Federico Milano

Álvaro Ortega, Federico Milano

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.

Bi Liu, Qi Huang, Junbo Zhao et al.

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.

Federico Milano

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.

Federico Milano, Georgios Tzounas, Ioannis Dassios et al.

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.

Federico Milano

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.

Ignacio Ponce, Federico Milano

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.

Federico Milano, Georgios Tzounas, Ioannis Dassios et al.

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.

Taulant Kerci, Angel Vaca, Andrew Groom et al.

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.

Taulant Kerci, Federico Milano

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.

César García-Veloso, Mario Paolone, Federico Milano

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.

Nikolas Sofos, Federico Milano

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.