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A prominent challenge to the safe and optimal operation of the modern power grid arises due to growing uncertainties in loads and renewables. Stochastic optimal power flow (SOPF) formulations provide a mechanism to handle these uncertainties by computing dispatch decisions and control policies that maintain feasibility under uncertainty. Most SOPF formulations consider simple control policies such as affine policies that are mathematically simple and resemble many policies used in current practice. Motivated by the efficacy of machine learning (ML) algorithms and the potential benefits of general control policies for cost and constraint enforcement, we put forth a deep neural network (DNN)-based policy that predicts the generator dispatch decisions in real time in response to uncertainty. The weights of the DNN are learnt using stochastic primal-dual updates that solve the SOPF without the need for prior generation of training labels and can explicitly account for the feasibility constraints in the SOPF. The advantages of the DNN policy over simpler policies and their efficacy in enforcing safety limits and producing near optimal solutions are demonstrated in the context of a chance constrained formulation on a number of test cases.

In recent years, several applications have been proposed in the context of distribution networks. Many of these can be formulated as an optimal power flow problem, a mathematical optimization program which includes a model of the steady-state physics of the electricity network. If the network loading is balanced and the lines are transposed, the network model can be simplified to a single-phase equivalent model. However, these assumptions do not apply to low-voltage distribution networks, so the network model should model the effects of phase unbalance correctly. In many parts of the world, the low-voltage distribution network has four conductors, i.e. three phases and a neutral. This paper develops OPF formulations for such networks, including transformers, shunts and voltage-dependent loads, in two variable spaces, i.e. current-voltage and power-voltage, and compares them for robustness and scalability. A case study across 128 low-voltage networks also quantifies the modelling error introduced by Kron reductions and its impact on the solve time. This work highlights the advantages of formulations in current-voltage variables over power-voltage, for four-wire networks., Comment: 10 pages, submitted to Power Systems Computation Conference 2022

We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy -- a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor's relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem size and is indeed useful in certifying global optimality of solutions for large-scale real world problems, e.g., the AC-OPF problem. In particular, we are able to certify 1% global optimality for a challenging AC-OPF instance with 6515 buses involving 14398 real variables and 63577 constraints., Comment: 18 pages, 5 figures, 4 tables

The Security-Constrained Economic Dispatch (SCED) is a fundamental optimization model for Transmission System Operators (TSO) to clear real-time energy markets while ensuring reliable operations of power grids. In a context of growing operational uncertainty, due to increased penetration of renewable generators and distributed energy resources, operators must continuously monitor risk in real-time, i.e., they must quickly assess the system's behavior under various changes in load and renewable production. Unfortunately, systematically solving an optimization problem for each such scenario is not practical given the tight constraints of real-time operations. To overcome this limitation, this paper proposes to learn an optimization proxy for SCED, i.e., a Machine Learning (ML) model that can predict an optimal solution for SCED in milliseconds. Motivated by a principled analysis of the market-clearing optimizations of MISO, the paper proposes a novel ML pipeline that addresses the main challenges of learning SCED solutions, i.e., the variability in load, renewable output and production costs, as well as the combinatorial structure of commitment decisions. A novel Classification-Then-Regression architecture is also proposed, to further capture the behavior of SCED solutions. Numerical experiments are reported on the French transmission system, and demonstrate the approach's ability to produce, within a time frame that is compatible with real-time operations, accurate optimization proxies that produce relative errors below $0.6\%$., Comment: The paper is the extension version of the paper submitted to PSCC2022

Mathematical models are just models. The desire to describe battery energy storage system (BESS) operation using computationally tractable model formulations has motivated a long-standing discussion in both the scientific and industrial communities. Linear BESS models are the most widely used so far. However, finding suitable linear BESS models has been controversial. This paper focuses on the description of linear BESS models. Four linear BESS formulations are presented, among the most popularly used. A new formulation is also proposed. The 5 BESS models are tested in 100 random BESS and 1.450 random samples of daily profiles of renewable generation. Two classical problems of power systems, namely, the set-point tracking problem and the transmission expansion planning problem, are selected for numerical analysis. Five thousand simulations are used to draw a better interpretation of each linear formulation presented and showcase specific challenges of BESS models. Practical recommendations are provided based on the findings., Power Systems Computation Conference (PSCC) 2022

We propose a novel feasible-path algorithm to solve the optimal power flow (OPF) problem for real-time use cases. The method augments the seminal work of Dommel and Tinney with second-order derivatives to work directly in the reduced space induced by the power flow equations. In the reduced space, the optimization problem includes only inequality constraints corresponding to the operational constraints. While the reduced formulation directly enforces the physical constraints, the operational constraints are softly enforced through Augmented Lagrangian penalty terms. In contrast to interior-point algorithms (state-of-the art for solving OPF), our algorithm maintains feasibility at each iteration, which makes it suitable for real-time application. By exploiting accelerator hardware (Graphic Processing Units) to compute the reduced Hessian, we show that the second-order method is numerically tractable and is effective to solve both static and real-time OPF problems., 7 pages, 3 figures, conference proceeding

We study the problem of phase optimization for electric-vehicle (EV) charging. We formulate our problem as a non-convex mixed-integer programming problem whose objective is to minimize the charging loss. Despite the hardness of directly solving this non-convex problem, we solve a relaxation of the original problem by proposing the PXA algorithm where "P", "X", and "A" stand for three variable matrices in the formed phase optimization problems. We show that under certain conditions, the solution is given by the PXA precisely converges to the global optimum. In addition, using the idea of model predictive control (MPC), we design the {PXA-MPC}, which is an online implementation of the PXA. Compared to other empirical phase balancing strategies, the PXA algorithm significantly improves the charging performance by maximizing energy delivery, minimizing charging price, and assisting future energy planning. The efficacy of our algorithm is demonstrated using data collected from a real-world adaptive EV charging network (ACN)., 8 pages, 6 figures, accepted by PSCC 2022

Power systems with high penetration of variable renewable generation are vulnerable to periods with low generation. An alternative to retain high dispatchable generation capacity is electric energy storage that enables utilization of surplus power, where the electric energy storage contributes to the security of supply. Such systems can be considered as energy-constrained, and the operation of the electric energy storage must balance the minimization of the current operating costs against the risk of not being able to meet the future demand. Safe and efficient operation requires stochastic methods with sufficient foresight. Operation dependent storage degradation is a complicating factor. This paper proposes a linear approximation of battery state-of-charge degradation and implements it in a stochastic dual dynamic programming based energy-management model in combination with cycling degradation. The long-term implications of degradation modeling in the daily operation are studied for a small Norwegian microgrid with variable renewable power generation and limited dispatchable generation capacity as well as battery and hydrogen storage to balance supply and demand. Our results show that the proposed strategy can prolong the expected battery lifetime by more than four years compared to the naive stochastic strategy but may cause increased degradation for other system resources. © 2022 The Authors Stochastic operation of energy constrained microgrids considering battery degradation

A critical aspect of power systems research is the availability of suitable data, access to which is limited by privacy concerns and the sensitive nature of energy infrastructure. This lack of data, in turn, hinders the development of modern research avenues such as machine learning approaches or stochastic formulations. To overcome this challenge, this paper proposes a systematic, data-driven framework for reconstructing high-fidelity time series, using publicly-available grid snapshots and historical data published by transmission system operators. The proposed approach, from geo-spatial data and generation capacity reconstruction, to time series disaggregation, is applied to the French transmission grid. Thereby, synthetic but highly realistic time series data, spanning multiple years with a 5-minute granularity, is generated at the individual component level.

This paper proposes a novel and simple linear model to capture line losses for use in linearized DC models, such as optimal power flow (DC-OPF) and security-constrained economic dispatch (SCED). The \textit{Line Loss Outer Approximation} (LLOA) model implements an outer approximation of the line losses lazily and typically terminates in a small number of iterations. Experiments on large-scale power systems demonstrate the accuracy and computational efficiency of LLOA and contrast it with classical line loss approaches. The results seem to indicate that LLOA is a practical and useful model for real-world applications, providing a good tradeoff between accuracy, computational efficiency, and implementation simplicity. In particular, the LLOA method may have significant advantages compared to the traditional loss factor formulation for multi-period, stochastic optimization problems where good reference points may not be available. The paper also provides a comprehensive overview and evaluation of approximation methods for line losses.

This paper proposes a market mechanism for multi-interval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energy-cycling functions that map prices to cycle depths. The resulting market-clearing process--implemented via convex programming--yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumer-based approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful cost, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.

Modern transmission network expansion planning (TNEP) is carried out with AC network model, which is able to handle voltage and voltage stability constraints. However, such a model requires optimization with iterative AC power flow model, which is computationally so demanding that most of the researchers have ignored the vital (N-1) security constraints. Therefore, the objective of this research work is to develop an efficient, two stage optimization strategy for solving this problem. In the first stage, a DC expansion planning problem is solved which provides an initial guess as well as some very good heuristics to reduce the number of power flow solutions for the second stage of AC transmission and reactive expansion planning. A modified artificial bee colony (MABC) algorithm is used to solve the resulting optimization problem. Static AC TNEP results for Garver 6 bus, IEEE 24 bus and IEEE 118 bus test systems have been obtained with the proposed and rigorous approaches and wherever possible, compared with similar results reported in literature to demonstrate the benefits of the proposed method. Also, multi-stage dynamic AC TNEP for the Garver 6 bus system is solved to show the applicability of the methodology to such problems., 14 pages, 3 figures

The blast wave of machine learning and artificial intelligence has also reached the power systems community, and amid the frenzy of methods and black-box tools that have been left in its wake, it is sometimes difficult to perceive a glimmer of Occam’s razor principle. In this letter, we use the unit commitment problem (UCP), an NP- hard mathematical program that is fundamental to power system operations, to show that simplicity must guide any strategy to solve it, in particular those that are based on learning from past UCP instances. To this end, we apply a naive algorithm to produce candidate solutions to the UCP and show, using a variety of realistically sized power systems, that we are able to find optimal or quasi-optimal solutions with remarkable speedups. To the best of our knowledge, this is the first work in the technical literature that quantifies how challenging learning the solution of the UCP actually is for real-size power systems. Our claim is thus that any sophistication of the learning method must be backed up with a statistically significant improvement of the results in this letter This work was supported in part by the Spanish Ministry of Science and Innovation through project PID2020-115460GB-I00, by the Anda-lusian Regional Government through project P20-00153, and by the Research Program for Young Talented Researchers of the University of Málaga under Project B1-2019-11. This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 755705). Finally, the authors thankfully acknowledge the computer re- sources, technical expertise, and assistance provided by the SCBI (Supercomputing and Bioinformatics) center of the University of Málaga. Funding for open access charge: Universidad de Málaga /CBUA

There are tradeoffs between current sharing among distributed resources and DC bus voltage stability when conventional droop control is used in DC microgrids. As current sharing approaches the setpoint, bus voltage deviation increases. Previous studies have suggested using secondary control utilizing linear controllers to overcome drawbacks of droop control. However, linear control design depends on an accurate model of the system. The derivation of such a model is challenging because the noise and disturbances caused by the coupling between sources, loads, and switches in microgrids are under-represented. This under-representation makes linear modeling and control insufficient. Hence, in this paper, we propose a robust adaptive control to adjust droop characteristics to satisfy both current sharing and bus voltage stability. First, the time-varying models of DC microgrids are derived. Second, the improvements for the adaptive control method are presented. Third, the application of the enhanced adaptive method to DC microgrids is presented to satisfy the system objective. Fourth, simulation and experimental results on a microgrid show that the adaptive method precisely shares current between two distributed resources and maintains the nominal bus voltage. Last, the comparative study validates the effectiveness of the proposed method over the conventional method., Comment: 11 pages, 15 figures

In this paper we present a formulation of the unit commitment problem with AC power flow constraints. It is solved by a Benders decomposition in which the unit commitment master problem is formulated as a mixed-integer problem with linearization of the power generation constraints for improved convergence. Semidefinite programming relaxation of the rectangular AC optimal power flow is used in the subproblem, providing somewhat conservative cuts. Numerical case studies, including a 6-bus and the IEEE 118-bus network, are provided to test the effectiveness of our proposal. We show in our numerical experiments that the use of such strategy improves the quality of feasibility and optimality cuts generated by the solution of the convex relaxation of the subproblem, therefore reducing the number of iterations required for algorithm convergence., Accepted for publication in the Electric Power Systems Research journal on November 11, 2020

In the Nordic countries (Sweden, Norway, Finland and Denmark), many interconnectors are formed by long High-Voltage Direct-Current (HVDC) lines. Every year, the operation of such interconnectors costs millions of Euros to Transmission System Operators (TSOs) due to the high amount of losses that are not considered while clearing the market. To counteract this problem, Nordic TSOs (Svenska kraftn\"at - Sweden, Statnett - Norway, Fingrid - Finland, Energinet - Denmark) have proposed to introduce linear HVDC loss factors in the market clearing algorithm. The assessment of such a measure requires a detailed model of the system under investigation. In this paper we develop and introduce a detailed market model of the Nordic countries and we analyze the impact of different loss factor formulations. We show that linear loss factors penalize one HVDC line over the other, and this can jeopardize revenues of merchant HVDC lines. In this regard, we propose piecewise-linear loss factors: a simple to implement but highly effective solution. Moreover, we demonstrate how the introduction of only HVDC loss factors is a partial solution, since it disproportionately increases the AC losses. Our results show that the inclusion of AC loss factors can eliminate this problem., Comment: Submitted to "XXI Power Systems Computation Conference" on October 4, 2019 - Revised on April 19, 2020 - Accepted on May 13, 2020

Computer simulations that demonstrate the valueof novel approaches are crucial to developing more flexibleand robust power systems operations with high penetrations ofrenewable energy at multiple geographic and temporal scales.However, optimization-based simulations that depend on forecastdata often face challenges in evaluating performance, reproducingresults, and testing under realistic simulation scenarios. In thispaper, we develop scientific computing best-practices for thevalidation and reproduction of power systems operational models.We then employ two case studies to demonstrate the proposedvalidation and reproduction framework., Comment: Accepted to 21st Power Systems Computation Conference

This paper presents a convex, multi-period, AC-feasible Optimal Power Flow (OPF) framework that robustly dispatches flexible demand-side resources in unbalanced distribution feeders against uncertainty in very-short timescale solar Photo-Voltaic (PV) forecasts. This is valuable for power systems with significant behind-the-meter solar PV generation as their operation is affected by uncertainty from forecasts of demand and solar PV generation. The aim of this work is then to ensure the feasibility and reliability of distribution system operation under high solar PV penetration. We develop and present a novel, robust OPF formulation that accounts for both the nonlinear power flow constraints and the uncertainty in forecasts. This is achieved by linearizing an optimal trajectory and using first-order methods to systematically tighten voltage bounds. Case studies on a realistic distribution feeder shows the effectiveness of a receding-horizon implementation.

There is an emerging need for efficient solutions to stochastic AC Optimal Power Flow ({AC-}OPF) to ensure optimal and reliable grid operations in the presence of increasing demand and generation uncertainty. This paper presents a highly scalable data-driven algorithm for stochastic AC-OPF that has extremely low sample requirement. The novelty behind the algorithm's performance involves an iterative scenario design approach that merges information regarding constraint violations in the system with data-driven sparse regression. Compared to conventional methods with random scenario sampling, our approach is able to provide feasible operating points for realistic systems with much lower sample requirements. Furthermore, multiple sub-tasks in our approach can be easily paralleled and based on historical data to enhance its performance and application. We demonstrate the computational improvements of our approach through simulations on different test cases in the IEEE PES PGLib-OPF benchmark library., a version of this paper will appear in the proceedings of the 21st Power Systems Computation Conference (PSCC 2020)

We derive the branch ampacity constraint associated to power losses for the convex optimal power flow (OPF) model based on the branch flow formulation. The branch ampacity constraint derivation is motivated by the physical interpretation of the transmission line {\Pi}-model and practical engineering considerations. We rigorously prove and derive: (i) the loop constraint of voltage phase angle, required to make the branch flow model valid for meshed power networks, is a relaxation of the original nonconvex alternating current optimal power flow (o-ACOPF) model; (ii) the necessary conditions to recover a feasible solution of the o-ACOPF model from the optimal solution of the convex second-order cone ACOPF (SOC-ACOPF) model; (iii) the expression of the global optimal solution of the o-ACOPF model providing that the relaxation of the SOC-ACOPF model is tight; (iv) the (parametric) optimal value function of the o-ACOPF or SOC-ACOPF model is monotonic with regarding to the power loads if the objective function is monotonic with regarding to the nodal power generations; (v) tight solutions of the SOC-ACOPF model always exist when the power loads are sufficiently large. Numerical experiments using benchmark power networks to validate our findings and to compare with other convex OPF models, are given and discussed., Comment: Accepted by the journal Electric Power Systems Research, dates: 14/6/2019, 18/9/2019, 07/05/2020, Email: zhao.yuan@epfl.ch