect to events and contingencies. Here we show that this is not always the case in the context of frequency control. In fact, our study shows that mainland AUS shows the highest frequency control strength during normal operating conditions, whereas the AIPS shows the highest relative frequency control strength for abnormal system conditions. The strength is, in particular, greatly influenced by different regulatory requirements and different system/ancillary services arrangements in each jurisdiction. The paper also provides possible mitigations to improve frequency control strength through grid codes and market rules.

This paper shows that the concept of complex frequency, originally introduced to characterize the dynamics of signals with complex values, constitutes a generalization of eigenvalues when applied to the states of linear time-invariant (LTI) systems. Starting from the definition of geometric frequency, which provides a geometrical interpretation of frequency in electric circuits that admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion, respectively, we show that complex frequency arises as its restriction to the two-dimensional Euclidean plane. For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order. The paper provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves. The paper also highlights that this equivalence does not generally hold for nonlinear systems. On the other hand, the geometric frequency of the system can always be defined, providing a geometrical interpretation of the system flow. A variety of examples based on linear and nonlinear circuits illustrate the proposed framework.

This paper focuses on power flow analysis through the lens of the Newton flow, a continuous-time formulation of Newton's method. Within this framework, we explore how quantized-state concepts, originally developed as an alternative to time discretization, can be incorporated to govern the evolution of the Newton flow toward the power flow solution. This approach provides a novel perspective on adaptive step-size control and shows how state quantization can enhance robustness in illconditioned cases. The performance of the proposed approach is discussed with the ACTIVSg70k synthetic test system.

In converter-dominated grids, the earliest post-fault interval is often limited by operating-quantity extraction, because cycle-based discrete Fourier transform (DFT) phasors can incur finite-window bias under fast control actions, current limiting, and waveform distortion. This paper develops a trajectory based, sub-cycle operating-quantity extractor that acts directly on instantaneous three-phase measurements. Leveraging the recently proposed concept of geometric frequency (GF) and using short-window least-squares differentiation, the proposed method separates the instantaneous stretching and rotation of the three phase trajectory and converts them into a closed-form equivalent series R–L drop. Resulting parameters are mapped onto the conventional R–X plane to drive standard mho and quadrilateral logic. A memory-polarized directional torque is also formed from instantaneous three-phase vectors to maintain sign consistency under voltage depression while reducing cycle-length reporting latency. Electromagnetic transient (EMT) case studies on a wind farm export corridor and real-time digital simulator (RTDS) based validation on a modified IEEE 9-bus transmission system verify the effectiveness of the proposed GF-based distance and directional protection method.

In power networks based on Inverter-Based Resources (IBRs), fast controllers cause frequency and voltage dynamics to overlap. Thus, it becomes critical to assess the overall dynamic performance of such networks through a combined system-wide metric. This letter presents a unified metric designed to evaluate dynamic performance in such cases. The proposed metric consists of a weighted sum of local voltage phasor variations at each bus, where the weights are the complex powers injected at the buses. The proposed metric is further decomposed into device-driven and network-driven components, enabling a more comprehensive assessment of grid dynamics. A case study based on a modified version of the IEEE 39-bus system is presented, in which synchronous machines are replaced by inverter-based resources. A sensitivity analysis of the R/X ratio is utilized to evaluate the metric in conventional grids, as well as in those characterized by strong voltage-frequency coupling with complex power flows.

Frequency control in power systems is implemented in a hierarchical structure traditionally known as primary frequency control (PFC), secondary frequency control (SFC) and tertiary control reserve (TCR) and, some jurisdictions, include time error control (TEC) as well. This hierarchical structure has been designed around a century ago based on timescales separation, that is, approximately an order of magnitude difference between each control structure. This paper argues, based on real-world observations as well as detailed dynamic simulations on a model of the All-Island power system (AIPS) of Ireland, that this frequency control structure is not necessary in current and future converter-dominated power grids. The paper proposes to redesign this structure by removing the SFC and TCR and rely on PFC and a real-time energy market. The PFC is responsible for addressing fast power imbalances in timescales of tens of ms to few minutes (e.g., 100 ms to 5 minutes) while the real-time energy market is responsible for addressing longer imbalances in timescales of minutes to hours (e.g., 5 minutes to 1 hour). TEC, on the other hand, is considered as optional.

This letter seeks to clarify the different existing definitions of both instantaneous complex phase and frequency as well as their equivalence under standard modeling assumptions considered for transmission systems, i.e. balanced positive sequence operation, sole presence of electro-mechanical transient dynamics and absence of harmonics and interharmonics. To achieve this, the two fundamental definitions, i.e., those based on either the use of (i) analytic signals or (ii) space vectors, together with the premises used for their formulation, are presented and their relationship shown. Lastly, a unified notation and terminology to avoid confusion is proposed.

A novel analytical framework for power system strength was recently introduced in the IEEE Transactions on Power Systems, providing a unified formulation for assessing voltage and frequency strength. Building upon this formulation, this paper addresses a series of practical challenges for translating the theoretical framework into a real-world application. In particular, simplified analytical solutions for network-wide bus-level strength metrics are provided, together with compact expressions to capture the impact of relevant devices on strength. In addition, novel normalized strength metrics at a device level are defined, enabling the comparison of strength across different systems. A strength source model is introduced to study the behavior of devices under varying strength conditions. Finally, the framework is implemented in a real-world study case, demonstrating its applicability and potential as a practical tool for a comprehensive strength assessment.

This paper proposes a novel formulation of effective regional inertia that explicitly accounts for both system topology and the spatial distribution of inertia. Unlike traditional approaches that model a region as an aggregated machine with an equivalent inertia, the proposed metric provides a topology-aware representation. The methodology builds on an analytical framework that extends classical slow coherency theory to address network partitioning and regional frequency stability. Based on these partitions, we develop a systematic procedure to evaluate the effective inertia of each region, enabling a more accurate interpretation of local inertial contributions, including those from virtual inertia provided by inverter-based resources (IBRs). Case studies on the IEEE 39-bus and 68-bus systems demonstrate that the integration of inertial devices does not uniformly improve system frequency response, underscoring the importance of the proposed metric for effective regional inertia assessment.

This paper utilizes the autocorrelation of frequency and voltage measurements to identify, quantify and classify local and global properties of power system dynamics. The analysis is based on measurements with various resolutions (20 ms, 1 s, and 1 min) from several nodes of the Irish All-Island Power System (AIPS). Simulations based on stochastic differential algebraic equations on an IEEE benchmark system support conclusions drawn from real-world data.

This paper presents a decentralized methodology for detecting and mitigating flapping phenomena in power systems, primarily caused by the operation of discrete devices. The proposed approach applies moving-window autocorrelation to local measurements, enabling each device to autonomously identify sustained oscillations. Upon detection, a probabilistic, device-specific mitigation strategy is executed. Flexible demand resources (DFRs), under-load tap changers (ULTCs), and automatic voltage regulators (AVRs) are utilised to illustrate the performance of the proposed approach to both discrete and continuous-operation devices. Results show that the proposed method is robust and properly distinguishes damped oscillations from persistent flapping, allowing devices to independently recognize problematic operating scenarios and implement corrective actions accordingly.