arXiv subjects

Elizabeth L. Ratnam

Publications and source records attributed to Elizabeth L. Ratnam.

16 recordsLinked to original sources

arXiv2026

An Electricity Market with Reactive Power Trading: Incorporating Dynamic Operating Envelopes

Electricity market design that accounts for grid constraints such as voltage and thermal limits at the distribution level can increase opportunities for the grid integration of Distributed Energy Resources (DERs). In this paper, we consider rooftop solar backed by battery storage connected to a distribution grid. We design an electricity market to support customers sharing rooftop generation in excess of their energy demand, where customers earn a profit through peer-to-peer (P2P) energy trading. Our proposed electricity market also incorporates P2P reactive power trading to improve the voltage profile across a distribution feeder. We formulate the electricity market as an optimization-based problem, where voltage and thermal limits across a feeder are managed through the assignment of customer-specific dynamic operating envelopes (DOEs). The electricity market equilibrium is referred to as a competitive equilibrium, which is equivalent to a Nash equilibrium in a standard game. Our proposed market design is benchmarked using the IEEE 13-node test feeder.

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arXiv2026

Equilibrium points and stability of synchronous machine systems

This paper investigates equilibrium points and stability in two synchronous machine configurations: (i) a single generator with an impedance load and (ii) two interconnected machines with co-located loads. We consider both abc and dq reference frames to show that the equilibrium condition reduces to a cubic polynomial in the single-machine case and to an 18th- degree polynomial in the two-machine case. For the single-machine system, Lyapunov stability analysis and linearization based stability analysis are carried out. For the two-machine system, local stability is assessed through linearization and eigenvalue analysis. Illustrative examples confirm the existence of multiple equilibria and illustrate the impact of parameter variation on stability. Our results provide insight into the stability of synchronous machine systems.

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arXiv2026

Negative Imaginary and Passivity Properties of Synchronous Machine Systems

The recent rapid proliferation of renewable energy is fundamentally changing the dynamic operations of power systems, necessitating new approaches to assess stability for these highly nonlinear systems. In this paper, we prove that synchronous machine systems, modeled in the nonlinear dq-frame, possess fundamental dissipativity properties. Specifically, we show passivity from current input to voltage output and a nonlinear negative imaginary property from torque input to rotor angle output. For the nonlinear system shifted around an equilibrium point, we derive explicit conditions for both passivity and the NI property to hold. Finally, we demonstrate that interconnection with passive droop controllers preserves these dissipativity properties with identical supply rates, thereby ensuring closed-loop stability.

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arXiv2025

Peer-to-Peer Energy Markets With Uniform Pricing: A Dynamic Operating Envelope Approach

The recent widespread adoption of rooftop solar backed by battery storage is enabling energy customers to both produce and consume electricity (i.e., prosumers of electricity). To facilitate prosumer participation in the electric grid, new market mechanisms are required. In this paper, we design peer-to-peer energy markets where prosumers trade their excess energy with peers to gain profit while satisfying the overall balance in electricity supply and demand. We first consider a market structure, considering the case where voltage and/or thermal constraints are binding. When such grid constraints are binding, market clearing prices can vary across locations. However, heterogeneous prices may be considered by regulators to lack fairness. To ensure uniform pricing, we design two peer-to-peer energy markets with dynamic operating envelopes (DOEs). DOEs enable us to decompose global voltage and thermal constraints across the power grid into local constraints for each prosumer, resulting in uniform prices across the grid. By means of numerical simulations on an IEEE 13-node feeder, we benchmark the proposed market-based approaches in the presence of binding voltage constraints.

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arXiv2025

Electric Grid Topology and Admittance Estimation: Quantifying Phasor-based Measurement Requirements

In this paper, we quantify voltage and current phasor-based measurement requirements for the unique estimation of the electric grid topology and admittance parameters. Our approach is underpinned by the concept of a rigidity matrix that has been extensively studied in graph rigidity theory. Specifically, we show that the rank of the rigidity matrix is the same as that of a voltage coefficient matrix in a corresponding electric power system. Accordingly, we show that there is a minimum number of measurements required to uniquely estimate the admittance matrix and corresponding grid topology. By means of a numerical example on the IEEE 4-node radial network, we demonstrate that our approach is suitable for applications in electric power grids.

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arXiv2024

On the Stability of Networked Nonlinear Negative Imaginary Systems with Applications to Electrical Power Systems

In the transition to achieving net zero emissions, it has been suggested that a substantial expansion of electric power grids will be necessary to support emerging renewable energy zones. In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Luré-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time.

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arXiv2024

Unified Control of Voltage, Frequency and Angle in Electrical Power Systems: A Passivity and Negative-Imaginary based Approach

This paper proposes a unified methodology for voltage regulation, frequency synchronization, and rotor angle control in power transmission systems considering a one-axis generator model with time-varying voltages. First, we formulate an output consensus problem with a passivity and negative-imaginary (NI) based control framework. We establish output consensus results for both networked passive systems and networked NI systems. Next, we apply the output consensus problem by controlling large-scale batteries co-located with synchronous generators -- using real-time voltage phasor measurements. By controlling the battery storage systems so as to dispatch real and reactive power, we enable simultaneous control of voltage, frequency, and power angle differences across a transmission network. Validation through numerical simulations on a four-area transmission network confirms the robustness of our unified control framework.

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arXiv2024

Competitive Equilibrium in Microgrids With Dynamic Loads

In this paper, we consider microgrids that interconnect prosumers with distributed energy resources and dynamic loads. Prosumers are connected through the microgrid to trade energy and gain profit while respecting the network constraints. We establish a local energy market by defining a competitive equilibrium which balances energy and satisfies voltage constraints within the microgrid for all time. Using duality theory, we prove that under some convexity assumptions, a competitive equilibrium is equivalent to a social welfare maximization solution. Additionally, we show that a competitive equilibrium is equivalent to a Nash equilibrium of a standard game. In general, the energy price for each prosumer is different, leading to the concept of locational prices. We investigate a case under which all prosumers have the same locational prices. Additionally, we show that under some assumptions on the resource supply and network topology, locational prices decay to zero after a period of time, implying the available supply will be more than the demand required to stabilize the system. Finally, two numerical examples are provided to validate the results, one of which is a direct application of our results on electric vehicle charging control.

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arXiv2023

A Nonlinear Negative Imaginary Systems Framework with Actuator Saturation for Control of Electrical Power Systems

In the transition to net zero, it has been suggested that a massive expansion of the electric power grid will be required to support emerging renewable energy zones. In this paper, we propose the use of battery-based feedback control and nonlinear negative imaginary systems theory to reduce the need for such an expansion by enabling the more complete utilization of existing grid infrastructure. By constructing a novel Lur'e-Postnikov-like Lyapunov function, a stability result is developed for the feedback interconnection of a nonlinear negative imaginary system and a nonlinear negative imaginary controller. Additionally, a new class of nonlinear negative imaginary controllers is proposed to deal with actuator saturation. We show that in this control framework, the controller eventually leaves the saturation boundary, and the feedback system is locally stable in the sense of Lyapunov. This provides theoretical support for the application of battery-based control in electrical power systems. Validation through simulation results for single-machine-infinite-bus power systems supports our results. Our approach has the potential to enable a transmission line to operate at its maximum power capacity, as stability robustness is ensured by the use of a feedback controller.

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arXiv2023

Design and Stability of Angle based Feedback Control in Power Systems: A Negative-Imaginary Approach

This paper considers a power transmission network characterized by interconnected nonlinear swing dynamics on generator buses. At the steady state, frequencies across different buses synchronize to a common nominal value such as $50$Hz or $60$Hz, and power flows on transmission lines are within steady-state envelopes. We assume that fast measurements of generator rotor angles are available. Our approach to frequency and angle control centers on equipping generator buses with large-scale batteries that are controllable on a fast timescale. We link angle based feedback linearization control with negative-imaginary systems theory. Angle based feedback controllers are designed using large-scale batteries as actuators and can be implemented in a distributed manner incorporating local information. Our analysis demonstrates the internal stability of the interconnection between the power transmission network and the angle based feedback controllers. This internal stability underscores the benefits of achieving frequency synchronization and preserving steady-state power flows within network envelopes through the use of feedback controllers. Our approach will enable transmission lines to be operated at maximum power capacity since stability robustness is ensured by the use of feedback controllers rather than conservative criteria such as the equal area criterion. By means of numerical simulations, we illustrate our results.

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arXiv2023

Transactive Multi-Agent Systems over Flow Networks

This paper presented insights into the implementation of transactive multi-agent systems over flow networks where local resources are decentralized. Agents have local resource demand and supply, and are interconnected through a flow network to support the sharing of local resources while respecting restricted sharing/flow capacity. We first establish a competitive market with a pricing mechanism that internalizes flow capacity constraints into agents' private decisions. We then demonstrate through duality theory that competitive equilibrium and social welfare equilibrium exist and agree under convexity assumptions, indicating the efficiency of the pricing mechanism. Additionally, a new social acceptance sharing problem is defined to investigate homogeneous pricing when the optimal sharing prices at all agents under competitive equilibrium are always equal for social acceptance. A conceptual computation method is proposed, prescribing a class of socially admissible utility functions to solve the social acceptance problem. A special case of linear-quadratic multi-agent systems over undirected star graphs is provided as a pedagogical example of how to explicitly prescribe socially admissible utility functions. Finally, extensive experiments are provided to validate the results.

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arXiv2022

Competitive Equilibrium for Dynamic Multi-Agent Systems: Social Shaping and Price Trajectories

In this paper, we consider dynamic multi-agent systems (MAS) for decentralized resource allocation. The MAS operates at a competitive equilibrium to ensure supply and demand are balanced. First, we investigate the MAS over a finite horizon. The utility functions of agents are parameterized to incorporate individual preferences. We shape individual preferences through a set of utility functions to guarantee the resource price at a competitive equilibrium remains socially acceptable, i.e., the price is upper-bounded by an affordability threshold. We show this problem is solvable at the conceptual level. Next, we consider quadratic MAS and formulate the associated social shaping problem as a multi-agent linear quadratic regulator (LQR) problem which enables us to propose explicit utility sets using quadratic programming and dynamic programming. Then, a numerical algorithm is presented for calculating a tight range of the preference function parameters which guarantees a socially accepted price. We investigate the properties of a competitive equilibrium over an infinite horizon. Considering general utility functions, we show that under feasibility assumptions, any competitive equilibrium maximizes the social welfare. Then, we prove that for sufficiently small initial conditions, the social welfare maximization solution constitutes a competitive equilibrium with zero price. We also prove for general feasible initial conditions, there exists a time instant after which the optimal price, corresponding to a competitive equilibrium, becomes zero. Finally, we specifically focus on quadratic MAS and propose explicit results.

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arXiv2022

Social Shaping of Dynamic Multi-Agent Systems over a Finite Horizon

This paper studies self-sustained dynamic multiagent systems (MAS) for decentralized resource allocation operating at a competitive equilibrium over a finite horizon. The utility of resource consumption, along with the income from resource exchange, forms each agent's payoff which is aimed to be maximized. Each utility function is parameterized by individual preferences which can be designed by agents independently. By shaping these preferences and proposing a set of utility functions, we can guarantee that the optimal resource price at the competitive equilibrium always remains socially acceptable, i.e., it never violates a given threshold that indicates affordability. First, we show this problem is solvable at the conceptual level under some convexity assumptions. Then, as a benchmark case, we consider quadratic MAS and formulate the associated social shaping problem as a multi-agent LQR problem which enables us to propose explicit utility sets using quadratic programming and dynamic programming. Finally, a numerical algorithm is presented for calculating the range of the preference function parameters which guarantee a socially accepted price. Some illustrative examples are given to examine the effectiveness of the proposed methods.

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arXiv2021

Social Shaping for Transactive Energy Systems

This paper considers the problem of shaping agent utility functions in a transactive energy system to ensure the optimal energy price at a competitive equilibrium is always socially acceptable, that is, below a prescribed threshold. Agents in a distributed energy system aim to maximize their individual payoffs, as a combination of the utility of energy consumption and the income/expenditure from energy exchange. The utility function of each agent is parameterized by individual preference vectors, with the overall system operating at competitive equilibriums. We show the social shaping problem of the proposed transactive energy system is conceptually captured by a set decision problem. The set of agent preferences that guarantees a socially acceptable price is characterized by an implicit algebraic equation for strictly concave and continuously differentiable utility functions. We also present two analytical solutions where tight ranges for the coefficients of linear-quadratic utilities and piece-wise linear utilities are established under which optimal pricing is proven to be always socially acceptable.

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arXiv2021

Towards Online Optimization for Power Grids

In this note, we discuss potential advantages in extending distributed optimization frameworks to enhance support for power grid operators managing an influx of online sequential decisions. First, we review the state-of-the-art distributed optimization frameworks for electric power systems, and explain how distributed algorithms deliver scalable solutions. Next, we introduce key concepts and paradigms for online optimization, and present a distributed online optimization framework highlighting important performance characteristics. Finally, we discuss the connection and difference between offline and online distributed optimization, showcasing the suitability of such optimization techniques for power grid applications.

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arXiv2019

A Survey of Algorithms for Distributed Charging Control of Electric Vehicles in Smart Grid

Electric vehicles (EVs) are an eco-friendly alternative to vehicles with internal combustion engines. Despite their environmental benefits, the massive electricity demand imposed by the anticipated proliferation of EVs could jeopardize the secure and economic operation of the power grid. Hence, proper strategies for charging coordination will be indispensable to the future power grid. Coordinated EV charging schemes can be implemented as centralized, decentralized, and hierarchical systems, with the last two, referred to as distributed charging control systems. This paper reviews the recent literature of distributed charging control schemes, where the computations are distributed across multiple EVs and/or aggregators. First, we categorize optimization problems for EV charging in terms of operational aspects and cost aspects. Then under each category, we provide a comprehensive discussion on algorithms for distributed EV charge scheduling, considering the perspectives of the grid operator, the aggregator, and the EV user. We also discuss how certain algorithms proposed in the literature cope with various uncertainties inherent to distributed EV charging control problems. Finally, we outline several research directions that require further attention.

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