arXiv subjects

Andrew L. Liu

Publications and source records attributed to Andrew L. Liu.

16 recordsLinked to original sources

arXiv2026

Carbon-Aware Data Center Workload Allocation: Emission Disclosure, Capacity Leasing, and Contract Reshuffling

The rapid adoption of AI has driven rapid growth in computational demand, with large language models (LLMs) at the forefront since ChatGPT's debut in 2022. Meanwhile, large amounts of renewable energy are ultimately curtailed due to transmission congestion and inadequate demand. This work develops a power market model that allows hyperscalers to spatially migrate LLM inference workloads to geo-distributed modular datacenters (MDCs) co-located with renewable generation at the edge of the network. We introduce the optimization problems faced by the hyperscaler and MDCs in addition to consumers, producers, and the electric grid operator, where the hyperscaler leases MDC capacity while ensuring that required service level objectives (SLOs) are met. The overall market model is formulated as a complementarity problem, for which we establish equilibrium existence and uniqueness of certain aggregate market quantities. We further show that bilateral contract allocations can vary while preserving the same physical market outcome, so cleaner contract-attributed procurement need not imply additional clean generation. Applying the model to the IEEE RTS-24 bus system, we find that even when MDCs disclose the CO$_2$ emissions associated with their energy supply, renting less polluting MDCs yields limited system emission reductions because of \textit{contract reshuffling}. This effect can be mitigated when conventional loads are supplied through forward contracts such as power purchase agreements. Interestingly, this also reduces system congestion as the hyperscaler becomes increasingly cost-aware.

eess.SY↗

arXiv2026

Learning in Stackelberg Markov Games

Designing socially optimal policies in multi-agent environments is a fundamental challenge in both economics and artificial intelligence. This paper studies a general framework for learning Stackelberg equilibria in dynamic and uncertain environments, where a single leader interacts with a population of adaptive followers. Motivated by pressing real-world challenges such as equitable electricity tariff design for consumers with distributed energy resources (such as rooftop solar and energy storage), we formalize a class of Stackelberg Markov games and establish the existence and uniqueness of stationary Stackelberg equilibria under mild continuity and monotonicity conditions. We then extend the framework to incorporate a continuum of agents via mean-field approximation, yielding a tractable Stackelberg-Mean Field Equilibrium (S-MFE) formulation. To address the computational intractability of exact best-response dynamics, we introduce a softmax-based approximation and rigorously bound its error relative to the true Stackelberg equilibrium. Our approach enables scalable and stable learning through policy iteration without requiring full knowledge of follower objectives. We validate the framework on an energy market simulation, where a public utility or a state utility commission sets time-varying rates for a heterogeneous population of prosumers. Our results demonstrate that learned policies can simultaneously achieve economic efficiency, equity across income groups, and stability in energy systems. This work demonstrates how game-theoretic learning frameworks can support data-driven policy design in large-scale strategic environments, with applications to real-world systems like energy markets.

eess.SY↗

arXiv2026

A Hybrid Mean Field Framework for Aggregators Participating in Wholesale Electricity Markets

The rapid growth of distributed energy resources (DERs), including rooftop solar and energy storage, is transforming the grid edge, where distributed technologies and customer-side systems increasingly interact with the broader power grid. DER aggregators, entities that coordinate and optimize the actions of many small-scale DERs, play a key role in this transformation. This paper presents a hybrid Mean-Field Control (MFC) and Mean-Field Game (MFG) framework for integrating DER aggregators into wholesale electricity markets. Unlike traditional approaches that treat market prices as exogenous, our model captures the feedback between aggregators' strategies and locational marginal prices (LMPs) of electricity. The MFC component optimizes DER operations within each aggregator, while the MFG models strategic interactions among multiple aggregators. To account for various uncertainties, we incorporate reinforcement learning (RL), which allows aggregators to learn optimal bidding strategies in dynamic market conditions. We prove the existence and uniqueness of a mean-field equilibrium and validate the framework through a case study of the Oahu Island power system. Results show that our approach reduces price volatility and improves market efficiency, offering a scalable and decentralized solution for DER integration in wholesale markets.

eess.SY↗

arXiv2026

Robust Generalization with Adaptive Optimal Transport Priors for Decision-Focused Learning

Few-shot learning requires models to generalize under limited supervision while remaining robust to distribution shifts. Existing Sinkhorn Distributionally Robust Optimization (DRO) methods provide theoretical guarantees but rely on a fixed reference distribution, which limits their adaptability. We propose a Prototype-Guided Distributionally Robust Optimization (PG-DRO) framework that learns class-adaptive priors from abundant base data via hierarchical optimal transport and embeds them into the Sinkhorn DRO formulation. This design enables few-shot information to be organically integrated into producing class-specific robust decisions that are both theoretically grounded and efficient, and further aligns the uncertainty set with transferable structural knowledge. Experiments show that PG-DRO achieves stronger robust generalization in few-shot scenarios, outperforming both standard learners and DRO baselines.

stat.ML↗

arXiv2026

Efficient Causal Structure Learning via Modular Subgraph Integration

Learning causal structures from observational data remains a fundamental yet computationally intensive task, particularly in high-dimensional settings where existing methods face challenges such as the super-exponential growth of the search space and increasing computational demands. To address this, we introduce VISTA (Voting-based Integration of Subgraph Topologies for Acyclicity), a modular framework that decomposes the global causal structure learning problem into local subgraphs based on Markov Blankets. The global integration is achieved through a weighted voting mechanism that penalizes low-support edges via exponential decay, filters unreliable ones with an adaptive threshold, and ensures acyclicity using a Feedback Arc Set (FAS) algorithm. The framework is model-agnostic, imposing no assumptions on the inductive biases of base learners, is compatible with arbitrary data settings without requiring specific structural forms, and fully supports parallelization. We also theoretically establish finite-sample error bounds for VISTA, and prove its asymptotic consistency under mild conditions. Extensive experiments on both synthetic and real datasets consistently demonstrate the effectiveness of VISTA, yielding notable improvements in both accuracy and efficiency over a wide range of base learners.

stat.ML↗

arXiv2025

Evaluating the Impact of Multiple DER Aggregators on Wholesale Energy Markets: A Hybrid Mean Field Approach

The integration of distributed energy resources (DERs) into wholesale energy markets can greatly enhance grid flexibility, improve market efficiency, and contribute to a more sustainable energy future. As DERs -- such as solar PV panels and energy storage -- proliferate, effective mechanisms are needed to ensure that small prosumers can participate meaningfully in these markets. We study a wholesale market model featuring multiple DER aggregators, each controlling a portfolio of DER resources and bidding into the market on behalf of the DER asset owners. The key of our approach lies in recognizing the repeated nature of market interactions the ability of participants to learn and adapt over time. Specifically, Aggregators repeatedly interact with each other and with other suppliers in the wholesale market, collectively shaping wholesale electricity prices (aka the locational marginal prices (LMPs)). We model this multi-agent interaction using a mean-field game (MFG), which uses market information -- reflecting the average behavior of market participants -- to enable each aggregator to predict long-term LMP trends and make informed decisions. For each aggregator, because they control the DERs within their portfolio under certain contract structures, we employ a mean-field control (MFC) approach (as opposed to a MFG) to learn an optimal policy that maximizes the total rewards of the DERs under their management. We also propose a reinforcement learning (RL)-based method to help each agent learn optimal strategies within the MFG framework, enhancing their ability to adapt to market conditions and uncertainties. Numerical simulations show that LMPs quickly reach a steady state in the hybrid mean-field approach. Furthermore, our results demonstrate that the combination of energy storage and mean-field learning significantly reduces price volatility compared to scenarios without storage.

eess.SY↗

arXiv2025

Decentralized Integration of Grid Edge Resources into Wholesale Electricity Markets via Mean-field Games

Grid edge resources refer to distributed energy resources (DERs) located on the consumer side of the electrical grid, controlled by consumers rather than utility companies. Integrating DERs with real-time electricity pricing can better align distributed supply with system demand, improving grid efficiency and reliability. However, DER owners, known as prosumers, often lack the expertise and resources to directly participate in wholesale energy markets, limiting their ability to fully realize the economic potential of their assets. Meanwhile, as DER adoption grows, the number of prosumers participating in the energy system is expected to increase significantly, creating additional challenges in coordination and market participation. To address these challenges, we propose a mean-field game framework that enables prosumers to autonomously learn optimal decision policies based on dynamic market prices and their variable solar generation. Our framework is designed to accommodate heterogeneous agents and demonstrates the existence of a mean-field equilibrium (MFE) in a wholesale energy market with many prosumers. Additionally, we introduce an algorithm that automates prosumers' resource control, facilitating real-time decision-making for energy storage management. Numerical experiments suggest that our approach converges towards an MFE and effectively reduces peak loads and price volatility, especially during periods of external demand or supply shocks. This study highlights the potential of a fully decentralized approach to integrating DERs into wholesale markets while improving market efficiency.

eess.SY↗

arXiv2024

Peer-to-Peer Energy Trading of Solar and Energy Storage: A Networked Multiagent Reinforcement Learning Approach

Utilizing distributed renewable and energy storage resources in local distribution networks via peer-to-peer (P2P) energy trading has long been touted as a solution to improve energy systems' resilience and sustainability. Consumers and prosumers (those who have energy generation resources), however, do not have the expertise to engage in repeated P2P trading, and the zero-marginal costs of renewables present challenges in determining fair market prices. To address these issues, we propose multi-agent reinforcement learning (MARL) frameworks to help automate consumers' bidding and management of their solar PV and energy storage resources, under a specific P2P clearing mechanism that utilizes the so-called supply-demand ratio. In addition, we show how the MARL frameworks can integrate physical network constraints to realize voltage control, hence ensuring physical feasibility of the P2P energy trading and paving way for real-world implementations.

eess.SY↗

arXiv2022

Optimal Retail Tariff Design with Prosumers: Pursuing Equity at the Expenses of Economic Efficiencies?

Distributed renewable resources owned by prosumers can be an effective way of fortifying grid resilience and enhancing sustainability. However, prosumers serve their own interests and their objectives are unlikely to align with that of society. This paper develops a bilevel model to study the optimal design of retail electricity tariffs considering the balance between economic efficiency and energy equity. The retail tariff entails a fixed charge and a volumetric charge tied to electricity usage to recover utilities' fixed costs. We analyze solution properties of the bilevel problem and prove an optimal rate design, which is to use fixed charges to recover fixed costs and to balance energy equity among different income groups. This suggests that programs similar to CARE (California Alternative Rate of Energy), which offer lower retail rates to low-income households, are unlikely to be efficient, even if they are politically appealing.

eess.SY↗

arXiv2021

A Distributed Algorithm for High-Dimension Convex Quadratically Constrained Quadratic Programs

We propose a Jacobi-style distributed algorithm to solve convex, quadratically constrained quadratic programs (QCQPs), which arise from a broad range of applications. While small to medium-sized convex QCQPs can be solved efficiently by interior-point algorithms, large-scale problems pose significant challenges to traditional algorithms that are mainly designed to be implemented on a single computing unit. The exploding volume of data (and hence, the problem size), however, may overwhelm any such units. In this paper, we propose a distributed algorithm for general, non-separable, large-scale convex QCQPs, using a novel idea of predictor-corrector primal-dual update with an adaptive step size. The algorithm enables distributed storage of data as well as parallel distributed computing. We establish the conditions for the proposed algorithm to converge to a global optimum, and implement our algorithm on a computer cluster with multiple nodes using Message Passing Interface (MPI). The numerical experiments are conducted on data sets of various scales from different applications, and the results show that our algorithm exhibits favorable scalability for solving large-scale problems.

math.OC↗

arXiv2021

A Nested Cross Decomposition Algorithm for Power System Capacity Expansion with Multiscale Uncertainties

Modern electric power systems have witnessed rapidly increasing penetration of renewable energy, storage, electrical vehicles and various demand response resources. The electric infrastructure planning is thus facing more challenges due to the variability and uncertainties arising from the diverse new resources. This study aims to develop a multistage and multiscale stochastic mixed integer programming (MM-SMIP) model to capture both the coarse-temporal-scale uncertainties, such as investment cost and long-run demand stochasticity, and fine-temporal-scale uncertainties, such as hourly renewable energy output and electricity demand uncertainties, for the power system capacity expansion problem. To be applied to a real power system, the resulting model will lead to extremely large-scale mixed integer programming problems, which suffer not only the well-known curse of dimensionality, but also computational difficulties with a vast number of integer variables at each stage. In addressing such challenges associated with the MM-SMIP model, we propose a nested cross decomposition algorithm that consists of two layers of decomposition, that is, the Dantzig-Wolfe decomposition and L-shaped decomposition. The algorithm exhibits promising computational performance under our numerical study, and is especially amenable to parallel computing, which will also be demonstrated through the computational results.

cs.DC↗

arXiv2021

A Distributed Algorithm for Multi-scale Multi-stage Stochastic Programs with Application to Electricity Capacity Expansion

This paper applies the N-block PCPM algorithm to solve multi-scale multi-stage stochastic programs, with the application to electricity capacity expansion models. Numerical results show that the proposed simplified N-block PCPM algorithm, along with the hybrid decomposition method, exhibits much better scalability for solving the resulting deterministic, large-scale block-separable optimization problem when compared with the ADMM algorithm and the PHA algorithm. The superiority of the algorithm's scalability is attributed to the two key features of the algorithm design: first, the proposed hybrid scenario-node-realization decomposition method with extended nonanticipativity constraints can decompose the original problem under various uncertainties of different temporal scales; second, when applying the N-block PCPM algorithm to solve the resulting deterministic, large-scale N-block convex optimization problem, the technique of orthogonal projection we exploit greatly simplifies the iteration steps and reduce the communication overhead among all computing units, which also contributes to the efficiency of the algorithm.

math.OC↗

arXiv2021

Distributed and Asynchronous Algorithms for N-block Convex Optimization with Coupling Constraints

This paper first proposes an N-block PCPM algorithm to solve N-block convex optimization problems with both linear and nonlinear constraints, with global convergence established. A linear convergence rate under the strong second-order conditions for optimality is observed in the numerical experiments. Next, for a starting point, an asynchronous N-block PCPM algorithm is proposed to solve linearly constrained N-block convex optimization problems. The numerical results demonstrate the sub-linear convergence rate under the bounded delay assumption, as well as the faster convergence with more short-time iterations than a synchronous iterative scheme.

math.OC↗

arXiv2020

A Distributed Optimization Approach to the Multi-Regional Day-Ahead Clearing Process in Electricity Markets

The implementation of electricity markets based on locational marginal pricing in a multi-settlement process has allowed wholesale competition, with pricing mechanisms that incentivize the optimal allocation of generation, transmission, and demand-response resources. While efficiency and reliability gains have been achieved in the US at a regional level, the lack of adequate inter-regional coordination mechanisms has limited broader gains. The shortcomings of cross-border coordination of electricity markets become more apparent as the industry transitions towards higher penetration of renewable resources, which tend to be concentrated in certain regions of the country, usually distant from load pockets. In addition to allowing market participants to respond to economic signals beyond the market region where their assets reside, a coordinated market solution would allow the extension of market mechanisms such as financial transmission rights and capacity contracts to cover transactions across borders; thereby, reducing price uncertainty around the investment in new generation resources. This paper presents a distributed solution to the unit commitment problem that allows for full coordination of the market clearing process across interconnected electricity markets.

eess.SY↗

arXiv2020

Multi-Agent Learning in Double-side Auctions forPeer-to-peer Energy Trading

Distributed energy resources (DERs), such as rooftop solar panels, are growing rapidly and are reshaping power systems. To promote DERs, feed-in-tariff (FIT) is usually adopted by utilities to pay DER owners certain fixed rates for supplying energy to the grid. An alternative to FIT is a market based approach; i.e., consumers and DER owners trade energy in an auction-based peer-to-peer (P2P) market, and the rates are determined by a market clearing process. However, the complexities in sucha market and agents' bounded rationality may invalidate many well-established theories on auction design and hinder market development. To address this issue, we propose an automated bidding framework in a repeated auction based on multi-armed bandit learning, which aims to minimize each bidder's cumulative regret. Numerical results indicate convergence of such a multi-agent learning game to a steady-state. For comparison purpose, we apply the framework to three different auction designs to realize a P2P market.

eess.SY↗

arXiv2015

Multi-Agent Decentralized Network Interdiction Games

In this work, we introduce decentralized network interdiction games, which model the interactions among multiple interdictors with differing objectives operating on a common network. As a starting point, we focus on decentralized shortest path interdiction (DSPI) games, where multiple interdictors try to increase the shortest path lengths of their own adversaries, who all attempt to traverse a common network. We first establish results regarding the existence of equilibria for DSPI games under both discrete and continuous interdiction strategies. To compute such an equilibrium, we present a reformulation of the DSPI games, which leads to a generalized Nash equilibrium problem (GNEP) with non-shared constraints. While such a problem is computationally challenging in general, we show that under continuous interdiction actions, a DSPI game can be formulated as a linear complementarity problem and solved by Lemke's algorithm. In addition, we present decentralized heuristic algorithms based on best response dynamics for games under both continuous and discrete interdiction strategies. Finally, we establish theoretical bounds on the worst-case efficiency loss of equilibria in DSPI games, with such loss caused by the lack of coordination among noncooperative interdictors, and use the decentralized algorithms to empirically study the average-case efficiency loss.

math.OC↗