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Electricity markets often utilize the DC approximation of the AC power flow equations to facilitate solving an otherwise complex nonconvex optimization problem. These DC power flow equations have analogies to DC circuit laws such as Kirchhoff's Laws, resulting in an intuitive understanding of power flows under this model. Variables derived from the Lagrangian dual of the DC optimal power flow problem, such as locational marginal prices (LMPs) and congestion cost, are less intuitive without an understanding of optimization theory. Even with this understanding, LMP behavior, such as the conditions in which negative prices occur or the impact of individual congested lines on network-wide LMPs, remain somewhat mysterious. In this paper, we show that prices also obey DC circuit laws, which can help facilitate an intuitive understanding of their behavior and relationships throughout a network without explicitly understanding duality. In particular, prices can be modeled as voltages, and their differences can be modeled as flows, allowing for a physical interpretation of prices. This analogy also lends itself to the use of well-understood DC circuit concepts such as superposition and Kirchhoff's Laws, which can further facilitate a clearer understanding of price behavior.

The adoption of electric vehicles (EVs), including electric taxis and buses, as a mode of transportation, is rapidly increasing in cities. In addition to providing economic and environmental benefits, these fleets can potentially participate in the energy arbitrage market by leveraging their mobile energy storage capabilities. This presents an opportunity for EV owners to contribute to a more sustainable and efficient energy system while also reducing their operational costs. The present study introduces deterministic and single-stage stochastic optimization frameworks that aim to maximize revenue by optimizing the charging, discharging, and travel of a fleet of electric vehicles in the context of uncertainty surrounding both spatial and temporal energy prices. The simulations are performed on a fleet of electric delivery trucks, which have to make deliveries to certain locations on specific dates. The findings indicate the promising potential of bidirectional electric vehicle charging as a mobile grid asset. However, it is important to note that significant revenue is only realized in scenarios where there is substantial variation in prices between different areas, and when these price variations can be accurately forecasted with a high level of confidence.

Linear approximations of the AC power flow equations are of great significance for the computational efficiency of large-scale optimal power flow (OPF) problems. Put differently, the feasibility of the obtained solution is essential for practical use cases of OPF. However, most studies focus on approximation error and come short of comprehensively studying the AC feasibility of different linear approximations of power flow. This paper discusses the merits of widely-used linear approximations of active power in OPF problems. The advantages and disadvantages of the linearized models are discussed with respect to four criteria; accuracy of the linear approximation, optimality, feasibility, and computation time. Each method is tested on five different systems.

The application of deep learning methods to speed up the resolution of challenging power flow problems has recently shown very encouraging results. However, power system dynamics are not snap-shot, steady-state operations. These dynamics must be considered to ensure that the optimal solutions provided by these models adhere to practical dynamical constraints, avoiding frequency fluctuations and grid instabilities. Unfortunately, dynamic system models based on ordinary or partial differential equations are frequently unsuitable for direct application in control or state estimates due to their high computational costs. To address these challenges, this paper introduces a machine learning method to approximate the behavior of power systems dynamics in near real time. The proposed framework is based on gradient-enhanced physics-informed neural networks (gPINNs) and encodes the underlying physical laws governing power systems. A key characteristic of the proposed gPINN is its ability to train without the need of generating expensive training data. The paper illustrates the potential of the proposed approach in both forward and inverse problems in a single-machine infinite bus system for predicting rotor angles and frequency, and uncertain parameters such as inertia and damping to showcase its potential for a range of power systems applications., Comment: 8 pages, 7 figures

Optimal Power Flow (OPF) is a fundamental problem in power systems. It is computationally challenging and a recent line of research has proposed the use of Deep Neural Networks (DNNs) to find OPF approximations at vastly reduced runtimes when compared to those obtained by classical optimization methods. While these works show encouraging results in terms of accuracy and runtime, little is known on why these models can predict OPF solutions accurately, as well as about their robustness. This paper provides a step forward to address this knowledge gap. The paper connects the volatility of the outputs of the generators to the ability of a learning model to approximate them, it sheds light on the characteristics affecting the DNN models to learn good predictors, and it proposes a new model that exploits the observations made by this paper to produce accurate and robust OPF predictions.

Increasing levels of renewable generation motivate a growing interest in data-driven approaches for AC optimal power flow (AC OPF) to manage uncertainty; however, a lack of disciplined dataset creation and benchmarking prohibits useful comparison among approaches in the literature. To instill confidence, models must be able to reliably predict solutions across a wide range of operating conditions. This paper develops the OPF-Learn package for Julia and Python, which uses a computationally efficient approach to create representative datasets that span a wide spectrum of the AC OPF feasible region. Load profiles are uniformly sampled from a convex set that contains the AC OPF feasible set. For each infeasible point found, the convex set is reduced using infeasibility certificates, found by using properties of a relaxed formulation. The framework is shown to generate datasets that are more representative of the entire feasible space versus traditional techniques seen in the literature, improving machine learning model performance., Comment: 5 pages, 3 figures, 2 Tables. Accepted at ISGT-NA'2022

Increasing amounts of distributed generation in distribution networks can provide both challenges and opportunities for voltage regulation across the network. Intelligent control of smart inverters and other smart building energy management systems can be leveraged to alleviate these issues. GridLearn is a multiagent reinforcement learning platform that incorporates both building energy models and power flow models to achieve grid level goals, by controlling behind-the-meter resources. This study demonstrates how multi-agent reinforcement learning can preserve building owner privacy and comfort while pursuing grid-level objectives. Building upon the CityLearn framework which considers RL for building-level goals, this work expands the framework to a network setting where grid-level goals are additionally considered. As a case study, we consider voltage regulation on the IEEE-33 bus network using controllable building loads, energy storage, and smart inverters. The results show that the RL agents nominally reduce instances of undervoltages and reduce instances of overvoltages by 34%.

While reinforcement learning (RL) is gaining popularity in energy systems control, its real-world applications are limited due to the fact that the actions from learned policies may not satisfy functional requirements or be feasible for the underlying physical system. In this work, we propose PROjected Feasibility (PROF), a method to enforce convex operational constraints within neural policies. Specifically, we incorporate a differentiable projection layer within a neural network-based policy to enforce that all learned actions are feasible. We then update the policy end-to-end by propagating gradients through this differentiable projection layer, making the policy cognizant of the operational constraints. We demonstrate our method on two applications: energy-efficient building operation and inverter control. In the building operation setting, we show that PROF maintains thermal comfort requirements while improving energy efficiency by 4% over state-of-the-art methods. In the inverter control setting, PROF perfectly satisfies voltage constraints on the IEEE 37-bus feeder system, as it learns to curtail as little renewable energy as possible within its safety set., Comment: Accepted at Twelfth ACM International Conference on Future Energy Systems (ACM e-Energy)

Using machine learning to obtain solutions to AC optimal power flow has recently been a very active area of research due to the astounding speedups that result from bypassing traditional optimization techniques. However, generally ensuring feasibility of the resulting predictions while maintaining these speedups is a challenging, unsolved problem. In this paper, we train a neural network to emulate an iterative solver in order to cheaply and approximately iterate towards the optimum. Once we are close to convergence, we then solve a power flow to obtain an overall AC-feasible solution. Results shown for networks up to 1,354 buses indicate the proposed method is capable of finding feasible, near-optimal solutions to AC OPF in milliseconds on a laptop computer. In addition, it is shown that the proposed method can find "difficult" AC OPF solutions that cause flat-start or DC-warm started algorithms to diverge.

Leveraging electrochemical and thermal energy storage systems has been proposed as a strategy to reduce peak power in data centers. Thermal energy storage systems, such as chilled water tanks, have gained increasing attention in data centers for load shifting due to their relatively small capital and operational costs compared to electrochemical energy storage. However, there are few studies investigating the possibility of utilizing thermal energy storage system with resources to provide ancillary services (e.g., frequency regulation) to the grid. This paper proposes a synergistic control strategy for the data center with a chilled water storage providing frequency regulation service by adjusting the chiller capacity, storage charging rate, and IT server CPU frequency. Then, a three-stage multi-market scheduling framework based on a model predictive control scheme is developed to minimize operational costs of data centers participating in both energy and regulation markets. The framework solves a power baseline scheduling problem, a regulation reserve problem, and a real-time power signal tracking problem sequentially. Simulation results show that utilizing the thermal energy storage can increase the regulation capacity bid, reduce energy costs and demand charges, and also harvest frequency regulation revenues. The proposed multi-market scheduling framework in a span of two days can reduce the operational costs up to 8.8% ($1,606.4) compared to the baseline with 0.2% (\$38.7) energy cost reduction, 6.5% (\$1,179.4) from demand reduction, and 2.1% (\$338.3) from regulation revenues.

Power grid operators typically solve large-scale, nonconvex optimal power flow (OPF) problems throughout the day to determine optimal setpoints for generators while adhering to physical constraints. Despite being at the heart of many OPF solvers, Newton-Raphson can be slow and numerically unstable. To reduce the computational burden associated with calculating the full Jacobian and its inverse, many Quasi-Newton methods attempt to find a solution to the optimality conditions by leveraging an approximate Jacobian matrix. In this paper, a Quasi-Newton method based on machine learning is presented which performs iterative updates for candidate optimal solutions without having to calculate a Jacobian or approximate Jacobian matrix. The proposed learning-based algorithm utilizes a deep neural network with feedback. With proper choice of weights and activation functions, the model becomes a contraction mapping and convergence can be guaranteed. Results shown for networks up to 1,354 buses indicate the proposed method is capable of finding approximate solutions to AC OPF very quickly.

In this paper, we discuss our approach and algorithmic framework for solving large-scale security constrained optimal power flow (SCOPF) problems. SCOPF is a mixed integer non-convex optimization problem that aims to obtain the minimum dispatch cost while maintaining the system N-1 secure. Finding a feasible solution for this problem over large networks is challenging and this paper presents contingency selection, approximation methods, and decomposition techniques to address this challenge in a short period of time. The performance of the proposed methods are verified through large-scale synthetic and actual power networks in the Grid Optimization (GO) competition organized by the U.S. Advanced Research Projects Agency-Energy (ARPA-E). As many prior works focus on small-scale systems and are not benchmarked using validated, publicly available datasets, we aim to present a practical solution to SCOPF that has been proven to achieve good performance on realistically sized (30,000 buses) networks.

The increasing adoption of smart meters introduces growing concerns about consumer privacy risks stemming from high resolution metering data. To counter these risks, there have been various works in actively shaping the grid-visible energy consumption profile using controllable loads such as energy storage systems (ESSs) and flexible consumer loads. In this paper, we compare the use of flexible thermal-based consumer loads (FTLs) against ESSs for consumer privacy protection. By first assuming ideal conditions, and subsequently bringing them closer to reality, the limitations of using FTLs for privacy protection are identified. Through theoretical analyses and realistic simulations, it is shown that, due to the limitations in the operation of FTLs, without significant over-sizing of systems and sacrifices in consumer comfort, FTLs of much higher equivalent energy storage capacity are required to afford the same level of protection as ESSs. Nonetheless, given their increasing ubiquity, controllable FTLs should be considered for use in consumer privacy protection., Comment: This is the accepted version of the article published in Applied Energy, and is shared under the Creative Commons CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

In this paper, we analyze the relationship between generation dispatch solutions produced by the DC optimal power flow (DC OPF) problem by the AC optimal power flow (AC OPF) problem. While there has been much previous work in analyzing the approximation error of the DC assumption, the AC feasibility has not been fully explored, although difficulty achieving AC feasibility is known in practice. Here, we consider the set of feasible points in a standard DC OPF problem and the set of feasible points in a standard AC OPF problem. Under some very light assumptions, we show that the intersection of these sets is the empty set; i.e., that no solution to the DC OPF problem will satisfy the AC power flow constraints. Then, it is demonstrated that even with generation adjustments in DC OPF to account for losses, DC OPF solutions are still not AC feasible.

We propose a novel data-driven method to accelerate the convergence of Alternating Direction Method of Multipliers (ADMM) for solving distributed DC optimal power flow (DC-OPF) where lines are shared between independent network partitions. Using previous observations of ADMM trajectories for a given system under varying load, the method trains a recurrent neural network (RNN) to predict the converged values of dual and consensus variables. Given a new realization of system load, a small number of initial ADMM iterations is taken as input to infer the converged values and directly inject them into the iteration. We empirically demonstrate that the online injection of these values into the ADMM iteration accelerates convergence by a significant factor for partitioned 14-, 118- and 2848-bus test systems under differing load scenarios. The proposed method has several advantages: it maintains the security of private decision variables inherent in consensus ADMM; inference is fast and so may be used in online settings; RNN-generated predictions can dramatically improve time to convergence but, by construction, can never result in infeasible ADMM subproblems; it can be easily integrated into existing software implementations. While we focus on the ADMM formulation of distributed DC-OPF in this paper, the ideas presented are naturally extended to other distributed optimization problems.

In this paper, we develop an online method that leverages machine learning to obtain feasible solutions to the AC optimal power flow (OPF) problem with negligible optimality gaps on extremely fast timescales (e.g., milliseconds), bypassing solving an AC OPF altogether. This is motivated by the fact that as the power grid experiences increasing amounts of renewable power generation, controllable loads, and other inverter-interfaced devices, faster system dynamics and quicker fluctuations in the power supply are likely to occur. Currently, grid operators typically solve AC OPF every 15 minutes to determine economic generator settings while ensuring grid constraints are satisfied. Due to the computational challenges with solving this nonconvex problem, many efforts have focused on linearizing or approximating the problem in order to solve the AC OPF on faster timescales. However, many of these approximations can be fairly poor representations of the actual system state and still require solving an optimization problem, which can be time consuming for large networks. In this work, we leverage historical data to learn a mapping between the system loading and optimal generation values, enabling us to find near-optimal and feasible AC OPF solutions on extremely fast timescales without actually solving an optimization problem.

A large amount of data has been generated by grid operators solving AC optimal power flow (ACOPF) throughout the years, and we explore how leveraging this data can be used to help solve future ACOPF problems. We use this data to train a Random Forest to predict solutions of future ACOPF problems. To preserve correlations and relationships between predicted variables, we utilize a multi-target approach to learn approximate voltage and generation solutions to ACOPF problems directly by only using network loads, without the knowledge of other network parameters or the system topology. We explore the benefits of using the learned solution as an intelligent warm start point for solving the ACOPF, and the proposed framework is evaluated numerically using multiple IEEE test networks. The benefit of using learned ACOPF solutions is shown to be solver and network dependent, but shows promise for quickly finding approximate solutions to the ACOPF problem.

Recently, the volatility associated with marginal prices has increased due to large scale integration of renewable generation. Price volatility is undesirable from a consumer perspective. To address this issue, we present a framework for hedging that uses duality theory for quantifying the amount of demand-side flexibility required for constraining marginal prices to the consumers maximum willingness to pay for electricity. Using our formulation, we investigate the ability of an Energy Storage System (ESS), as a demand-side flexibility source, to hedge against electricity price volatility across a multi-time period horizon while accounting for its inter-temporal constraints. Additionally, we analyze the economical benefit that operating the ESS under information forecasts brings to the consumers., Comment: Submitted to IEEE Power and Energy Society General Meeting 2019

Locational Marginal Price (LMP) is a dual variable associated with supply-demand matching and represents the cost of delivering power to a particular location if the load at that location increases. In recent times it become more volatile due to increased integration of renewables that are intermittent. The issue of price volatility is further heightened during periods of grid congestion. Motivated by these problems, we propose a market design where, by constraining dual variables, we determine the amount of demand-side flexibility required to limit the rise of LMP. Through our proposed approach a price requesting load can specify its maximum willingness to pay for electricity and through demand-side flexibility hedge against price volatility. For achieving this, an organizational structure for flexibility management is proposed that exhibits the coordination required between the Distribution System Operator (DSO), an aggregator and the price requesting load. To demonstrate the viability of our proposed formulation, we run an illustrative simulation under infinite and finite line capacities.

This paper considers distribution systems with a high penetration of distributed, renewable generation and addresses the problem of incorporating the associated uncertainty into the optimal operation of these networks. Joint chance constraints, which satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints, leading to a chance-constrained optimal power flow problem. Departing from the computationally-heavy scenario-based approaches or approximations that transform the joint constraint into conservative deterministic constraints, this paper develops a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and accurately and efficiently approximates bounds on the joint chance constraint iteratively. In particular, the proposed framework improves upon the classic methods based on the union bound (or Boole's inequality) by generating a much less conservative set of single chance constraints that also guarantees the satisfaction of the original joint constraint. The proposed framework is evaluated numerically using the IEEE 37-node test feeder, focusing on the problem of voltage regulation in distribution grids., Comment: 10 pages

In this paper we provide non-simultaneous charging and discharging guarantees for a linear energy storage system (ESS) model for a model predictive control (MPC) based home energy management system (HEMS) algorithm. The HEMS optimally controls the residential load and residentially-owned power sources, such as photovoltaic (PV) power generation and energy storage, given residential customer preferences such as energy cost sensitivity and ESS lifetime. Under certain problem formulations with a linear ESS model, simultaneous charging and discharging can be observed as the optimal solution when there is high penetration of PV power. We present analysis for a proposed HEMS optimization formulation that ensures non-simultaneous ESS charging and discharging operation for a linear ESS model that captures both charging and discharging efficiency of the ESS. The energy storage system model behavior guarantees are shown for various electricity pricing schemes such as time of use (TOU) pricing and net metering. Simulation results demonstrating desirable ESS behavior are provided for each electricity pricing scheme., Comment: Revised manuscript submitted to IEEE Transactions on Control Systems Technology on 19 April 2019

We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The framework explicitly combines multi-stage feedback policies with any forecasting method and historical forecast error data. The objective is to determine power scheduling policies for controllable devices in a power network to balance operational cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include both nominal power schedules and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming the uncertainties across the networks follow prescribed probability distributions, we consider ambiguity sets of distributions centered around a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real unknown data-generating distribution, we formulate a multi-stage distributionally robust OPF problem to compute optimal control policies that are robust to both forecast errors and sampling errors inherent in the dataset. Two specific data-based distributionally robust stochastic OPF problems are proposed for distribution networks and transmission systems., Comment: arXiv admin note: text overlap with arXiv:1706.04267

This is the second part of a two-part paper on data-based distributionally robust stochastic optimal power flow (OPF). The general problem formulation and methodology have been presented in Part I [1]. Here, we present extensive numerical experiments in both distribution and transmission networks to illustrate the effectiveness and flexibility of the proposed methodology for balancing efficiency, constraint violation risk, and out-of-sample performance. On the distribution side, the method mitigates overvoltages due to high photovoltaic penetration using local energy storage devices. On the transmission side, the method reduces N-1 security line flow constraint risks due to high wind penetration using reserve policies for controllable generators. In both cases, the data-based distributionally robust model predictive control (MPC) algorithm explicitly utilizes forecast error training datasets, which can be updated online. The numerical results illustrate inherent tradeoffs between the operational costs, risks of constraints violations, and out-of-sample performance, offering systematic techniques for system operators to balance these objectives.

We propose a data-driven method to solve a stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The objective is to determine power schedules for controllable devices in a power network to balance operation cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include scheduled power output adjustments and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming the uncertainties across the networks follow prescribed probability distributions, we assume the distributions are only observable through a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real data-generating distribution, we formulate a distributionally robust optimization OPF problem to search for power schedules and reserve policies that are robust to sampling errors inherent in the dataset. A simple numerical example illustrates inherent tradeoffs between operation cost and risk of constraint violation, and we show how our proposed method offers a data-driven framework to balance these objectives.

This paper examines distribution systems with a high integration of distributed energy resources (DERs) and addresses the design of local control methods for real-time voltage regulation. Particularly, the paper focuses on proportional control strategies where the active and reactive output-powers of DERs are adjusted in response to (and proportionally to) local changes in voltage levels. The design of the voltage-active power and voltage-reactive power characteristics leverages suitable linear approximations of the AC power-flow equations and is network-cognizant; that is, the coefficients of the controllers embed information on the location of the DERs and forecasted non-controllable loads/injections and, consequently, on the effect of DER power adjustments on the overall voltage profile. A robust approach is pursued to cope with uncertainty in the forecasted non-controllable loads/power injections. Stability of the proposed local controllers is analytically assessed and numerically corroborated.

Evolving power systems with increasing levels of stochasticity call for a need to solve optimal power flow problems with large quantities of random variables. Weather forecasts, electricity prices, and shifting load patterns introduce higher levels of uncertainty and can yield optimization problems that are difficult to solve in an efficient manner. Efficient solution methods for single chance constraints in optimal power flow problems have been considered in the literature; however, joint chance constraints have predominantly been solved via scenario-based approaches or by utilizing the overly conservative Boole's inequality as an upper bound. In this paper, joint chance constraints are used to solve an AC optimal power flow problem which maintain desired levels of voltage magnitude in distribution grids under high penetrations of photovoltaic systems. A tighter version of Boole's inequality is derived and used to provide a new upper bound on the joint chance constraint, and simulation results are shown demonstrating the benefit of the proposed upper bound., Comment: 8 Pages; Submitted to IEEE Transactions on Smart Grid. arXiv admin note: text overlap with arXiv:1601.07263 by other authors

The marginal price of electricity traditionally depends on the dual variables associated with relevant optimization goals. Particularly, in the optimal power flow realm, prices represent the cost of supplying an additional unit of power at each bus; for the economic dispatch case, dual variables represent the cost of supplying an additional unit of power to the whole system. Dual variables are useful for many additional tasks, including the analysis of system congestion and the determination of the cost of load adjustments. In this letter, departing from conventional optimal power flow analysis, it is shown how constraints on relevant dual variables affect the prices of electricity, allowing for market settings and demand response programs that accept bids and caps on individual prices., Comment: 2 pages