AC Optimal Power Flow Problem Research Articles

Discrete optimal power flow with prohibited zones, multiple-fuel options, and practical operational rules for control devices

Although the optimal power flow (OPF) problem has been extensively studied, solving realistic OPF models that accurately represent the operating behavior of power system components remains challenging. This paper proposes a novel model for the AC OPF problem, aiming to minimize the fuel costs of thermal units while taking into account valve-point loading effects (VPLE), prohibited operation zones (POZ), multiple fuel options (MFO), and operational rules associated with the discrete tap ratios of on-load tap changer (OLTC) transformers and with the discrete shunt susceptances of capacitor/reactor banks. These rules are represented using complementarity constraints. We propose a solution approach that integrates several strategies to address the non-smooth features of the objective function related to VPLE, the disjoint constraints and functions tied to POZ and MFO, the discrete characteristics of the reactive control variables, and the complementarity constraints governing operational rules linked to voltage control devices such as OLTC transformers and capacitor/reactor banks. The resulting optimization problem is designed to be compatible with commercial solver packages. Numerical tests on the IEEE 30, 118, and 300-bus systems aim to examine the cumulative impact of these operational factors on the optimal solution. The solution strategy proposed has demonstrated its effectiveness in solving the proposed OPF problem within reasonable computation times.

Data-driven stochastic AC-OPF using Gaussian process regression

At present, electricity generation is responsible for more than a quarter of the greenhouse gas emissions in the US. Integrating significant amounts of renewable energy sources into a power grid is probably the most accessible way to reduce carbon emissions from power grids and slow down climate change. Unfortunately, the most accessible renewable power sources, such as wind and solar, are highly intermittent and thus bring a lot of uncertainty to power grid operation and challenge existing optimization and control policies. The chance-constrained (CC) alternating current optimal power flow (AC-OPF) framework finds the minimum cost of generation dispatch, maintaining the power grid operation within security limits with a prescribed probability. Unfortunately, the AC-OPF problem’s chance-constrained extension is non-convex, computationally challenging, and requires knowledge of system parameters and also needs additional assumptions to be made about the behavior of renewable generation probability distribution. Known linear and convex approximations to the above problems, though tractable, are too conservative for operational practice and do not consider uncertainty in system parameters. This paper presents an alternative data-driven approach for solving the stochastic AC-OPF problem, based on Gaussian process regression (GPR) to close this gap. The Gaussian process (GP) approach learns a simple yet non-convex data-driven approximation to the AC power flow equations that can incorporate uncertain inputs. The latter is then used to determine the solution of CC-OPF efficiently, by accounting for both input and parameter uncertainty. The practical efficiency of the proposed approach using different approximations for GP-uncertainty propagation is validated and illustrated using a number of standard IEEE test cases.

A Nested Decomposition Method for the AC Optimal Power Flow of Hierarchical Electrical Power Grids

The interconnected electrical power grids are operated by multilevel control centers with hierarchical architecture. Since large-scale renewables are integrated in different voltage levels, the traditional isolated operation of interconnected power grids is uneconomical and meets operational risk due to lack of coordination. Therefore, the coordination between control centers in different levels and areas are indispensable. However, existing distributed approaches may either encounter numerical problems or converge slowly when the nonlinear AC optimal power flow (ACOPF) models are applied. Moreover, the total iteration number presents an exponential increase with the levels of hierarchical grids. In order to improve computational efficiency, this paper proposes a nested decomposition method for the coordinated operation of multilevel ACOPF problem, which has superlinear convergence and can achieve the optimal solution (KKT point). During each iteration, a projection function, which embodies the optimal objective value of a lower level power grid projected onto its boundary variable space, is computed with second-order exactness. Thus, the proposed method can be applied to nonlinear continuous optimizations with high efficiency. The paper also provides a rigorous proof for its convergence under the condition that the lower level optimization problems are convex with continuously differentiable objectives and constraints. Numerical tests are conducted with three trilevel power grid of different scales, which verify that the computational efficiency and scalability of the proposed algorithm are superior to those of existing methods.

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

Low Carbon Electric Vehicle Charging Coordination in Coupled Transportation and Power Networks

As a part of the global decarbonization agenda, the electrification of the transport sector involving the large-scale integration of electric vehicles (EV) constitutes one of the key initiatives. However, the upstream generation carbon emissions associated with EV charging demand are still accountable which should not be overlooked. In this context, efficient coordination of EV flows and their charging demand in the coupled transportation and power networks promise significant decarbonization potential. Nevertheless, such potential can only be realized under adequate incentive mechanisms. In this article, a novel low carbon EV charging coordination approach in coupled transportation and power network is proposed and formulated as a bi-level optimization. In the upper level, an AC optimal power flow problem is formulated and solved to determine the optimal operation for power system. Then the carbon emission flow tracing is performed to compute rational locational-differentiated price signals. Given price-based incentives, the lower level employs traffic user equilibrium to describe the distribution of EV path flow and charging demands taking into account the uncertain traffic condition, and the aggregate charging demands are fed back to the upper level. The bi-level problem is solved iteratively through a modified particle swarm algorithm with enhanced convergence properties. Case studies demonstrate the effectiveness of the proposed coordination method in effectively mitigating the global carbon emission of the coupled networks.

DeepOPF: A Feasibility-Optimized Deep Neural Network Approach for AC Optimal Power Flow Problems

To cope with increasing uncertainty from renewable generation and flexible load, grid operators need to solve alternative current optimal power flow (AC-OPF) problems more frequently for efficient and reliable operation. In this article, we develop a deep neural network (DNN) approach, called DeepOPF, for solving AC-OPF problems in a fraction of the time used by conventional iterative solvers. A key difficulty for applying machine learning techniques for solving AC-OPF problems lies in ensuring that the obtained solutions respect the equality and inequality physical and operational constraints. Generalized a prediction-and-reconstruction procedure in our previous studies, DeepOPF first trains a DNN model to predict a set of independent operating variables and then directly compute the remaining ones by solving the power flow equations. Such an approach not only preserves the power-flow balance equality constraints but also reduces the number of variables to be predicted by the DNN, cutting down the number of neurons and training data needed. DeepOPF then employs a penalty approach with a zero-order gradient estimation technique in the training process toward guaranteeing the inequality constraints. We also drive a condition for tuning the DNN size according to the desired approximation accuracy, which measures its generalization capability. It provides theoretical justification for using DNN to solve AC-OPF problems. Simulation results for IEEE 30/118/300-bus and a synthetic 2000-bus test cases demonstrate the effectiveness of the penalty approach. They also show that DeepOPF speeds up the computing time by up to two orders of magnitude as compared to a state-of-the-art iterative solver, at the expense of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&lt; $</tex-math></inline-formula> 0.2% cost difference.

An Improved Dynamic Programming-Based Decentralized Algorithm for AC-OPF of Radially Connected Multi-Level Networks

In this paper, an improved dynamic programming (IDP)-based decentralized algorithm is proposed for the AC-optimal power flow (AC-OPF) of radially connected transmission-distribution-microgrid networks (RCTDMNs). Most of the existing decentralized algorithms need to iterate among networks, which may lead to additional computational and information exchange burden. Different from these algorithms, we decompose the original AC-OPF problem into small-scale subproblems based on dynamic programming (DP) theory, and then solve them in a decentralized manner without iteration among networks. In order to improve the computational performance of DP, its value functions are directly obtained by solving Galerkin projection equations (GPEs) instead of traversing the whole state space. Moreover, a parallel technique is proposed to further accelerate the proposed IDP algorithm when dealing with the AC-OPF problem of RCTDMNs. Case studies on various test systems demonstrate the effectiveness of the proposed method.

ConvOPF-DOP: A Data-Driven Method for Solving AC-OPF Based on CNN Considering Different Operation Patterns

AC optimal power flow (AC-OPF) is a significant problem in the economic operation of power systems. Traditional AC-OPF calculation methods only consider a specific operation pattern, which has limitations. We propose ConvOPF-DOP, a novel data-driven approach based on Convolutional Neural Network (CNN), to solve the AC-OPF problem in different operation patterns. This study uses the cluster analysis method to obtain the network structure and operating condition information from historical data. In this way, even when the network structure is unknown, the corresponding network topology label can also be identified from the collected data. Then we use the network topology label and load data as input data, optimize hyperparameters of the CNN model by the Bayesian optimization method, and train the CNN model to learn the relationship between the input data and the voltage output. Finally, the Power Flow (PF) calculation method is used to get the remaining optimal solutions. The effectiveness and superiority of ConvOPF-DOP are verified through 30-bus systems in four different operation patterns. ConvOPF-DOP brings 350× speed increase compared with the traditional method while ensuring high accuracy of generated optimal solutions.

AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization

We address the issue of computing a global minimizer of the AC Optimal Power Flow problem. We introduce valid inequalities to strengthen the Semidefinite Programming relaxation, yielding a novel Conic Programming relaxation. Leveraging these Conic Programming constraints, we dynamically generate Mixed-Integer Linear Programming (MILP) relaxations, whose solutions asymptotically converge to global minimizers of the AC Optimal Power Flow problem. We apply this iterative MILP scheme on the IEEE PES PGLib [2] benchmark and compare the results with two recent Global Optimization approaches.

Data-driven alternating current optimal power flow: A Lagrange multiplier based approach

This paper proposes a data-driven Alternating Current Optimal Power Flow (AC-OPF) method assisted by Lagrange multipliers. Stacked Extreme Learning Machine (SELM) is introduced for AC-OPF learning to avoid the time-consuming training and hyperparameter adjustment process of deep neural networks. Instead of incorporating the prior physical information into the neural networks algorithm, we developed a new neural network structure for the SELM learning based on the Lagrange multipliers of the AC-OPF problem. Case studies of several IEEE benchmark systems demonstrate that the AC-OPF learning performance is improved by introducing additional Lagrange multipliers information, while the proposed method outperforms other alternatives with almost 99% learning accuracy and fast computation.

Physics-Informed Neural Networks for AC Optimal Power Flow

This paper introduces, for the first time to our knowledge, physics-informed neural networks to accurately estimate the AC-Optimal Power Flow (AC-OPF) result and delivers rigorous guarantees about their performance. Power system operators, along with several other actors, are increasingly using Optimal Power Flow (OPF) algorithms for a wide number of applications, including planning and real-time operations. However, in its original form, the AC OPF problem is often challenging to solve as it is non-linear and non-convex. Besides the large number of approximations and relaxations, recent efforts have also been focusing on Machine Learning approaches, especially neural networks. So far, however, these approaches have only partially considered the wide number of physical models available during training. And, more importantly, they have offered no guarantees about potential constraint violations of their output. Our approach (i) introduces a framework to incorporate AC power flow equations inside neural network training and (ii) integrates methods that rigorously determine and reduce the worst-case constraint violations across the entire input domain, while maintaining the optimality of the prediction. We demonstrate how physics-informed neural networks achieve higher accuracy and lower constraint violations than standard neural networks, and show how we can further reduce the worst-case violations for all neural networks.

Cross-layer design for real-time grid operation: Estimation, optimization and power flow

In this paper, we propose a combined Online Feedback Optimization (OFO) and dynamic estimation approach for a real-time power grid operation under time-varying conditions. A dynamic estimation uses grid measurements to generate the information required by an OFO controller, that incrementally steers the controllable power injections set-points towards the solutions of a time-varying AC Optimal Power Flow (AC-OPF) problem. More concretely, we propose a quadratic programming-based OFO that guarantees satisfying the grid operational constraints, like admissible voltage limits. Within the estimation, we design an online power flow solver that efficiently computes power flow approximations in real time. Finally, we certify the stability and convergence of this combined approach under time-varying conditions, and we validate its effectiveness on a simulation with a test feeder and high resolution consumption data.

Learning to accelerate globally optimal solutions to the AC Optimal Power Flow problem

We propose machine learning-based (ML) methods to accelerate convergence to global solutions for the AC Optimal Power Flow (AC-OPF) problem. In particular, for the non-convex AC-OPF problem, optimality-based bound tightening (OBBT) has been observed to be a very effective approach for tightening the variable domains, thus leading to tight convex relaxations that are nearly global optimum solutions. However, by construction, OBBT is computationally expensive even on medium-scaled power networks. To address this issue, we propose a novel ML-based policy to replace the exhaustive algorithm of OBBT by choosing a subset of variables whose tightening of bounds can still contribute to the best improvement of the convex relaxation of the AC-OPF problem. To this end, we leverage historical data of load profiles for a test system to learn a map between the system loading and subset selection of variables which will need to participate in the OBBT algorithm, thus enabling us to find near-global optimal solutions at faster run-times. Finally, we present detailed numerical studies on a few medium-sized benchmark instances, on which we observe up to 6.3× speed-up in OBBT run times.

Optimal power flow under stochastic N-1 disruptions in electric transmission grids

We consider a multi-period optimal power flow (OPF) problem in transmission systems with a generic network topology subject to stochastic N−1 contingencies (disruptions). We propose a multi-stage stochastic convex programming formulation for such a disruption problem and solve it using an efficient decomposition algorithm. We compare the disruption optimization formulation with the conventional N−1 security-constrained OPF formulation and show that our proposed model can utilize the generation resource more efficiently in a realistic contingency setting. We also present computational results comparing multiple convex relaxation schemes of the non-convex alternating current OPF (ACOPF) problem in transmission networks under stochastic disruptions. Through this, we provide insights on the performance of the policy trained by convex relaxations of an ACOPF problem.

Two-stage homotopy method to incorporate discrete control variables into AC-OPF

Alternating-Current Optimal Power Flow (AC-OPF) is an optimization problem critical for planning and operating the power grid. The problem is traditionally formulated using only continuous variables. Typically, control devices with discrete-valued settings, which provide valuable flexibility to the network and improve resilience, are omitted from AC-OPF formulations due to the difficulty of integrality constraints. We propose a two-stage homotopy algorithm to solve the AC-OPF problem with discrete-valued control settings. This method does not rely on prior knowledge of control settings or other initial conditions. The first stage relaxes the discrete settings to continuous variables and solves the optimization using a robust homotopy technique. Once the solution has been obtained using relaxed models, the second homotopy problem gradually transforms the relaxed settings to their nearest feasible discrete values. We test the proposed algorithm on several large networks with switched shunts and adjustable transformers and show it can outperform a similar state-of-the-art solver.

Asymptotically tight conic approximations for chance-constrained AC optimal power flow

The increasing penetration of renewable energy in power systems calls for secure and reliable system operations under significant uncertainty. To that end, the chance-constrained AC optimal power flow (CC-ACOPF) problem has been proposed. Most research in the literature of CC-ACOPF focuses on one-sided chance constraints; however, two-sided chance constraints (TCCs), albeit more complex, provide more accurate formulations as both upper and lower bounds of the chance constraints are enforced simultaneously. In this paper, we introduce a fully two-sided CC-ACOPF problem (TCC-ACOPF), in which the active/reactive generation, voltage, and power flow all remain within their upper/lower bounds simultaneously with a predefined probability. Instead of applying Bonferroni approximation or scenario-based approaches, we present an efficient second-order cone programming (SOCP) approximation of the TCCs under Gaussian Mixture (GM) distribution via a piecewise linear (PWL) approximation. Compared to the conventional normality assumption for forecast errors, the GM distribution adds an extra level of accuracy representing the uncertainties. Moreover, we show that our SOCP formulation has adjustable rates of accuracy and its optimal value enjoys asymptotic convergence properties. Furthermore, an algorithm is proposed to speed up the solution procedure by optimally selecting the PWL segments. Finally, we demonstrate the effectiveness of our proposed approaches with both real historical data and synthetic data on the IEEE 30-bus and 118-bus systems. We show that our formulations provide significantly more robust solutions (about 60% reduction in constraint violation) compared to other state-of-art ACOPF formulations.

Multi-objective ACOPF using distributed gradient dynamics

This paper proposes a distributed gradient dynamics approach for solving Multi-Objective AC Optimal Power Flow (MO-ACOPF) problem. The fuel cost, environmental pollution, and network loss are considered as objective functions. The non-convex MO-ACOPF problem is reformulated as a gradient dynamical system. We derive condition under which the equilibrium point of the proposed gradient dynamical system is the saddle point of the MO-ACOPF Lagrangian function. We propose a distributed solution algorithm for derived MO-ACOPF gradient dynamical system. In the distributed solution algorithm, each bus in the power network is assumed as a local agent, which only uses information from its neighbor agents to compute optimal operating points. We prove convergence of our distributed algorithm to the equilibrium point of the MO-ACOPF gradient dynamical system (and accordingly to the primal and dual optimum of original MO-ACOPF). Accordingly, our proposed approach has the following main advantages: (a) it is a distributed and decentralized algorithm and accordingly it needs less computational time as compared to the centralized model, (b) it has the proof of convergence to the global optimal solution, and (c) it is suitable for varying loads in power networks. The simulation results based on standard IEEE 14-, 30- ,39-, 57-, 118- and 300-bus test systems in different scenarios are carefully studied. The proposed distributed gradient dynamics provides the best results compared to the relevant literature as illustrated by simulation results.

Security Constrained Reactive Power Scheduling Considering N-1 Contingency of Transmission Lines

This paper presents a methodology for reactive power scheduling (RPS) of power system in the form of AC optimal power flow (AC-OPF) problem. The objective function is minimization of system total active power losses. The OPF optimally determines reactive power output of generating units and synchronous condensers, tap-changers ratio, shunt capacitor banks and reactors. The effect of tap-changer is modeled in the active and reactive power flow of transformer. The proposed method grantees secure operation of system in normal operating condition and also in contingency of transmission line outage. The validity of proposed method is studied based on IEEE RTS 24-bus. Results show the capability of suggested AC-OPF for RPS of system in base case as well as contingency of single line outage.

Strategic bidding in electricity markets with convexified AC market-clearing process

This paper presents a framework to solve the strategic bidding problem of participants in an electricity market cleared by employing the full AC Optimal Power Flow (ACOPF) problem formulation. Traditionally, the Independent System Operators (ISOs) leveraged DC Optimal Power Flow (DCOPF) problem formulation to settle the electricity market. The main quest of this work is to find what would be the challenges and opportunities if ISOs leveraged the full ACOPF as the Market-Clearing Problem (MCP)? This paper presents tractable mathematical programming with equilibrium constraints for the convexified AC market-clearing problem. Market participants maximize their profit via strategic bidding while considering the reactive power dispatch of generation units. The equilibrium constraints are procured by presenting the dual form of the relaxed ACOPF problem. The strategic bidding problem with an ACOPF-based MCP improves the exactness of the Location Marginal Prices (LMPs) and profit of market participants compared to the one with DCOPF. It is shown that the strategic bidding problem with DCOFP-based MCP is unable to model the limitation of reactive power support. The presented results display cases where the proposed strategic bidding method renders 52.3% more profit for the Generation Company (GENCO) than the DCOPF-based MCP model. The proposed strategic bidding framework also addresses the challenges of coupling real and reactive power dispatch of generation constraints, ramping constraints, demand response implications with curtailable and time shiftable loads, and AC line flow constraints. Therefore, the presented method will help market participants leverage the more accurate ACOPF model in the strategic bidding problem.

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.