DC Optimal Power Flow Problem Research Articles

Tight and Compact Sample Average Approximation for Joint Chance-Constrained Problems with Applications to Optimal Power Flow

In this paper, we tackle the resolution of chance-constrained problems reformulated via sample average approximation. The resulting data-driven deterministic reformulation takes the form of a large-scale mixed-integer program (MIP) cursed with Big-Ms. We introduce an exact resolution method for the MIP that combines the addition of a set of valid inequalities to tighten the linear relaxation bound with coefficient strengthening and constraint screening algorithms to improve its Big-Ms and considerably reduce its size. The proposed valid inequalities are based on the notion of k-envelopes and can be computed off-line using polynomial-time algorithms and added to the MIP program all at once. Furthermore, they are equally useful to boost the strengthening of the Big-Ms and the screening rate of superfluous constraints. We apply our procedures to a probabilistically constrained version of the DC optimal power flow problem with uncertain demand. The chance constraint requires that the probability of violating any of the power system’s constraints be lower than some positive threshold. In a series of numerical experiments that involve five power systems of different size, we show the efficiency of the proposed methodology and compare it with some of the best performing convex inner approximations currently available in the literature. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported in part by the European Research Council under the EU Horizon 2020 research and innovation program [Grant 755705], in part by the Spanish Ministry of Science and Innovation [Grant AEI/10.13039/501100011033] through project PID2020-115460GB-I00, and in part by the Junta de Andalucía and the European Regional Development Fund through the research project P20_00153. Á. Porras is also financially supported by the Spanish Ministry of Science, Innovation and Universities through the University Teacher Training Program with fellowship number FPU19/03053. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0302 .

Assessing the impacts of nonideal communications on distributed optimal power flow algorithms

Power system operators are increasingly looking toward distributed optimization to address various challenges facing electric power systems. To assess their capabilities in environments with nonideal communications, this paper investigates the impacts of data quality on the performance of distributed optimization algorithms. Specifically, this paper compares the performance of the Alternating Direction Method of Multipliers (ADMM), Analytical Target Cascading (ATC), and Auxiliary Problem Principle (APP) algorithms in the context of DC Optimal Power Flow (DC OPF) problems. Using several test systems, this paper characterizes the performance of these algorithms in terms of their convergence rates and solution quality under three data quality nonidealities: (1) additive Gaussian noise, (2) bad data (large error), and (3) intermittent communication failure.

A Convex Neural Network Solver for DCOPF With Generalization Guarantees

The dc optimal power flow (DCOPF) problem is a fundamental problem in power systems operations and planning. With high penetration of uncertain renewable resources in power systems, DCOPF needs to be solved repeatedly for a large amount of scenarios, which can be computationally challenging. As an alternative to iterative solvers, neural networks are often trained and used to solve DCOPF. These approaches can offer orders of magnitude reduction in computational time, but they cannot guarantee generalization, and small training error does not imply small testing errors. In this work, we propose a novel algorithm for solving DCOPF that guarantees the generalization performance. First, by utilizing the convexity of the DCOPF problem, we train an input convex neural network. Second, we construct the training loss based on Karush–Kuhn–Tucker optimality conditions. By combining these two techniques, the trained model has provable generalization properties, where small training error implies small testing errors. In experiments, our algorithm significantly outperforms other machine learning methods.

Feasible region-based heuristics for optimal transmission switching

In this paper, we develop a optimal transmission switching (OTS) heuristic based on DC optimal power flow (OPF) and assess the efficacy of the approach when implemented within AC OPF. Traditional formulations of the OTS problem can result in hundreds or thousands of binary variables for large networks, making the OTS problem challenging to solve on fast timescales even for relatively small networks. Here, we identify which constraints and therefore which variables are constraining the DC OPF feasible region, and rank them based on their impact on the cost function. We develop a heuristic algorithm which iteratively removes these constraints and solves a series of standard DC OPF problems. The heuristic is tested on a variety of PGlib networks and the results show that the algorithm can provide substantial cost decreases without having to solve any mixed integer programs. We provide additional insights about the OTS problem, including identifying scenarios outside congestion where OTS can prove useful. Lastly, the performance of the DC-based heuristic is shown when the line switching decisions are implemented within AC OPF.

A Scalable Lower Bound for the Worst-Case Relay Attack Problem on the Transmission Grid

We consider a bilevel attacker–defender problem to find the worst-case attack on the relays that control transmission grid components. The attacker infiltrates some number of relays and renders all of the components connected to them inoperable with the goal of maximizing load shed. The defender responds by minimizing the resulting load shed, redispatching using a DC optimal power flow (DCOPF) problem on the remaining network. Though worst-case interdiction problems on the transmission grid have been studied for years, there remains a need for exact and scalable methods. Methods based on using duality on the inner problem rely on the bounds of the dual variables of the defender problem in order to reformulate the bilevel problem as a mixed integer linear problem. Valid dual bounds tend to be large, resulting in weak linear programming relaxations and, hence, making the problem more difficult to solve at scale. Often smaller heuristic bounds are used, resulting in a lower bound. In this work, we also consider a lower bound, but instead of bounding the dual variables, we drop the constraints corresponding to Ohm’s law, relaxing DCOPF to capacitated network flow. We present theoretical results showing that, for uncongested networks, approximating DCOPF with network flow yields the same set of injections and, thus, the same load shed, which suggests that this restriction likely gives a high-quality lower bound in the uncongested case. Furthermore, we show that, in the network flow relaxation of the defender problem, the duals are bounded by one, so we can solve our restriction exactly. Finally, because the big-M values in the linearization are equal to one and network flow has a well-known structure, we see empirically that this formulation scales well computationally with increased network size. Through empirical experiments on 16 networks with up to 6,468 buses, we find that this bound is almost always as tight as we can get from guessing the dual bounds even for congested networks in which the theoretical results do not hold. In addition, calculating the bound is approximately 150 times faster than achieving the same bound with the reformulation guessing the dual bounds.

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.

Computationally efficient method for placing reactive power sources against contingencies

This paper presents a computationally efficient algorithm to find the locations and required reactive power injections for a minimum number of reactive power compensators under a set of considered contingencies. The problem is formulated and solved in two stages. The first stage solves the security-constrained DC optimal power flow problem to find the active power generation dispatch. Then, the second stage determines the best locations for a minimum number of reactive power compensators to assist the reactive power generation dispatch in order to maintain bus voltages within limits and to ensure voltage stability under the considered set of contingencies. Effectiveness and computational performance of the proposed method are illustrated by extensive simulations carried out on PEGASE 2869-bus system.

Iterative Bidding in Electricity Markets: Rationality and Robustness

This paper studies an electricity market consisting of an independent system operator (ISO) and a group of generators. The goal is to solve the dc optimal power flow (DC-OPF) problem: have the generators collectively meet the power demand while minimizing the aggregate generation cost and respecting line flow limits in the network. The ISO by itself cannot solve the DC-OPF problem as generators are strategic and do not share their cost functions. Instead, each generator submits to the ISO a bid, consisting of the price per unit of electricity at which it is willing to provide power. Based on the bids, the ISO decides how much production to allocate to each generator to minimize the total payment while meeting the load and satisfying the line limits. We provide a provably correct, decentralized iterative scheme, termed Bid Adjustment Algorithm , for the resulting Bertrand competition game. Regarding convergence, we show that the algorithm takes the generators’ bids to any desired neighborhood of the efficient Nash equilibrium at a linear convergence rate. As a consequence, the optimal production of the generators converges to the optimizer of the DC-OPF problem. Regarding robustness, we show that the algorithm is robust to affine perturbations in the bid adjustment scheme and that there is no incentive for any individual generator to deviate from the algorithm by using an alternative bid update scheme. We also establish the algorithm robustness to collusion, i.e., as long as each bus with generation has a generator following the strategy, there is no incentive for any group of generators to share information with the intent of tricking the system to obtain a higher payoff.

DC Optimal Power Flow With Joint Chance Constraints

Managing uncertainty and variability in power injections has become a major concern for power system operators due to increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies all the constraints jointly with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an S$\ell _1$QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms.

Solution to DC Optimal Power Flow Problems with Demand Response via Distributed Asynchronous Primal-Dual Algorithms

As distributed energy resources increase, distributed optimization for power system operation and demand response (DR) have been intensively investigated in recent years. This paper presents a dist...

Application of metaheuristic algorithms in DC-optimal power flow

This paper presents the application of two metaheuristic algorithms (Exchange Market Algorithm and Ant Lion Optimizer) in solving the DC optimal power flow problem. The objective of this study is to minimize fuel costs associated with electricity generation. Advanced Interactive Multidimensional Modelling Software is also used to solve the same optimization problem and the results obtained from using this method are used to validate those from the two metaheuristic algorithms. The three methods have been implemented on the standard IEEE 14- and 30-bus system, as well as the 62-bus Indian utility system. From the analysis, the results obtained prove the robustness and effectiveness of using both algorithms to solve DC optimal power flow and even more complex optimization problems.

Fast Power System Cascading Failure Path Searching With High Wind Power Penetration

Cascading failures have become a severe threat to interconnected modern power systems. The ultrahigh complexity of the interconnected networks is the main challenge toward the understanding and management of cascading failures. In addition, high penetration of wind power integration introduces large uncertainties and further complicates the problem into a massive scenario simulation problem. This paper proposes a framework that enables a fast cascading path searching under high penetration of wind power. In addition, we ease the computational burden by formulating the cascading path searching problem into a Markov chain searching problem and further use a dictionary-based technique to accelerate the calculations. In detail, we first generate massive wind generation and load scenarios. Then, we utilize the Markov search strategy to decouple the problem into a large number of DC power flow (DCPF) and DC optimal power flow (DCOPF) problems. The major time-consuming part, the DCOPF and the DCPF problems, is accelerated by the dynamic construction of a line status dictionary (LSD). The information in the LSD can significantly ease the computation burden of the following DCPF and DCOPF problems. The proposed method is proven to be effective by a case study of the IEEE RTS-79 test system and an empirical study of China's Henan Province power system.

Correct the Piecewise Linearization Error of Quadratic Objective Function in DC OPF Model

In power industries and academia, piecewise linearization methods are widely used in optimal power flow (OPF) problems to approximate the nonlinear objective functions. Better computational performance can be achieved. However, limited attention has been paid to the approximation error. In electricity markets, this error could cause distortions of market clearing results and deficits in social welfare. To address this issue, we provide a criterion for the desired number of piecewise segments. We then propose an efficient method for correcting the piecewise linearization error of the DC OPF problems.

Solution to DC Optimal Power Flow Problems via a Fully-Distributed Accelerated ADMM

Solution to DC Optimal Power Flow Problems via a Fully-Distributed Accelerated ADMM

Optimal placement and control variable setting of power flow controllers in multi-terminal HVDC grids for enhancing static security

This research proposes an approach to select an optimal place and control variable setting for recently proposed Power Flow Controller (PFC)s including series, cascaded, and interline PFCs in Multi-Terminal High-Voltage DC (MT-HVDC) grids based on sensitivity analysis technique to enhance static security. To do so, appropriate static power injection models of the PFCs are obtained. Accordingly, sensitivity relationships between the defined power performance index and each PFCs control variable are obtained. It is for the optimal PFCs placement purpose in order to utilize maximum capacities of the transmission grid. Optimal settings for the PFC as well as the controllable voltage source converters are computed in turn by applying sequential quadratic programming solver to the developed security-based DC optimal power-flow problem which includes several corrective constraints. The study scope is single contingencies affecting HVDC lines with the objective of eliminating the consequent overloaded HVDC lines and thereby enhancing the MT-HVDC grid security by controlling the power flows within the MT-HVDC grid. Static simulations are performed considering two generic four-terminal and eighth-terminal MT-HVDC test grids in order to demonstrate the proposed methods effectiveness and authenticate its robustness. The obtained results indicate that the MT-HVDC grid can remain secure under single HVDC line contingencies in most cases by implementing the proposed method.

A Fully-Decentralized Consensus-Based ADMM Approach for DC-OPF With Demand Response

This paper discusses a consensus-based alternating direction method of multipliers (ADMMs) approach for solving the dynamic dc optimal power flow (DC-OPF) problem with demand response in a distributed manner. In smart grid, emerging techniques together with distributed nature of data and information, significantly increase the complexity of power systems operation and stimulate the needs for distributed optimization. In this paper, the distributed DC-OPF approach solves local OPF problems of individual subsystems in parallel, which are coordinated via global consensus variables (i.e., phase angles on boundary buses of adjacent subsystems). Three distributed DC-OPF algorithms are discussed with different convergence performance and/or communication requirement. All three distributed algorithms can effectively handle prevailing constraints for the transmission network, generating units, and demand response in individual subsystems, while the global convergence can be guaranteed. In particular, based on the traditional distributed ADMM approach, a fully decentralized algorithm without the central controller is proposed in Algorithm 2 with a new communication strategy, in which only limited information on boundary buses are exchanged among adjacent subsystems. In addition, the accelerated ADMM is discussed in Algorithm 3 for improving the convergence performance. In recognizing distributed OPF approaches in literature, one major research focus on this paper is to quantify the impact of key parameters and subsystem partitioning strategies on the convergence performance and the data traffic via numerical case studies. A general guidance for subsystem partitioning is proposed and verified via large-scale power systems.

On fully distributed dual first order methods for convex network optimization

This paper provides distributed optimization methods to carry out fully distributed computations in convex network optimization problems. The network constraints are moved into the cost using the Lagrange multipliers. For solving the dual problem we propose a fully distributed dual fast gradient algorithm having sublinear rate of convergence in terms of dual suboptimality but also in terms of primal suboptimality and feasibility violation for the last generated primal iterate. The convergence analysis combines the Lipschitz property of dual function with fast gradient schemes. The main application we consider is the DC optimal power flow problem from an electric energy system. Our approach allows to carry out an optimal dispatch for each bus to be done independently and in parallel while still achieving the global economical optimum of the whole electric system. The newly developed algorithms can be run distributively, as we show on several numerical simulations using classical IEEE bus test cases.

Semidefinite Relaxation of Optimal Power Flow for AC–DC Grids

The proliferation of technologies operating on dc power has motivated the system planners toward integration of dc and ac grids. The optimal power flow (OPF) analysis is widely used to determine the economically efficient operating points of the power grids. The OPF problem in ac–dc grids is a non-convex optimization problem due to the nonlinear power flow equations and the operating constraints imposed by the ac–dc converters. In this paper, we study the OPF problem in ac–dc grids to address the non-convexity of the problem. The objective of the ac–dc OPF problem is to jointly minimize the generation cost and the losses on the lines and converters. The optimization problem is subject to the ac and dc power flow constraints, the limits of the voltages and line flows, and the operating limits of the converters. We use convex relaxation techniques and transform the problem to a semidefinite program. We derive a sufficient condition for zero relaxation gap to obtain the global optimal solution. Simulations are performed on an IEEE 118-bus test system connected to sample dc grids. We show that the zero relaxation gap condition holds for the case study and the global optimal solution can be obtained.

Real-Time Distributed Control of Battery Energy Storage Systems for Security Constrained DC-OPF

The introduction of fast-response battery energy storage system (BESS) provides a number of advantages to address the new challenges of smart grids, including improving the reliability and security of the power grids. This paper proposes a real-time distributed control algorithm to the corrective security-constrained DC optimal power flow problem for transmission network. The objective is to minimize the adjustment of the BESSs, while maintaining the supply-demand balance and ensuring no security constraint violations in the post-contingency state for the short-term period. Comparing to the conventional centralized methods, only simple local computation and information exchange with neighbors are required to update the local control signal, which leads to fast response of the BESSs to alleviate the impact of the transmission line outage. Real-time simulation results of the modified 6-bus and 24-bus systems demonstrate that the dynamic performances of proposed distributed algorithm satisfy standard requirements and indicate its applicability for practical power grid.

Determining Optimal Buses for Implementing Demand Response as an Effective Congestion Management Method

Demand response (DR) can be counted as an effective congestion management method in power systems. However, appointing the optimal buses for the demand response programs (DRPs) implementation is one of the main challenges for the power system operators. In this paper, a new procedure has been developed by which the optimal locations and times of DRPs implementation are determined. Optimal buses are identified based on the power transfer distribution factors (PTDFs), available transfer capability, and dynamic dc optimal power flow problem. Applying the developed method results in reducing lines' congestion, increasing customers, and independent system operator's benefits, improving load curve's characteristics, preventing line outages and black outs, and consequently increasing the network reliability. The proposed method has been applied on the IEEE 39-bus New England test system. Results indicate the effectiveness and practical benefits of the proposed method.