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Economic withholding in electricity markets refers to generators bidding higher than their true marginal fuel cost, and is a typical approach to exercising market power. However, existing market designs require storage to design bids strategically based on their own future price predictions, motivating storage to conduct economic withholding without assuming market power. As energy storage takes up more significant roles in wholesale electricity markets, understanding its motivations for economic withholding and the consequent effects on social welfare becomes increasingly vital. This paper derives a theoretical framework to study the economic capacity withholding behavior of storage participating in competitive electricity markets and validate our results in simulations based on the ISO New England system. We demonstrate that storage bids can reach unbounded high levels under conditions where future price predictions show bounded expectations but unbounded deviations. Conversely, in scenarios with peak price limitations, we show the upper bounds of storage bids are grounded in bounded price expectations. Most importantly, we show that storage capacity withholding can potentially lower the overall system cost when price models account for system uncertainties. Our paper reveals energy storage is not a market manipulator but an honest player contributing to the social welfare. It helps electricity market researchers and operators better understand the economic withholding behavior of storage and reform market policies to maximize storage contributing to a cost-efficient decolonization.

This paper considers the control of AC-AC inter-linking converters (ILCs) in a multi-grid network. We overview the control schemes in the literature and propose a passivity framework for the stabilization of multi-grid networks, considering both AC grid-following and AC grid-forming behavior for the ILC connections. We then analyze a range of AC/AC interlinking converter control methods derived from the literature and propose suitable controllers for this purpose including both AC grid-forming and grid-following behavior. The controller we propose is partially grid-forming; in particular, it is based on a combination of a grid-following and a grid-forming converter to improve the stability properties of the network. Simulation results and theoretical analysis confirm that the proposed ILC control designs are appropriate for the multi-grid network., Comment: 11 pages, 11 figures

Passivity-based approaches have been suggested as a solution to the problem of decentralised control design in many multi-agent network control problems due to the plug- and-play functionality they provide. However, it is not clear if these controllers are optimal at a network level due to their inherently local formulation, with designers often relying on heuristics to achieve desired global performance. On the other hand, solving for an optimal controller is not guaranteed to produce a passive system. In this paper, we address these dual problems by using inverse optimal control theory to formulate a set of sufficient local conditions, which when satisfied ensure that the resulting decentralised control policies are the solution to a network optimal control problem, while at the same time satisfying appropriate passivity properties. These conditions are then reformulated into a set of linear matrix inequalities (LMIs) which can be used to yield such controllers for linear systems. The proposed approach is demonstrated through a DC microgrid case study. The results substantiate the feasibility and efficacy of the presented method., Comment: 8 pages, 5 figures, submitted to 62nd IEEE Conference on Decision and Control (CDC 2023)

This paper presents a comprehensive convergence analysis for the mirror descent (MD) method, a widely used algorithm in convex optimization. The key feature of this algorithm is that it provides a generalization of classical gradient-based methods via the use of generalized distance-like functions, which are formulated using the Bregman divergence. Establishing convergence rate bounds for this algorithm is in general a non-trivial problem due to the lack of monotonicity properties in the composite nonlinearities involved. In this paper, we show that the Bregman divergence from the optimal solution, which is commonly used as a Lyapunov function for this algorithm, is a special case of Lyapunov functions that follow when the Popov criterion is applied to an appropriate reformulation of the MD dynamics. This is then used as a basis to construct an integral quadratic constraint (IQC) framework through which convergence rate bounds with reduced conservatism can be deduced. We also illustrate via examples that the convergence rate bounds derived can be tight., Comment: Accepted Version. arXiv admin note: substantial text overlap with arXiv:2204.00502

We consider the problem of synchronization of coupled oscillators in a Kuramoto-type model with lossy couplings. Kuramoto models have been used to gain insight on the stability of power networks which are usually nonlinear and involve large scale interconnections. Such models commonly assume lossless couplings and Lyapunov functions have predominantly been employed to prove stability. However, coupling conductances can impact synchronization. We therefore consider a more advanced Kuramoto model that includes coupling conductances, and is characterized by nonhomogeneous coupling weights and noncomplete coupling graphs. Lyapunov analysis once such coupling conductances and aforementioned properties are included becomes nontrivial and more conventional energy-like Lyapunov functions are not applicable or are conservative. Small-signal analysis has been performed for such models, but due to the fact that we have convergence to a manifold, stability analysis via a linearization is on its own inconclusive for the nonlinear model. In this paper, we provide a formal derivation using centre manifold theory that if a particular condition on the equilibrium point associated with the coupling conductances and susceptances holds, then the synchronization manifold for the nonlinear system considered is asymptotically stable. Our analysis is demonstrated with simulations., Comment: 6 pages, 1 figure

Distributed secondary frequency control for power systems, is a problem that has been extensively studied in the literature, and one of its key features is that an additional communication network is required to achieve optimal power allocation. Therefore, being able to provide stability guarantees in the presence of communication delays is an important requirement. Primal-dual and distributed averaging proportional-integral (DAPI) protocols, respectively, are two main control schemes that have been proposed in the literature. Each has its own relative merits, with the former allowing to incorporate general cost functions and additional operational constraints, and the latter being more straightforward in its implementation. Although delays have been addressed in DAPI schemes, there are currently no theoretical guarantees for the stability of primal-dual schemes for frequency control, when these are subject to communication delays. In fact, simulations illustrate that even small delays can destabilize such schemes. In this paper, we show how a novel formulation of prima-dual schemes allows to construct a distributed algorithm with delay independent stability guarantees. We also show that this algorithm can incorporate many of the key features of these schemes such as tie-line power flow requirements, generation constraints, and the relaxation of demand measurements with an observer layer. Finally, we illustrate our results through simulations on a 5-bus example and on the IEEE-39 test system., Comment: Accepted by IEEE Transactions on Automatic Control

This paper is concerned with convergence analysis for the mirror descent (MD) method, a well-known algorithm in convex optimization. An analysis framework via integral quadratic constraints (IQCs) is constructed to analyze the convergence rate of the MD method with strongly convex objective functions in both continuous-time and discrete-time. We formulate the problem of finding convergence rates of the MD algorithms into feasibility problems of linear matrix inequalities (LMIs) in both schemes. In particular, in continuous-time, we show that the Bregman divergence function, which is commonly used as a Lyapunov function for this algorithm, is a special case of the class of Lyapunov functions associated with the Popov criterion, when the latter is applied to an appropriate reformulation of the problem. Thus, applying the Popov criterion and its combination with other IQCs, can lead to convergence rate bounds with reduced conservatism. We also illustrate via examples that the convergence rate bounds derived can be tight., Comment: 6 pages, 3 figures

This paper deals with dynamic networks in which the causality relations between the vertex signals are represented by linear time-invariant transfer functions (modules). Considering an acyclic network where only a subset of its vertices are measured and a subset of the vertices are excited, we explore conditions under which all the modules are identifiable on the basis of measurement data. Two sufficient conditions are presented in the paper, where the first condition concerns an identifiability analysis that needs to be performed for each vertex, while the second condition, based on the concept of tree/anti-tree covering, results from a graphical synthesis approach to allocate actuators and sensors in an acyclic network for achieving generic identifiability.

This work is associated with the use of parallel feedforward compensators (PFCs) for the problem of output synchronization over heterogeneous agents and the benefits this approach can provide. Specifically, it addresses the addition of stable PFCs on agents that interact with each other using diffusive couplings. The value in the application of such PFC is twofold. Firstly, it has been an issue that output synchronization among passivity-short systems requires global information for the design of controllers in the cases when initial conditions need to be taken into account, such as average consensus and distributed optimization. We show that a stable PFC can be designed to passivate a passivity-short system while its output asymptotically vanishes as its input tends to zero. As a result, output synchronization is achieved among these systems by fully distributed controls without altering the original consensus results. Secondly, in the literature of output synchronization over signed weighted graphs, it is generally required that the graph Laplacian be positive semidefinite, i.e., $L \geq 0$ for undirected graphs or $L + L^T \geq 0$ for balanced directed graphs. We show that the PFC serves as output feedback to the communication graph to enhance the robustness against negative weight edges. As a result, output synchronization is achieved over a signed weighted and balanced graph, even if the corresponding Laplacian is not positive semidefinite., Comment: 10 pages, 8 figures, Systems & Control Letters

Grid-forming inverter-based autonomous microgrids present new operational challenges as the stabilizing rotational inertia of synchronous machines is absent. The design of efficient control policies for grid-forming inverters is, however, a non-trivial problem where multiple performance objectives need to be satisfied, including voltage/frequency regulation, current limiting capabilities, as well as active power sharing and a scalable operation. We propose in this paper a novel control architecture for frequency and voltage control which allows current limitation via an inner loop, active power sharing via a distributed secondary control policy and scalability by satisfying a passivity property. In particular, the frequency controller employs the inverter output current and angle to provide an angle droop-like policy which improves its stability properties. This also allows to incorporate a secondary control policy for which we provide an analytical stability result which takes line conductances into account (in contrast to the lossless line assumptions in literature). The distinctive feature of the voltage control scheme is that it has a double loop structure that uses the DC voltage in the feedback control policy to implement a power balancing strategy to improve performance. The performance of the control policy is illustrated via simulations with detailed nonlinear models in a realistic setting., Comment: 13 pages, 11 figures

We consider the problem of ensuring stability in a DC microgrid by means of decentralized conditions. Such conditions are derived which are formulated as input-output properties of locally defined subsystems. These follow from various decompositions of the microgrid and corresponding properties of the resulting representations. It is shown that these stability conditions can be combined together by means of appropriate homotopy arguments, thus reducing the conservatism relative to more conventional decentralized approaches that often rely on a passivation of the bus dynamics. Examples are presented to demonstrate the efficiency and the applicability of the results derived.

This paper considers dynamic networks where vertices and edges represent manifest signals and causal dependencies among the signals, respectively. We address the problem of how to determine if the dynamics of a network can be identified when only partial vertices are measured and excited. A necessary condition for network identifiability is presented, where the analysis is performed based on identifying the dependency of a set of rational functions from excited vertices to measured ones. This condition is further characterised by using an edge-removal procedure on the associated bipartite graph. Moreover, on the basis of necessity analysis, we provide a necessary and sufficient condition for identifiability in circular networks.

Microgrids are increasingly recognized as a key technology for the integration of distributed energy resources into the power network, allowing local clusters of load and distributed energy resources to operate autonomously. However, microgrid operation brings new challenges, especially in islanded operation as frequency and voltage control are no longer provided by large rotating machines. Instead, the power converters in the microgrid must coordinate to regulate the frequency and voltage and ensure stability. We consider the problem of designing controllers to achieve these objectives. Using passivity theory to derive decentralized stability conditions for the microgrid, we propose a control design method for grid-forming inverters. For the analysis we use higher-order models for the inverters and also advanced dynamic models for the lines with an arbitrarily large number of states. By satisfying the decentralized condition formulated, plug-and-play operation can be achieved with guaranteed stability, and performance can also be improved by incorporating this condition as a constraint in corresponding optimization problems formulated. In addition, our control design can improve the power sharing properties of the microgrid compared to previous non-droop approaches. Finally, realistic simulations confirm that the controller design improves the stability and performance of the power network., Comment: 14 pages, 10 figures

Inverter-based microgrids are an important technology for sustainable electrical power systems and typically use droop-controlled grid-forming inverters to interface distributed energy resources to the network and control the voltage and frequency. Ensuring stability of such microgrids is a key issue, which requires the use of appropriate models for analysis and control system design. Full-order detailed models can be more difficult to analyze and increase computational complexity, hence a number of reduced-order models have been proposed in the literature which present various trade-offs between accuracy and complexity. However, these simplifications present the risk of failing to adequately capture important dynamics of the microgrid. Therefore, there is a need for a comprehensive review and assessment of their relative quality, which is something that has not been systematically carried out thus far in the literature and we aim to address in this paper. In particular, we review various inverter-based microgrid reduced-order models and investigate the accuracy of their predictions for stability via a comparison with a corresponding detailed average model. Our study shows that the simplifications reduced order models rely upon can affect their accuracy in various regimes of the line R/X ratios, and that inappropriate model choices can result in substantially inaccurate stability results. Finally, we present recommendations on the use of reduced order models for the stability analysis of microgrids., Comment: 13 pages, 12 figures

Thermostatically controlled loads (TCLs) can provide ancillary services to the power network by aiding existing frequency control mechanisms. TCLs are, however, characterized by an intrinsic limit cycle behavior which raises the risk that these could synchronize when coupled with the frequency dynamics of the power grid, i.e. simultaneously switch, inducing persistent and possibly catastrophic power oscillations. To address this problem, schemes with a randomized response time in their control policy have been proposed in the literature. However, such schemes introduce delays in the response of TCLs to frequency feedback that may limit their ability to provide fast support at urgencies. In this paper, we present a deterministic control mechanism for TCLs such that those switch when prescribed frequency thresholds are exceeded in order to provide ancillary services to the power network. For the considered scheme, we provide analytic conditions which ensure that synchronization is avoided. In particular, we show that as the number of loads tends to infinity, there exist arbitrarily long time intervals where the frequency deviations are arbitrarily small. Our analytical results are verified with simulations on the Northeast Power Coordinating Council (NPCC) 140-bus system, which demonstrate that the proposed scheme offers improved frequency response compared to conventional implementations., Comment: 10 pages, 5 figures, 1 table

Load side participation can provide valuable support to the power network in case of urgencies. On many occasions, loads are naturally represented by on and off states. However, the use of on-off loads for frequency control can lead to chattering and undesirable limit cycle behavior, which are issues that need to be resolved for such loads to be used for network support. This paper considers the problem of primary frequency regulation with ancillary service from on-off loads in power networks and establishes conditions that lead to convergence guarantees and an appropriate power allocation within the network. In particular, in order to assist existing frequency control mechanisms, we consider loads that switch when prescribed frequency thresholds are exceeded. Such control policies are prone to chattering, which limits their practicality. To resolve this issue, we consider loads that follow a decentralized hysteretic on-off policy, and show that chattering is not observed within such a setting. Hysteretic loads may exhibit, however, limit cycle behavior, which is undesirable. To address this, we propose an adapted hysteretic control scheme for which we provide convergence guarantees. Furthermore, we consider a mixed-integer optimization problem for power allocation and propose a suitable design of the control policy such that the cost incurred at equilibrium is within $\epsilon$ from the optimal cost, providing a non conservative value for $\epsilon$. The practicality of our analytic results is demonstrated with numerical simulations on the Northeast Power Coordinating Council (NPCC) 140-bus system., Comment: 16 pages, 9 figures

Hybrid AC/DC networks are a key technology for future electrical power systems, due to the increasing number of converter-based loads and distributed energy resources. In this paper, we consider the design of control schemes for hybrid AC/DC networks, focusing especially on the control of the interlinking converters (ILC(s)). We present two control schemes: firstly for decentralized primary control, and secondly, a distributed secondary controller. In the primary case, the stability of the controlled system is proven in a general hybrid AC/DC network which may include asynchronous AC subsystems. Furthermore, it is demonstrated that power-sharing across the AC/DC network is significantly improved compared to previously proposed dual droop control. The proposed scheme for secondary control guarantees the convergence of the AC system frequencies and the average DC voltage of each DC subsystem to their nominal values respectively. An optimal power allocation is also achieved at steady-state. The applicability and effectiveness of the proposed algorithms are verified by simulation on a test hybrid AC/DC network in MATLAB / Simscape Power Systems.

During the last decades, significant advances have been made in the area of power system stability and control. Nevertheless, when this analysis is carried out by means of decentralized conditions in a general network, it has been based on conservative assumptions such as the adoption of lossless networks. In the current paper, we present a novel approach for decentralized stability analysis and control of power grids through the transformation of both the network and the bus dynamics into the system reference frame. In particular, the aforementioned transformation allows us to formulate the network model as an input-output system that is shown to be passive even if the network's lossy nature is taken into account. We then introduce a broad class of bus dynamics that are viewed as multivariable input/output systems compatible with the network formulation, and appropriate passivity conditions are imposed on those that guarantee stability of the power network. We discuss the opportunities and advantages offered by this approach while explaining how this allows the inclusion of advanced models for both generation and power flows. Our analysis is verified through applications to the Two Area Kundur and the IEEE 68-bus test systems with both primary frequency and voltage regulation mechanisms included.

We consider the problem of distributed secondary frequency regulation in power networks such that stability and an optimal power allocation are attained. This is a problem that has been widely studied in the literature, and two main control schemes have been proposed, usually referred to as 'primal-dual' and 'distributed averaging proportional-integral (DAPI)' respectively. However, each has its limitations, with the former requiring knowledge of uncontrollable demand, which can be difficult to obtain in real time, and with the existing literature on the latter being based on static models for generation and demand. We propose a novel control scheme that overcomes these issues by making use of generation measurements in the control policy. In particular, our analysis allows distributed stability and optimality guarantees to be deduced with practical measurement requirements and permits a broad range of linear generation dynamics, that can be of higher order, to be incorporated in the power network. We show how the controller parameters can be selected in a computationally efficient way by solving appropriate linear matrix inequalities (LMIs). Furthermore, we demonstrate how the proposed analysis applies to several examples of turbine governor models. The practicality of our analysis is demonstrated with simulations on the Northeast Power Coordinating Council (NPCC) 140-bus system that verify that our proposed controller achieves convergence to the nominal frequency and an economically optimal power allocation., Comment: 11 pages, 3 figures, 1 table

The problem of decentralized frequency control in power networks has received an increasing attention in recent years due to its significance in modern power systems and smart grids. Nevertheless, generation dynamics including turbine-governor dynamics, in conjunction with nonlinearities associated with generation and power flow, increase significantly the complexity in the analysis, and are not adequately addressed in the literature. In this paper we show how incremental secant gain conditions can be used in this context to deduce decentralized stability conditions with reduced conservatism. Furthermore, for linear generation dynamics, we establish Popov-like conditions that are able to reduce the conservatism even further by incorporating additional local information associated with the coupling strength among the bus dynamics. Various examples are discussed throughout the paper to demonstrate the significance of the results presented.

We study the problem of decentralized secondary frequency regulation in power networks where ancillary services are provided via on-off load-side participation. We initially consider on-off loads that switch when prescribed frequency thresholds are exceeded, together with a large class of passive continuous dynamics for generation and demand. The considered on-off loads are able to assist existing secondary frequency control mechanisms and return to their nominal operation when the power system is restored to its normal operation, a highly desirable feature which minimizes users disruption. We show that system stability is not compromised despite the switching nature of the loads. However, such control policies are prone to chattering, which limits the practicality of these schemes. As a remedy to this problem, we propose a hysteretic on-off policy where loads switch on and off at different frequency thresholds and show that stability guarantees are retained when the same decentralized passivity conditions for continuous generation and demand hold. Several relevant examples are discussed to demonstrate the applicability of the proposed results. Furthermore, we verify our analytic results with numerical investigations on the Northeast Power Coordinating Council (NPCC) 140-bus system.

In part I we considered the problem of convergence to a saddle point of a concave-convex function via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of subgradient dynamics that provide a restriction in an arbitrary convex domain. We show that despite the nonlinear and non-smooth character of these dynamics their $\omega$-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.

We consider the problem of convergence to a saddle point of a concave-convex function via gradient dynamics. Since first introduced by Arrow, Hurwicz and Uzawa in [1] such dynamics have been extensively used in diverse areas, there are, however, features that render their analysis non trivial. These include the lack of convergence guarantees when the function considered is not strictly concave-convex and also the non-smoothness of subgradient dynamics. Our aim in this two part paper is to provide an explicit characterization to the asymptotic behaviour of general gradient and subgradient dynamics applied to a general concave-convex function. We show that despite the nonlinearity and non-smoothness of these dynamics their $\omega$-limit set is comprised of trajectories that solve only explicit linear ODEs that are characterized within the paper. More precisely, in Part I an exact characterization is provided to the asymptotic behaviour of unconstrained gradient dynamics. We also show that when convergence to a saddle point is not guaranteed then the system behaviour can be problematic, with arbitrarily small noise leading to an unbounded variance. In Part II we consider a general class of subgradient dynamics that restrict trajectories in an arbitrary convex domain, and show that when an equilibrium point exists their limiting trajectories are solutions of subgradient dynamics on only affine subspaces. The latter is a smooth class of dynamics with an asymptotic behaviour exactly characterized in Part I, as solutions to explicit linear ODEs. These results are used to formulate corresponding convergence criteria and are demonstrated with several examples and applications presented in Part II.

We present a systematic method for designing distributed generation and demand control schemes for secondary frequency regulation in power networks such that stability and an economically optimal power allocation can be guaranteed. A dissipativity condition is imposed on net power supply variables to provide stability guarantees. Furthermore, economic optimality is achieved by explicit decentralized steady state conditions on the generation and controllable demand. We discuss how various classes of dynamics used in recent studies fit within our framework and give examples of higher order generation and controllable demand dynamics that can be included within our analysis. In case of linear dynamics, we discuss how the proposed dissipativity condition can be efficiently verified using an appropriate linear matrix inequality. Moreover, it is shown how the addition of a suitable observer layer can relax the requirement for demand measurements in the employed controller. The efficiency and practicality of the proposed results are demonstrated with a simulation on the Northeast Power Coordinating Council (NPCC) 140-bus system.

We present a method to design distributed generation and demand control schemes for primary frequency regulation in power networks that guarantee asymptotic stability and ensure fairness of allocation. We impose a passivity condition on net power supply variables and provide explicit steady state conditions on a general class of generation and demand control dynamics that ensure convergence of solutions to equilibria that solve an appropriately constructed network optimization problem. We also show that the inclusion of controllable demand results in a drop in steady state frequency deviations. We discuss how various classes of dynamics used in recent studies fit within our framework and show that this allows for less conservative stability and optimality conditions. We illustrate our results with simulations on the IEEE 68 bus system and observe that both static and dynamic demand response schemes that fit within our framework offer improved transient and steady state behavior compared with control of generation alone. The dynamic scheme is also seen to enhance the robustness of the system to time-delays.

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied primarily on the conditions of homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems by deriving convergence results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results shows that if the undelayed system has a convergent trajectory that is unbounded in all components as $t \to -\infty$, then the system is globally asymptotically stable for arbitrary bounded time-varying delays. This follows from a more general result derived in the paper that allows to quantify delay-independent regions of attraction, which can be used to prove global asymptotic stability for various classes of systems. These also recover various known delay-independent stability results that are discussed within the paper., Comment: 8 pages