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.

This paper discusses the impact of discrete secondary controllers on the dynamic response of power systems. The idea of the paper originates from the observation that there is a range of values, from few tens of seconds to few minutes, of the execution cycles of conventional automatic generation control (AGC) that leads to a limit cycle. Below and above this range the system is stable. This is certainly not a problem in practice as the AGC updates the power set points of generating units every few seconds. However, this phenomenon has interesting consequences if one considers real-time electricity markets with short dispatch periods (i.e., 5 minutes) as these markets can be modeled as a sort of AGC. The paper first provides a formal analogy between conventional AGC and real-time electricity markets. Then it shows that the discretization-driven instability exists if the system includes a real-time electricity market modeled as secondary frequency controller. Finally, the paper discusses the impact of the combined effect of high wind generation shares and discrete secondary controllers on power system dynamics.

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.

This paper analyses an emerging real-world phenomena in inverter-based renewable-dominated power systems, namely, asymmetry of frequency distribution. The paper first provides a rationale on why asymmetry reduces the "quality" of the frequency control and system operation. Then it provides qualitative theoretical insights that explain asymmetry in terms of the nonlinearity of real-world power systems and associated models. In particular network losses and pitch angle-based frequency control of wind power plants are discussed. Then the paper proposes a nonlinear compensation control to reduce the asymmetry as well as a statistical metric based on the frequency probability distribution to quantify the level of asymmetry in a power system. Real-world data obtained from the Irish and Australian transmission systems serve to support the theoretical appraisal, whereas simulations based on an IEEE benchmark system show the effectiveness of the proposed nonlinear compensation. The case study also shows that, while automatic generation control reduces asymmetry, frequency control limits and droop-based frequency support provided by wind generation using a tight deadband of ±15 mHz, namely active power control, leads to a significant increase in the asymmetry of the frequency probability distribution.