AC OPF Research Articles

Virtual power plant optimal dispatch considering power-to-hydrogen systems

Power-to-Hydrogen (P2H) clean systems have been increasingly adopted for Virtual Power Plant (VPP) to drive system decarbonization. However, current models for the joint operation of VPP and P2H often disregard the full impact on grid operation or hydrogen supply to multiple consumers. This paper contributes with a VPP operating model considering a full Alternating Current Optimal Power Flow (AC OPF) while integrating different paths for the use of green hydrogen, such as supplying hydrogen to a Combined Heat and Power (CHP), industry and local hydrogen consumers. The proposed framework is tested using a 37-bus distribution grid and the results illustrate the benefits that a P2H plant can bring to the VPP in economic, grid operation and environmental terms. An important conclusion is that depending on the prices of the different hydrogen services, the P2H plant can increase the levels of self-sufficiency and security of supply of the VPP, decrease the operating costs, and integrate more renewables.

A risk-aware and second-order cone programming model for wind expansion planning considering correlated uncertainties using the copula method

Integration of large-scale renewable energy sources (RESs) into existing power grids is challenging for power system operators and planners due to the intermittent and uncertain characteristics of RESs and the limitation of transmission capacity. This paper presents a new risk-based framework to obtain the optimal placement and sizing of wind farms in the bulk power system considering the impact of mutual correlation between wind farms’ power productions in different candidate buses. In this regard, a second-order cone programming (SOCP) optimization problem is developed to minimize the total expected social cost including the operation cost of conventional generators, wind curtailment cost, and the investment cost of newly installed wind farms. In the proposed model, the correlated uncertainties of wind power production are incorporated using the copula method. Also, the conditional value-at-risk (CVaR) measure is utilized to manage the risk of the model associated with uncertainties. The proposed planning model is then implemented into the Garver 6-bus and IEEE 118-bus test systems. Given the results, employing SOCP-based AC OPF in the presented planning model results in a precise solution, demonstrating an 87% reduction in running time. Moreover, the numerical results illustrate that by considering the correlation between wind farms’ power production and reducing the risk parameter, there is a significant increase in the installed capacity of wind farms.

Dynamic Relationship Between KKT Saddle Solutions and Optimal Solutions in AC OPF Problems

Many KKT condition-based methods for solving AC OPF models can compute KKT-saddle solutions that are feasible solutions but not local optimal solutions. This paper explores a dynamic relationship between KKT-saddle solutions and optimal solutions in AC OPF problems and presents a “buy one, get two free” theorem by showing that if one KKT saddle is computed, then this KKT saddle is dynamically connected to two local optimal solutions. In addition, it is shown that by following the unstable manifold of the computed KKT saddle, one can easily compute two local optimal solutions. This “buy one, get two free” property occasionally degenerates into a “buy one, get one free” property when the unstable equilibrium point (UEP) associated with the computed KKT saddle does not lie on a quasi-stability boundary. Numerical demonstration of these two properties is conducted on several test systems including the IEEE 118-bus system. The robustness of these two properties under the perturbations of AC OPF models is analytically and numerically shown.

Binding constraints를 고려한 딥러닝 기반 최적조류계산 모델 및 대형계통 적용 사례 연구

Recently, research using machine learning and deep learning is being conducted to quickly solve the AC optimal power flow (AC OPF) problem. The problem with the existing research is that, unlike the actual system, it did not consider the generator start-up status according to demand change by using a test system in which the minimum power generation of the generator is zero. This paper proposes a deep learning-based AC OPF model and post-processing for a simulated large-scale system in Korea. It is a difficult problem to predict in Korea

Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality

We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power-generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs (MINLPs). The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second order cone programs as well as primal and dual cutting planes. For example, objective and no-good cuts help to compute good feasible solutions in which outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems in which discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high-quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP solvers. Our method outperforms all other algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research has been funded by the Federal Ministry of Education and Research of Germany [Grant 05M18WEB]. This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. The authors thank the Deutsche Forschungsgemeinschaft for support within projects A05, B06, B07, and B10 of the Sonderforschungsbereich/Transregio 154 “Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks.” This work has been supported by the Federal Ministry for Economic Affairs and Energy, Germany [Grant 03El1036A]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1270 .

A Data-driven AC Optimal Power Flow Using Extreme Learning Machine

With the increasing integration of renewable energy (RE), AC Optimal Power Flow (AC OPF) becomes a necessary foundation for electricity system which has a high level of renewable energy generation. However, most current studies of AC OPF are not applicable due to the requirements of high computation speed in practical applications. To this end, we propose a data-driven method using Extreme Learning Machine (ELM) for getting the AC OPF optimal solution in a faster way. This approach can map the relationship between the optimal operation results and variations of REs and loads, avoiding the time-consuming solving process of AC OPF. Moreover, an ELM network structure suitable for AC OPF is designed to significantly improve the computational speed of ACOPF with acceptable accuracy. This method was applied to the RTS-79 system for improving the calculation efficiency of the AC OPF.

A stochastic distributed control approach for load restoration of networked microgrids with mobile energy storage systems

In response to extreme events, substantial research efforts have focused on developing load restoration strategies for networked microgrids (MGs). However, existing distributed control approaches only consider independent time periods, which cannot capture the time-coupled flexibility of storage units. Additionally, mobile energy storage systems (MESSs) have been deployed for resilience enhancement due to their advantages in mobility and flexibility. However, existing research on networked MGs utilizes simplistic energy management approaches for MG modeling without detailed network structures, which cannot capture the mobility of MESSs. To address these issues, this paper proposes a three-stage stochastic distributed control approach based on rolling optimization to enhance the resilience of networked MGs with MESSs. Specifically, a stochastic linearized OPF is formulated in the first stage to capture the flexibility of MESSs and uncertainties, while a consensus algorithm is utilized in the second stage to calculate the power exchange among MGs. Finally, a detailed non-linear AC OPF algorithm is employed in the third stage to capture technical constraints relating to stability properties towards accurate optimization results. Case studies considering load distinction, multiple line outages, and limited generation resources are developed to demonstrate the effectiveness of the proposed distributed control approach in accurate and timely decision-making.

A data-driven multi-stage stochastic robust optimization model for dynamic optimal power flow problem

The uncertainty caused by the distributed generations(DG) with inconspicuous patterns has been an essential subject in the optimization scheduling for the distribution network. We propose a novel data-driven approach to deal with the dynamic optimal power flow(DOPF) problem which contains uncertain variables with their unknown probability distribution. The data-driven model is made to learn the joint probability distribution of the uncertain variables and use robust optimization(RO) to solve the multi-stage stochastic linear DOPF by averaging the worst case from each uncertainty set. In contrast to the motivation for traditional RO to find solutions that perform well on the worst-case realization, our proposed approach adds robustness to the historical data as a tool to avoid overfitting as the number of data points tends to infinity. The application verification for the AC OPF problem is presented for the IEEE-33 system. The simulation verifies the feasibility and robustness of the proposed approach and its results are compared with those of other data-driven stochastic optimization methods. We prove that the proposed approach can effectively solve the overvoltage problem caused by the high permeability of photovoltaic generation and achieve a better out-of-sample performance guarantee, and also has obvious economic advantages over other data-driven methods.

DNN-based policies for stochastic AC OPF

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.

Approximations for generalized unsplittable flow on paths with application to power systems optimization

The Unsplittable Flow on a Path (UFP) problem has garnered considerable attention as a challenging combinatorial optimization problem with notable practical implications. Steered by its pivotal applications in power engineering, the present work formulates a novel generalization of UFP, wherein demands and capacities in the input instance are monotone step functions over the set of edges. As an initial step towards tackling this generalization, we draw on and extend ideas from prior research to devise a quasi-polynomial time approximation scheme (QPTAS) under the premise that the demands and capacities lie in a quasi-polynomial range. Second, retaining the same assumption, an efficient logarithmic approximation is introduced for the single-source variant of the problem. Finally, we round up the contributions by designing a (kind of) black-box reduction that, under some mild conditions, allows to translate LP-based approximation algorithms for the studied problem into their counterparts for the Alternating Current Optimal Power Flow (AC OPF) problem -- a fundamental workflow in operation and control of power systems.

Emulating AC OPF Solvers With Neural Networks

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 letter, 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.

Optimal Distributed Energy Resource Coordination: A Decomposition Method Based on Distribution Locational Marginal Costs

In this paper, we consider the day-ahead operational planning problem of a radial distribution network hosting Distributed Energy Resources (DERs) including rooftop solar and storage-like loads, such as electric vehicles. We present a novel decomposition method that is based on a centralized AC Optimal Power Flow (AC OPF) problem interacting iteratively with self-dispatching DER problems adapting to real and reactive power Distribution Locational Marginal Costs. We illustrate the applicability and tractability of the proposed method on an actual distribution feeder, while modeling the full complexity of spatiotemporal DER capabilities and preferences, and accounting for instances of non-exact AC OPF convex relaxations. We show that the proposed method achieves optimal Grid-DER coordination, by successively improving feasible AC OPF solutions, and discovers spatiotemporally varying marginal costs in distribution networks that are key to optimal DER scheduling by modeling losses, ampacity and voltage congestion, and, most importantly, dynamic asset degradation.

Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap

We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality. We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems.Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm.

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions – Part I: Model Description and the Algorithm

The existing research on market price-affecting agents, i.e. price makers, neglects or simplifies the nature of AC power flows in the power system as it predominantly relies on DC power flows. This paper proposes a novel bilevel formulation based on the smoothing technique, where any price-affecting strategic player can be modelled in the upper level, while the market clearing problem in the lower level uses convex quadratic transmission AC optimal power flow (AC OPF), with the goal of achieving accuracy close to the one of the exact nonlinear formulations. Achieving convexity in the lower level is the foundation for bilevel modeling since traditional single-level reduction techniques do not hold for nonconvex models. The bilevel market participation problem with the AC OPF formulation in the lower level is transformed into a single-level problem and solved using multiple techniques such as the primal-dual counterpart, the strong duality theorem, the McCormick envelopes, the complementary slackness, the penalty factor, the interaction discretization as well as the proposed smoothing techniques. Due to an extensive amount of information and descriptions, the overall work is presented as a two-part paper. This first part provides a literature overview, positions the work and presents the model and the solution algorithm, while the solution techniques and case studies are provided in the accompanying paper.

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions – Part II: Solution Techniques and Case Study

This is a second part of the research on AC optimal power flow being used in the lower level of the bilevel strategic bidding or investment models. As an example of a suitable upper-level problem, we observe a strategic bidding of energy storage and propose a novel formulation based on the smoothing technique. After presenting the idea and scope of our work, as well as the model itself and the solution algorithm in the companion paper (Part I), this paper presents a number of existing solution techniques and the proposed one based on smoothing the complementary conditions. The superiority of the proposed algorithm and smoothing techniques is demonstrated in terms of accuracy and computational tractability over multiple transmission networks of different sizes and different OPF models. The results indicate that the proposed approach outperforms all other options in both metrics by a significant margin. This is especially noticeable in the metric of accuracy where out of total 422 optimizations over 9 meshed networks the greatest AC OPF error is 0.023% that is further reduced to 3.3e-4% in the second iteration of our algorithm.

A Novel Tractable Methodology to Stochastic Multi-Period AC OPF in Active Distribution Systems Using Sequential Linearization Algorithm

The ongoing transition of passive distribution networks to active distribution systems (ADSs) calls for the development of sophisticated and tractable tools to address the needs and challenges of future ADSs. In this regard, this work proposes a novel tractable methodology for flexibility procurement via stochastic multi-period AC optimal power flow (S-MP-OPF). The first novelty of the proposed methodology is the formulation of a new mixed-integer linear programming (MILP) model, which develops novel linear approximations for all nonlinear constraints (i.e., active and reactive power flows and branch currents) that ensure tractability. The proposed linearization approach relies on the second-order Taylor series expansion of trigonometric terms, does not hinge on the flat-voltage, near-voltage and small angle assumptions, and formulates the linear approximations using square of voltage magnitude and voltage angle difference variables. The second novelty is a sophisticated heuristic sequential linearization algorithm (SLA) which incorporates and improves the accuracy of the MILP model in an iterative manner. The bench-marking of proposed SLA is done on a 34-bus ADS for a comprehensive set of flexible options, whereas its versatility and scalability is demonstrated on real-world 31-bus weakly-meshed and 191-bus radial ADSs. Extensive numerical analyses show that the proposed SLA leads to a highly accurate and feasible solution with small optimality gap within few iterations, under normal and stressed operating conditions, outperforms alternative methods in terms of solution accuracy and computational efficiency, and is scalable to large S-MP-OPF problems.

An Exact Sequential Linear Programming Algorithm for the Optimal Power Flow Problem

Despite major advancements in nonlinear programming (NLP) and convex relaxations, most system operators around the world still predominantly use some form of linear programming (LP) approximation of the AC power flow equations. This is largely due to LP technology's superior reliability and computational efficiency, especially in real-time market applications, security-constrained applications, and extensions involving integer variables, in addition to its ability to readily generate locational marginal prices (LMP) for market applications. In the aim of leveraging the advantages of LP while retaining the accuracy of NLP interior-point methods (IPMs), this paper proposes a sequential linear programming (SLP) approach consisting of a sequence of carefully constructed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">supporting hyperplanes and halfspaces</i> . The algorithm is numerically demonstrated to converge on 138 test cases with up the 3375 buses to feasible high-quality solutions (i) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">without</i> AC feasibility restoration (i.e., using LP solvers exclusively), (ii) in computation times generally within the same order of magnitude as those from a state-of-the-art NLP solver, and (iii) with robustness against the choice of starting point. In particular, the (relative) optimality gaps and the mean constraint violations are on average around <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-3}$</tex-math></inline-formula> % and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-7}$</tex-math></inline-formula> , respectively, under a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single</i> parameter setting for all the 138 test cases. To the best of our knowledge, the proposed SLP approach is the first to use LP exclusively to reach <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">feasible</i> and high-quality solutions to the nonconvex AC OPF in a reliable way, which paves the way for system and market operators to keep using their LP solvers but now with the ability to accurately capture transmission losses, price reactive power (Q-LMP), and obtain more accurate LMP.

Microgrid Operation and Control: Challenges and expected functionalities

This article considers several functionalities expected from the emerging microgrids and systems of microgrids. These performance objectives are then related to several modeling- and controlrelated challenges and open R&D questions that must be studied. The challenges are illustrated on Sheriff and Banshee microgrids, which are IEEE standards for testing microgrid control. I suggest that an extended ac optimal power flow (AC OPF) can be used to increase ranges of intermittent resources and responsive demand that can be used in a reliable way and without requiring excessive load shedding. The emphasis is on optimizing voltage-controlled equipment available at the inverter-controlled diverse distributed energy resources (DERs), including electric vehicle (EV) control of small residential end users, and the grid itself. The second question concerns enabling stability of microgrids over broad ranges of system inputs, and I briefly present a novel multilayered interactive modeling in transformed energy state space for stable control design. The article closes by revealing the importance of operating and control paradigm change from today's hierarchical control to interactive dynamic multilayered protocols, which also would provide financial incentives for reliance on control.

Day-ahead Coordinated Operation of a Wind-Storage System Considering Wind Forecast Uncertainty

This paper proposes an optimal operation for coordinated Battery Energy Storage (BES) and wind generation in a day-ahead market under wind uncertainty. A comprehensive AC Optimal Power Flow (AC OPF) model was established to incorporate wind and storage into a power system. To take into account wind forecast uncertainty, preprocessing technique, time series model, and fast forward selection method were applied for scenario generation and reduction processes. Tests were performed on a modified IEEE 14-bus system and the results show that the use of BESs is an alternative to guarantee a more efficient and flexible operation of wind power plants.

A Three-Level Planning Model for Optimal Sizing of Networked Microgrids Considering a Trade-Off Between Resilience and Cost

Extreme events can cause severe power system damage. Resilience-driven operation of networked microgrids (MGs) has been heavily studied in literature. There is, though, little research considering the influence of resilience on decision making for planning. In this paper, a three-level model is suggested to solve the optimal sizing problem of networked MGs considering both resilience and cost. In the first level, a meta-heuristic technique based on an adaptive genetic algorithm (AGA) is utilized to tackle the normal sizing problem, while a time-coupled AC OPF is utilized to capture stability properties for accurate decision-making. The second and third levels are combined as a defender-attacker-defender model. In the former, the suggested AGA is utilized to generate attacking plans capturing load profile uncertainty and contingencies for load shedding maximization, while a multi-objective optimization problem is suggested for the latter to obtain a trade-off between cost and resilience. Simulations considering meshed networks and load distinction into critical and non-critical are developed to demonstrate algorithm effectiveness on capturing resilience at the planning stage and optimally sizing multiple parameters. The results indicate that higher resilience levels lead to higher investment cost, while sizing networked MGs leads to decreased investment in comparison with standalone MGs sizing.