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We consider the problem of controlling the frequency of low-inertia power systems via inverter-based resources (IBRs) that are weakly connected to the grid. We propose a novel grid-forming control strategy, the so-called frequency shaping control, that aims to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. Our solution relaxes several existing assumptions in the literature and is able to navigate tradeoffs between peak power requirements and maximum frequency deviations. Finally, we analyze the robustness to imperfect knowledge of network parameters, while particularly highlighting the importance of accurate estimation of these parameters.

Efforts to utilize 100% renewable energy in community microgrids require new approaches to energy markets and transactions to efficiently address periods of scarce energy supply. In this paper we contribute to the promising approach of peer-to-peer (P2P) energy trading in two main ways: analysis of a centralized, welfare-maximizing economic dispatch that characterizes optimal price and allocations, and a novel P2P system for negotiating energy trades that yields physically feasible and at least weakly Pareto-optimal outcomes. Our main results are 1) that optimal pricing is insufficient to induce agents with batteries to take optimal actions, 2) a novel P2P algorithm to address this while keeping private information, 3) a formal proof that this algorithm converges to the centralized solution in the case of two agents negotiating for a single period, and 4)numerical simulations of the P2P algorithm performance with up to 10 agents and 24 periods that show it converges on average to total welfare within 0.1% of the social optimum in on the order of 10s to 100s of iterations, increasing with the number of agents, time periods, and total storage capacity.

With the increasing interest in converter-fed islanded microgrids, particularly for resilience, it is becoming more critical to understand the dynamical behavior of these systems. This paper takes a holistic view of grid-forming converters and considers control approaches for both modeling and regulating the DC-link voltage when the DC-source is a battery energy storage system. We are specifically interested in understanding the performance of these controllers, subject to large load changes, for decreasing values of the DC-side capacitance. We consider a fourth, second, and zero-order model of the battery; and establish that the zero-order model captures the dynamics of interest for the timescales considered for disturbances examined. Additionally, we adapt a grid search for optimizing the controller parameters of the DC/DC controller and show how the inclusion of AC side measurements into the DC/DC controller can improve its dynamic performance. This improvement in performance offers the opportunity to reduce the DC-side capacitance given an admissible DC voltage transient deviation, thereby, potentially allowing for more reliable capacitor technology to be deployed.

We consider the problem of look-ahead economic dispatch (LAED) with uncertain renewable energy generation. The goal of this problem is to minimize the cost of conventional energy generation subject to uncertain operational constraints. The risk of violating these constraints must be below a given threshold for a family of probability distributions with characteristics similar to observed past data or predictions. We present two data-driven approaches based on two novel mathematical reformulations of this distributionally robust decision problem. The first one is a tractable convex program in which the uncertain constraints are defined via the distributionally robust conditional-value-at-risk. The second one is a scalable robust optimization program that yields an approximate distributionally robust chance-constrained LAED. Numerical experiments on the IEEE 39-bus system with real solar production data and forecasts illustrate the effectiveness of these approaches. We discuss how system operators should tune these techniques in order to seek the desired robustness-performance trade-off and we compare their computational scalability.

This paper investigates the input-output performance of secondary frequency controllers through the control-theoretic notion of $H_2$ norms. We consider a quadratic objective accounting for the cost of reserve procurement and provide exact analytical formulae for the performance of continuous-time aggregated averaging controllers. Then, we contrast it with distributed averaging controllers -- seeking optimality conditions such as identical marginal costs -- and primal-dual controllers which have gained attention as systematic techniques to design distributed algorithms solving convex optimization problems. Our conclusion is that while the performance of aggregated averaging controllers, such as gather & broadcast, is independent of the system size and driven predominantly by the control gain, the plain vanilla closed-loop primal-dual controllers scale poorly with size and do not offer any improvement over feedforward primal-dual controllers. Finally, distributed averaging-based controllers scale sub-linearly with size and are independent of system size in the high-gain limit.

The electric power system is witnessing a shift in the technology of generation. Conventional thermal generation based on synchronous machines is gradually being replaced by power electronics interfaced renewable generation. This new mode of generation, however, lacks the natural inertia and governor damping which are quintessential features of synchronous machines. The loss of these features results in increasing frequency excursions and, ultimately, system instability. Among the numerous studies on mitigating these undesirable effects, the main approach involves virtual inertia emulation to mimic the behavior of synchronous machines. In this work, explicit models of grid-following and grid-forming virtual inertia (VI) devices are developed for inertia emulation in low-inertia systems. An optimization problem is formulated to optimize the parameters and location of these devices in a power system to increase its resilience. Finally, a case study based on a high-fidelity model of the South-East Australian system is used to illustrate the effectiveness of such devices.

Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input-output performance becomes more important. We analyze the input-output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit expressions for the saddle-point H2 norm under a relevant input-output configuration. We then proceed to derive analogous results for regularized and augmented versions of the saddle-point algorithm. We observe some rather peculiar effects -- a modest amount of regularization significantly improves the transient performance, while augmentation does not necessarily offer improvement. We then propose a distributed dual version of the algorithm which overcomes some of the performance limitations imposed by augmentation. Finally, we apply our results to a resource allocation problem to compare the input-output performance of various centralized and distributed saddle-point implementations and show that distributed algorithms may perform as well as their centralized counterparts.

One of the recognized principal issues brought along by the steadfast migration towards power electronic interfaced energy sources is the loss of rotational inertia. In conventional power systems, the inertia of the synchronous machines plays a crucial role in safeguarding against any drastic variations in frequency by acting as a buffer in the event of large and sudden power generation-demand imbalances. In future power electronic-based power systems, the same role can be played by strategically located virtual inertia devices. However, the question looms large as to how the system operators would procure and pay for these devices. In this article, we propose a market mechanism inspired by the ancillary service markets in power supply. We consider a linear network reduced power system model along with a robust $\mathcal{H}_2$ performance metric penalizing the worst-case primary control effort. With a social welfare maximization problem for the system operator as a benchmark, we construct a market mechanism in which bids are invited from agents providing virtual inertia, who in turn are compensated via a Vickrey-Clarke-Groves payment rule. The resulting mechanism ensures truthful bidding to be the dominant bidding strategy and guarantees non-negative payoffs for the agents. A three-region case study is considered in simulations, and a comparison with a regulatory approach to the same problem is presented.

A major transition in the operation of electric power grids is the replacement of synchronous machines by distributed generation connected via power electronic converters. The accompanying "loss of rotational inertia" and the fluctuations by renewable sources jeopardize the system stability, as testified by the ever-growing number of frequency incidents. As a remedy, numerous studies demonstrate how virtual inertia can be emulated through various devices, but few of them address the question of "where" to place this inertia. It is however strongly believed that the placement of virtual inertia hugely impacts system efficiency, as demonstrated by recent case studies. In this article, we carry out a comprehensive analysis in an attempt to address the optimal inertia placement problem. We consider a linear network-reduced power system model along with an H2 performance metric accounting for the network coherency. The optimal inertia placement problem turns out to be non-convex, yet we provide a set of closed-form global optimality results for particular problem instances as well as a computational approach resulting in locally optimal solutions. Further, we also consider the robust inertia allocation problem, wherein the optimization is carried out accounting for the worst-case disturbance location. We illustrate our results with a three-region power grid case study and compare our locally optimal solution with different placement heuristics in terms of different performance metrics.