Emma Tegling

The Price of Synchrony: Evaluating the Resistive Losses in Synchronizing Power Networks

This paper investigates the resistive power losses that are incurred in keeping a network of synchronous generators in a synchronous state. These losses arise due to the transient powerflow fluctuations that occur when the system is perturbed from a synchronous state by a small transient event or in the face of persistent stochastic disturbances. We call these losses the “price of synchrony,” as they reflect the real power-flow costs incurred in resynchronizing the system. In the case of small fluctuations at each generator node, we show how the total networks resistive losses can be quantified using an H2 norm of a linear system of coupled swing equations subject to distributed disturbances. This norm is shown to be a function of transmission-line and generator properties, to scale unboundedly with network size, and to be weakly dependent on the network topology. This conclusion differentiates the price of synchrony from typical power systems stability notions, which show highly connected networks to be more coherent and, thus, easier to synchronize. In particular, the price of synchrony is more dependent on a networks size than its topology. We discuss possible implications of these results in terms of the design of future power grids, which are expected to have highly distributed generation resources leading to larger networks with the potential for greater transient losses.

Improving performance of droop-controlled microgrids through distributed PI-control

This paper investigates transient performance of inverter-based microgrids in terms of the resistive power losses incurred in regulating frequency under persistent stochastic disturbances. We model the inverters as second-order oscillators and compare two algorithms for frequency regulation: the standard frequency droop controller and a distributed proportional-integral (PI) controller. The transient power losses can be quantified using an input-output ℋ2 norm. We show that the distributed PI-controller, which has previously been proposed for secondary frequency control (the elimination of static errors), also has the potential to significantly improve performance by reducing transient power losses. This loss reduction is shown to be larger in a loosely interconnected network than in a highly interconnected one, whereas losses do not depend on connectivity if standard droop control is employed. Moreover, our results indicate that there is an optimal tuning of the distributed PI-controller for loss reduction. Overall, our results provide an additional argument in favor of distributed algorithms for secondary frequency control in microgrids.

Performance metrics for droop-controlled microgrids with variable voltage dynamics

This paper investigates the performance of a microgrid with droop-controlled inverters in terms of the total power losses incurred in maintaining synchrony under persistent small disturbances. The inverters are modeled with variable frequencies and voltages under droop control. For small fluctuations from a steady state, these transient power losses can be quantified by an input-output H2 norm of a linear system subject to distributed disturbances. We evaluate this H2 norm under the assumption of a dominantly inductive network with identical inverters. The results indicate that while phase synchronization, in accordance with previous findings, produces losses that scale with a networks size but only weakly depend on its connectivity, the losses associated with the voltage control will be larger in a highly connected network than in a loosely connected one. The typically higher rate of convergence in a highly interconnected network thus comes at a cost of higher losses associated with the power flows used to reach the steady state.

Coherence in synchronizing power networks with distributed integral control

We consider frequency control of synchronous generator networks and study transient performance under both primary and secondary frequency control. We model random step changes in power loads and evaluate performance in terms of expected deviations from a synchronous frequency over the synchronization transient; what can be thought of as lack of frequency coherence. We compare a standard droop control strategy to two secondary proportional integral (PI) controllers: centralized averaging PI control (CAPI) and distributed averaging PI control (DAPI). We show that the performance of a power system with DAPI control is always superior to that of a CAPI controlled system, which in turn has the same transient performance as standard droop control. Furthermore, for a large class of network graphs, performance scales unfavorably with network size with CAPI and droop control, which is not the case with DAPI control. We discuss optimal tuning of the DAPI controller and describe how inter-nodal alignment of the integral states affects performance. Our results are demonstrated through simulations of the Nordic power grid.

On the Coherence of Large-Scale Networks With Distributed PI and PD Control

We consider distributed control of double-integrator networks, where agents are subject to stochastic disturbances. We study performance of such networks in terms of coherence, defined through an

Performance and scalability of voltage controllers in multi-terminal HVDC networks

\mathcal {H}_{2}

An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum

norm metric that represents the variance of nodal state fluctuations. Specifically, we address known performance limitations of the standard consensus protocol, which cause this variance to scale unboundedly with network size for a large class of networks. We propose distributed proportional integral and proportional derivative controllers that relax these limitations and achieve bounded variance, in cases where agents can access an absolute measurement of one of their states. This case applies to, for example, frequency control of power networks and vehicular formation control with limited sensing. We discuss optimal tuning of the controllers with respect to network coherence and demonstrate our results in simulations.

On performance limitations of large-scale networks with distributed feedback control

In this paper, we compare the transient performance of a multi-terminal high-voltage DC (MTDC) grid equipped with a slack bus for voltage control to that of two distributed control schemes: a standard droop controller and a distributed averaging proportional-integral (DAPI) controller. We evaluate performance in terms of an ℋ2 metric that quantifies expected deviations from nominal voltages, and show that the transient performance of a droop or DAPI controlled MTDC grid is always superior to that of an MTDC grid with a slack bus. In particular, by studying systems built up over lattice networks, we show that the ℋ2 norm of a slack bus controlled system may scale unboundedly with network size, while the norm remains uniformly bounded with droop or DAPI control. We simulate the control strategies on radial MTDC networks to demonstrate that the transient performance for the slack bus controlled system deteriorates significantly as the network grows, which is not the case with the distributed control strategies.

Noise-Induced Limitations to the Scalability of Distributed Integral Control

We apply an operator-theoretic viewpoint to a class of non-smooth dynamical systems that are exposed to event-triggered state resets. The considered benchmark problem is that of a pendulum which receives a downward kick at certain fixed angles. The pendulum is modeled as a hybrid automaton and is analyzed from both a geometric perspective and the formalism of Koopman operator theory. A connection is drawn between these two interpretations of a dynamical system by establishing a link between the spectral properties of the Koopman operator and the geometric properties in the state-space.