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 power flow 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 term 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 network's 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 network's 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.

Abstract-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.

We study fundamental performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of whether dynamic feedback controllers perform better than static (memoryless) ones when subject to locality constraints. We consider distributed linear consensus and vehicular formation control problems modeled over toric lattice networks. For the resulting spatially invariant systems we study the large-scale asymptotics (in network size) of global performance metrics that quantify the level of network coherence. With static feedback from relative state measurements, such metrics are known to scale unfavorably in lattices of low spatial dimensions, preventing, for example, a 1-dimensional string of vehicles to move like a rigid object. We show that the same limitations in general apply also to dynamic feedback control that is locally of first order. This means that the addition of one local state to the controller gives a similar asymptotic performance to the memoryless case. This holds unless the controller can access noiseless measurements of its local state with respect to an absolute reference frame, in which case the addition of controller memory may fundamentally improve performance. In simulations of platoons with 20-200 vehicles we show that the performance limitations we derive manifest as unwanted accordion-like motions. Similar behaviors are to be expected in any network that is embeddable in a low-dimensional toric lattice, and the same fundamental limitations would apply. To derive our results, we present a general technical framework for the analysis of stability and performance of spatially invariant systems in the limit of large networks.

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 internodal alignment of the integral states affects performance. Our results are demonstrated through simulations of the Nordic power grid.M. Andreasson, deceased, was with the School of Electrical Engineering and the

Abstract-We consider distributed control of doubleintegrator networks, where agents are subject to stochastic disturbances. We study performance of such networks in terms of coherence, defined through an H2 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 (PI) and proportional derivative (PD) 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. I. INTRODUCTIONThe problem of distributed control of networked systems has been extensively studied over the past decades [1]- [3]. A well-studied subproblem in this context is that of consensus, that is, to drive the network of agents to the same state. When the system is subject to external disturbances or uncertainties, there are, however, limitations to the performance achievable through distributed control. In particular, there are limits on how close the agents of a large-scale network can come to a state of consensus, or what is often called the coherence of the network. Such fundamental limitations were studied in [4], where scalings of measures of network coherence with network size were established for first-and secondorder consensus networks. It was shown that reasonable performance in sparse networks, such as vehicle platoons, requires that each agent can access measurements of its own states with respect to a global reference frame; what we refer to as absolute feedback.The importance of absolute feedback for network coherence was also recognized in [5]-[7], where the impact of leaders (select agents with access to absolute measurements) was studied in first-order consensus networks. In this work, we will focus on problems of second-order consensus, and do not consider adding absolute measurements to improve performance, which would require increasing the agents' sensory capabilities. Instead, we propose the use of alternative controller structures, namely controllers with integral and derivative action in addition to standard proportional control.Proportional-integral-derivative (PID) control has been studied for consensus networks in [8] and the role of integral