Virtual Inertia Emulation (VIE) and traditional Active Power Droop Control (APDC) are among the most common approaches for regulating active power output of inverter-based generators. Furthermore, it has been shown that, under certain conditions, these two methods can be equivalent. However, neither those studies, nor analyses comparing control schemes with respect to their dynamical properties, have investigated impact converter operation mode. This paper explores subject by investigating...

We consider energy storage systems having nonlinear efficiency functions, which are becoming increasingly important as shown in several recent works, and propose an optimal solution based on Pontryagin's minimum principle. A central challenge such problems is the hard limits state variable, restrict use of To address this challenge, we to include capacity constraints objective function with a proper weighting constant. show that approach allows formulation problem classical principle,...

The flow of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional notation="LaTeX">$k$</tex-math></inline-formula> -contracting system, with notation="LaTeX">$k\in \lbrace 1,\ldots,n\rbrace$</tex-math></inline-formula> , contracts parallelotopes. For notation="LaTeX">$k=1$</tex-math></inline-formula> this reduces to a standard contracting system. One reason for the...

We consider a Lurie system obtained via connection of linear time-invariant and nonlinear feedback function. Such systems often have more than single equilibrium are thus not contractive with respect to any norm. derive new sufficient condition for k-contraction system. For k = 1, our reduces the standard stability based on bounded real lemma small gain condition. 2, guarantees well-ordered asymptotic behaviour closed-loop system: every solution converges an equilibrium, which is necessarily...

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that logarithmic norm (also called matrix measure) the $k$-additive compound Jacobian uniformly negative. However, this may be difficult to check practice because an $n\times n$ has dimensions $\binom{n}{k}\times \binom{n}{k}$. For $A$, we prove duality relation between $k$ $(n-k)$ compounds $A$. We use derive does not require...

Recently there have been extensive research efforts to identify possible adverse effects of distributed sources and power electronics based devices when integrated into existing grids, where two main challenges are low rotational inertia stability. This paper studies the dynamics stability simple systems: an ideal source a synchronous machine, both connected infinite bus. The objective in cases is determine analytically minimal storage device that necessary for One this analysis educational...

Decreasing costs of distributed generation and storage, alongside increasing network charges, provide consumers with a growing incentive to defect from the main grid. On large scale, this may lead price inflation, hindrance energy transition, even "death spiral" - domino effect disconnections. Here, we develop game-theoretic framework that demonstrates how conflicting interests among an aspect previous studies overlooked complex dynamics grid defection. Our results reveal although individual...

We propose an optimal control method for storage systems that are affected by the power grid's ramp constraints. Such problems becoming increasingly important in recent years due to integration of renewable energy sources, which often leads duck curve effects. The main idea proposed is limit number computations forward recursion stage applying ongoing trimming process, reduces discrete values being scanned. As a result, practical range significantly reduced, lower computational burden. To...

We consider the problem of making a networked system contracting by designing "minimal effort" local controllers. Our method combines hierarchical contraction characterization and matrix-balancing approach to stabilizing Metzler matrix via minimal diagonal perturbations. demonstrate our controllers that render contractive network FitzHugh-Nagumo neurons with general topology interactions.

The ribosome flow model (RFM) is a phenomenological for the of particles along one-dimensional chain

The flow of contracting systems contracts 1dimensional polygons (i.e. lines) at an exponential rate. One reason for the usefulness is that many interconnections sub-systems yield overall system. A recent generalization called k-contracting systems, where k ∈{1,...,n}. such k-dimensional rate, and in particular they reduce to when = 1. Here, we analyze serial 1-contracting 2-contracting systems. We provide conditions guaranteeing have a well-ordered asymptotic behaviour, demonstrate...

With increasing penetration of distributed and renewable sources into power grids, the dynamic behavior large-scale systems is becoming increasingly complex. These recent developments have led to several models attempting simplify analysis phenomena, among them are based on dq0 transformation. A question that often arises when modeling interconnected quantities how choose reference frame. One approach model network its components using a transformation unified However, no generator large...

This paper presents necessary and sufficient conditions for a linear three-phase circuit to have time-invariant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$dq0$</tex-math></inline-formula> model. More specifically, we show that with state input matrices can be partitioned into three-by-three circulant at the notation="LaTeX">$abc$</tex-math></inline-formula> frame of reference, transformed model in...

Compound matrices play an important role in many fields of mathematics and have recently found new applications systems control theory. However, the explicit formulas for these compounds are non-trivial not always easy to use. Here, we derive multiplicative additive a matrix using Kronecker products sums. This provides approach based on well-known powerful theory We demonstrate several formulas, including deriving expression compound product two matrices.

Multistationarity - the existence of multiple equilibrium points is a common phenomenon in dynamical systems from variety fields, including neuroscience, opinion dynamics, biology, and power systems. A recently proposed generalization contraction theory, called $k$-contraction, promising approach for analyzing asymptotic behaviour multistationary In particular, all bounded trajectories time-invariant 2-contracting system converge to an point, but may have where more than one locally stable....

We present a new sufficient condition for finite-gain $L_2$ input-to-output stability of networked system. The requires matrix, that combines information on the gains sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). show result generalizes standard small gain theorem negative feedback connection two sub-systems. An important advantage is known conditions DTDS can applied derive stability. demonstrate this using several examples. also necessary matrix rank...

We derive a sufficient condition guaranteeing that singularly perturbed linear time-varying system is strongly monotone with respect to matrix cone $C$ of rank $k$. This implies the inherits asymptotic properties systems are $C$, which include convergence set equilibria when $k=1$, and Poincar\'e-Bendixson property $k=2$. extend this result nonlinear compact convex state-space. demonstrate our theoretical results using simple numerical example.

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -contraction is that logarithmic norm (also called matrix measure) the -additive compound Jacobian uniformly negative. However, this computation may be difficult to perform analytically expensive numerically because an...

The following activity is one of a series developed for junior high school students (age 12-15). Each the activities involves historical discussion subject related to certain curriculum topic, and takes form teacher directed presentation/discussion with accompanying transparencies worksheet. choice development was guided by general principles. connection between regular selected meant give relevance history also, motivate deepen student understanding matter. structure enables change, omit,...

In the field of power system dynamics, a main challenge is to properly tune inverter control parameters, where one important parameter inverter's output impedance, physical or virtual. this work, we provide proof showing that minimal impedance must be used in order keep stable. Although result well-known among practicing engineers, our reveals fundamental property which causes instabililty, namely pole excess dynamic model. To prove claim model as signal flow diagram, impedances inverters...