Optimal Power Flow Formulation Research Articles

Surrogate Formulation for Chance-Constrained DC Optimal Power Flow With Affine Control Policy

Surrogate Formulation for Chance-Constrained DC Optimal Power Flow With Affine Control Policy

Chance Constrained MDP Formulation and Bayesian Advantage Policy Optimization for Stochastic Dynamic Optimal Power Flow

Chance Constrained MDP Formulation and Bayesian Advantage Policy Optimization for Stochastic Dynamic Optimal Power Flow

Impact of phase selection on accuracy and scalability in calculating distributed energy resources hosting capacity

Hosting capacity (HC) and dynamic operating envelopes (DOEs), defined as dynamic, time-varying HC, are calculated using three-phase optimal power flow (OPF) formulations. Due to the computational complexity of such optimisation problems, HC and DOE are often calculated by introducing certain assumptions and approximations, including the linearised OPF formulation, which we implement in the Python-based tool ppOPF. Furthermore, we investigate how assumptions of the distributed energy resource (DER) connection phase impact the objective function value and computational time in calculating HC and DOE in distribution networks of different sizes. The results are not unambiguous and show that it is not possible to determine the optimal connection phase without introducing binary variables since, no matter the case study, the highest objective function values are calculated with mixed integer OPF formulations. The difference is especially visible in a real-world low-voltage network in which the difference between different scenarios is up to 14 MW in a single day. However, binary variables make the problem computationally complex and increase computational time to several hours in the DOE calculation, even when the optimality gap different from zero is set.

Enhancing Distribution Grid Efficiency and Congestion Management through Optimal Battery Storage and Power Flow Modeling

The significant growth in demand for electricity has led to increasing congestion on distribution networks. The challenge is twofold: it is needed to expand and modernize our grid to meet this increased demand but also to implement smart grid technologies to improve the efficiency and reliability of electricity distribution. In order to mitigate these congestions, novel approaches by using flexibility sources such as battery energy storage can be used. This involves the use of battery storage systems to absorb excess energy at times of low demand and release it at peak times, effectively balancing the load and reducing the stress on the grid. In this paper, two optimal power flow formulations are discussed: the branch flow model (non-convex) and the relaxed bus injection model (convex). These formulations determine the optimal operation of the flexibility sources, i.e., battery energy storage, with the objective of minimizing power losses while avoiding congestions. Furthermore, a comparison of the performance of these two formulations is performed, analyzing the objective function results and the flexibility operation. For this purpose, a real Spanish distribution network with its corresponding load data for seven days has been used.

Capability Curve Modeling for Hydro-Power Generators in Optimal Power Flow Problems

With the growing emphasis on sustainability in the power sector, it becomes imperative to ensure that every component of the power system operates optimally and efficiently. More sustainable power system operation can be achieved by relying on accurate models that ensure resources, especially those from renewable sources like hydro-power, are utilized to their fullest potential. In many optimization problems based on optimal power flow formulations, the steady-state operation characteristics of hydro-power plants are modeled in an approximate manner, which could potentially lead to solutions that do not fully exploit their capabilities or even to solutions that jeopardize their stable operation. This work proposes a formulation for the complete capability curve of the plant, including the exact modeling of its generator stability limits. The ability of the proposed formulation to reproduce the plant operation boundaries is appropriately demonstrated through a test case. Furthermore, two approximate formulations commonly used in the literature are solved, highlighting their limitations. It is concluded that the complete representation of the capability curve can improve the quality of solutions provided by OPF-based problems.

An AC optimal power flow framework for active–reactive power scheduling considering generator capability curve

AbstractThis paper presents a simplified optimal power flow (OPF) framework to facilitate co‐optimised active and reactive power scheduling for synchronous generators and price‐sensitive demands. The proposed framework creates an opportunity for generators and loads to simultaneously participate in a combined market for active and reactive power. The co‐optimisation of active and reactive power generation is constrained by the interdependence of the active and reactive power capacities, which is represented by the generator capability curve. Thus, a detailed mathematical derivation of the opportunity costs across various regions of the generator capability curve is presented. This study considers a detailed generator capability curve that considers the armature, field, under‐excitation, and prime mover limits. The interdependence of active and reactive power consumption for demand is modelled using the concept of power‐factor. The OPF problem for generator and load scheduling is formulated as a non‐linear optimisation task, leveraging the inherent properties of the generator capability curve, that is, piecewise smoothness, continuity, and the monotonically increasing slope magnitudes. Furthermore, to simplify the OPF formulation, the non‐linear capability curve is represented as a combination of the linear curves. To demonstrate the effectiveness of the proposed OPF methodologies, suitable case studies are conducted using different test systems.

A Consensus-Based Distributed Secondary Control Optimization Strategy for Hybrid Microgrids

In this paper, a new consensus-based distributed secondary control (DSC) strategy is proposed for frequency control, dc-voltage regulation, and optimal dispatch (OD) in isolated hybrid ac/dc-microgrids (HMGs). At the same time, all the units are maintained within limits. The proposed control scheme dispatches the distributed generators (DGs) within the microgrid in compliance with the Karush-Kuhn-Tucker (KKT) conditions of a linear optimal power flow (OPF) formulation. Furthermore, the power through the interlinking converters (ICs) is also dispatched to help minimize the total operation cost of the microgrid. The controllers rely on local measurements and information from neighbouring devices at both sides of the microgrid (DGs and ICs). Thus, unlike conventional methods, the microgrid is considered a single entity and not as three independent systems interacting with one another, and the OD is calculated considering both the ac-DGs and dc-DGs. Extensive simulations demonstrate a good performance of the controller amid load step changes and unit congestion, driving the system to an optimal economic operation.

A second-order conic approximation to solving the optimal power flow problem in bipolar DC networks while considering a high penetration of distributed energy resources

This article presents an optimal power flow (OPF) formulation based on the branch flow model for bipolar DC grids with asymmetric loading. The proposal considers the power injection formulation of the OPF to obtain the branch flow model equations that allow solving the power flow problem in bipolar DC networks. Furthermore, a nonlinear programming (NLP) optimization model is deduced and transformed into a second-order conic programming one, which involves convex optimization and guarantees a global optimum for the relaxed model, the uniqueness of this solution, and fast convergence. Finally, an approximation based on balanced operation is proposed to simplify the OPF. Several case studies in two test systems composed of 21 and 85 nodes with a bipolar structure demonstrate that the proposal is accurate, fast, and close to the exact solution of the NLP model when compared with competitive literature reports. All optimization models were implemented in the Python language.

Optimal Power Flow for Hybrid AC/DC Electrical Networks Configured With VSC-MTDC Transmission Lines and Renewable Energy Sources

This article presents the single objective optimal power flow (OPF) formulation incorporating both renewable energy sources, and voltage source converter-based multiterminal direct current transmission lines, simultaneously. To solve the formulated OPF problem, powerful metaheuristic optimization algorithms including adaptive guided differential evolution, marine predators algorithm, atom search optimization, stochastic fractal search (SFS), and fitness-distance balance-based SFS (FDB-SFS) are employed. The performance of the algorithms is tested for the minimization of fuel cost, pollutant emissions of thermal generators, voltage deviation, and active power loss in a modified IEEE 30-bus power network. The simulation results give that FDB-SFS achieved the best results on the fuel cost (786.5361 $/h), the fuel cost with valve point effect (815.6644 $/h), and the fuel cost with emission-carbon tax (820.5991 $/h). In addition, FDB-SFS reduced voltage deviation and active power loss values by 14.2587% and 6.7438% compared to SFS. The nonparametric Wilcoxon and Friedman statistical test results confirmed that FDB-SFS is an effective and robust algorithm that can be used in the optimization of the introduced OPF problem.

Optimal Power Flow in Distribution Network: A Review on Problem Formulation and Optimization Methods

Distributed generators (DGs) have a high penetration rate in distribution networks (DNs). Understanding their impact on a DN is essential for achieving optimal power flow (OPF). Various DG models, such as stochastic and forecasting models, have been established and are used for OPF. While conventional OPF aims to minimize operational costs or power loss, the “Dual-Carbon” target has led to the inclusion of carbon emission reduction objectives. Additionally, state-of-the-art optimization techniques such as machine learning (ML) are being employed for OPF. However, most current research focuses on optimization methods rather than the problem formulation of the OPF. The purpose of this paper is to provide a comprehensive understanding of the OPF problem and to propose potential solutions. By delving into the problem formulation and different optimization techniques, selecting appropriate solutions for real-world OPF problems becomes easier. Furthermore, this paper provides a comprehensive overview of prospective advancements and conducts a comparative analysis of the diverse methodologies employed in the field of optimal power flow (OPF). While mathematical methods provide accurate solutions, their complexity may pose challenges. On the other hand, heuristic algorithms exhibit robustness but may not ensure global optimality. Additionally, machine learning techniques exhibit proficiency in processing extensive datasets, yet they necessitate substantial data and may have limited interpretability. Finally, this paper concludes by presenting prospects for future research directions in OPF, including expanding upon the uncertain nature of DGs, the integration of power markets, and distributed optimization. The main objective of this review is to provide a comprehensive understanding of the impact of DGs in DN on OPF. The article aims to explore the problem formulation of OPF and to propose potential solutions. By gaining in-depth insight into the problem formulation and different optimization techniques, optimal and sustainable power flow in a distribution network can be achieved, leading to a more efficient, reliable, and cost-effective power system. This offers tremendous benefits to both researchers and practitioners seeking to optimize power system operations.

Assessment of Converter Performance in Hybrid AC-DC Power System under Optimal Power Flow with Minimum Number of DC Link Control Variables

This paper presents a strategy to evaluate the performances of converter stations under the optimized operating points of hybrid AC-DC power systems with a reduced number of DC link variables. Compared to previous works reported with five DC-side control variables (CVs), the uniqueness of the presented optimal power flow (OPF) formulation lies within the selection of only two DC-side control variables (CVs), such as the inverter voltage and current in the DC link, apart from the conventional AC-side variables. Previous research has mainly been focused on optimizing hybrid power system performance through OPF-based formulations, but has mostly ignored the associated converter performances. Hence, in this study, converter performance, in terms of ripple and harmonics in DC voltage and AC current and the utilization of the converter infrastructure, is evaluated. The minimization of active power loss is taken as an objective function, and the problem is solved for a modified IEEE 30 bus system using a recently developed and very efficient Archimedes optimization algorithm (AOA). Case studies are performed to assess the efficacy of the presented OPF model in power systems, as well as converter performance. Furthermore, the results are extended to assess the applicability of the proposed model to the allocation of photovoltaic (PV)-type distributed generations (DGs) in hybrid AC-DC systems. The average improvement in power loss is found to be around 7.5% compared to the reported results. Furthermore, an approximate 10% improvement in converter power factor and an approximate 50% reduction in ripple factor are achieved.

Fairness-Aware Distributed Energy Coordination for Voltage Regulation in Power Distribution Systems

The accelerating deployment of solar photovoltaics into low-voltage distribution networks can cause reverse power flow and overvoltage problems. However, if coordinated properly, the real and reactive power flexibility of these resources enables distribution operators to manage their networks more efficiently. Existing literature is rich in droop-based control (Volt-Watt and Volt-VAr) and optimization-based distributed energy coordination for four-quadrant control of photovoltaics to prevent overvoltage issues. While optimal coordination can effectively mitigate overvoltage, it treats resources at sensitive parts of the grid unfairly. To address this concern, we propose a distributed optimal power flow formulation that incorporates fairness in curtailing photovoltaic generation and utilizes the reactive power capability of smart inverters. Fair curtailment of photovoltaic systems is studied with aggregation at each of two layers in a distribution network: 1) area-level fairness and 2) feeder-level fairness to demonstrate the flexibility of the proposed approach in resource aggregation. To explore the trade-off, the fairness-aware control actions are compared against the performance of a centralized controller that aims to maximize the aggregate PV generation without incorporating fairness. Simulation results show that introducing area-level fairness increased curtailment by 1.11 percentage points, and feeder-level fairness increased curtailment by 4.7 percentage points compared to a fairness-agnostic control.

Economic and environmental dispatch of power systems: Minimising CO2 emissions by integrating renewable energy sources

This paper analyses the impact of the integration of renewable energy sources on the economic and environmental operation of power systems (PS). The models of renewable energy sources, as well as the generation cost function and Carbon Dioxide (CO2) emissions are integrated in an Optimal Power Flow (OPF) formulation in order to obtain a non-linear multi-objective optimisation model, which considers economic and environmental aspects. This OPF model, having integrated the models of renewable energy sources, allows to evaluate the impact that these energy sources have on the minimisation of generation cost and CO2 emissions. In this work, the multi-objective OPF problem is solved using weighting factors for the two functions considered in the global objective function. Several case studies are carried out to evaluate the effect of the integration of renewable energy sources on the economic and environmental operation of power systems. The results of the case studies show that renewable energy sources reduce generation cost and CO2 emissions.

Integration of Centralized and Distributed Methods to Mitigate Voltage Unbalance Using Solar Inverters

Growing penetrations of single-phase distributed generation such as rooftop solar photovoltaic (PV) systems can increase voltage unbalance in distribution grids. However, PV systems are also capable of providing reactive power compensation to reduce unbalance. In this paper, we compare two methods to mitigate voltage unbalance with solar PV inverters: a centralized optimization-based method utilizing a three-phase optimal power flow formulation and a distributed approach based on Steinmetz design. While the Steinmetz-based method is computationally simple and does not require extensive communication or full network data, it generally leads to less unbalance improvement and more voltage constraint violations than the optimization-based method. In order to improve the performance of the Steinmetz-based method without adding the full complexity of the optimization-based method, we propose an integrated method that incorporates design parameters computed from the set-points generated by the optimization-based method into the Steinmetz-based method. We test and compare all methods on a large three-phase distribution feeder with time-varying load and PV data. The simulation results indicate trade-offs between the methods in terms of computation time, voltage unbalance reduction, and constraint violations. We find that the integrated method can provide a good balance between performance and information/communication requirements.

Optimization of Distributed Energy Resources in Distribution Networks: Applications of Convex Optimal Power Flow Formulations in Distribution Networks

The increasing penetration of distributed energy resources (DER) has imposed several challenges in the analysis and operation distribution networks. In the last decade, the implementation of battery energy storage systems (BESS) in electric networks has caught the interest in research since the results have shown multiple positive effects when deployed optimally. However, a complex formulation of the optimization problem regarding DER implementations can easly become nonconvex, thus limiting the scope of the results (quality) and affecting the computational efficiency. In this paper, convex formulations of the optimal power flow (OPF) problem for DER (PV and BESS) installations in San Andres distribution network were implemented to minimize power losses. Four main simulation cases were stablished covering convex power flow analysis, convex optimal power flow to locate and size dispatchable and nondispatchable DER (PV) with demand and irradiation profiles, and the inclusion of aggregated and distributed PV and BESS systems. The computational realization of those formulations was made with three different software packages: CVX in python, CPLEX, and MATLAB (MATPOWER/PSO). The results show that installing DER capacity (both PV and/or BESS) can substantially improve the power losses in distribution networks, especially if considered distributed instead of aggregated, achieving reductions up to 75% for power losses and up to 57% for conventional generation utilization. Besides power losses, it was observed that optimal DER (PV and BESS) implementations behaved similarly as demand response signals do, potentially increasing economic benefits for DISCOs. In terms of computational efficiency, convex formulations help substantially to reduce computation times, especially if scalability is to be considered when distributed DER installations were studied.

Optimal Adaptive Droop Design via a Modified Relaxation of the OPF

The ever-increasing penetration of renewable energy resources (RESs) in power distribution networks has brought, among others, the challenge of maintaining the grid voltages within the secure region. Employing droop voltage regulators on the RES’s inverters is an efficient and low-cost solution to reach this objective. However, fixing droop parameters or optimizing them only for overvoltage conditions does not provide the required robustness and optimality under changing operating conditions. In this article, a convex optimization approach is proposed for reconfiguring <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$ </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">$Q$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> droop regulators during online operation. The objective is to minimize power curtailment, power losses, and voltage deviation subject to electrical security constraints. This enables to optimally operate the grid with high RES penetration under variable conditions while preserving electrical security constraints (e.g., current and voltage limits). As a first contribution, a mixed-integer linear model of the droop characteristics is developed. According to this model, a droop-regulated generation unit is represented as a constant-power generator in parallel with a constant-impedance load. As a second contribution, a modified augmented relaxed optimal power flow (MAR-OPF) formulation is proposed to guarantee that the electrical security constraints are respected in the presence of constant-impedance loads in the network. Sufficient conditions for the feasibility of the MAR-OPF solution are provided. Those conditions can be checked a priori and are valid for several real distribution networks. Furthermore, an iterative approach is proposed to derive an approximate solution to the MAR-OPF that is close to the global optimal one. The performance of the MAR-OPF approach and the accuracy of the proposed model are evaluated on standard 34- and 85-bus test networks.

A generalised approach for efficient computation of look ahead security constrained optimal power flow

We consider a generalised comprehensive Look-ahead Security-constrained Optimal Power Flow (LASCOPF) formulation under the N−1 contingency criterion over multiple dispatch intervals. We observe that the number of decision variables varies quadratically with the number of intervals. To improve scalability, we propose a reduced LASCOPF formulation for which the number of decision variables varies only linearly. We extend these formulations to the N−k contingency criterion. For reduced LASCOPF we observe that the number of decision variables varies with the number of k-permutations of contingencies. To improve scalability, we propose a formulation that is further reduced to vary only with the number of k-combinations. Also, we show that our formulations can be extended simply to model recovery from the corresponding outages. Furthermore, we present LASCOPF under the N−1 contingency criterion using DC and AC power flow under generator contingencies. We prove that, barring borderline cases, solving the reduced formulation is equivalent to solving the comprehensive formulation. We extend these results to the N−k contingency criterion. Finally, we present numerical results on the IEEE 14 bus, IEEE 30 bus and IEEE 300 bus test cases, and the 1354 bus part of the European power system using AC power flow to demonstrate the computational advantage of the reduced formulations under the N−1 and N−2 contingency criteria.

A Novel Scalable Reconfiguration Model for the Postdisaster Network Connectivity of Resilient Power Distribution Systems.

The resilient operation of power distribution networks requires efficient optimization models to enable situational awareness. One of the pivotal tools to enhance resilience is a network reconfiguration to ensure secure and reliable energy delivery while minimizing the number of disconnected loads in outage conditions. Power outages are caused by natural hazards, e.g., hurricanes, or system malfunction, e.g., line failure due to aging. In this paper, we first propose a distribution-network optimal power flow formulation (DOPF) and define a new resilience evaluation indicator, the demand satisfaction rate (DSR). DSR is the rate of satisfied load demand in the reconfigured network over the load demand satisfied in the DOPF. Then, we propose a novel model to efficiently find the optimal network reconfiguration by deploying sectionalizing switches during line outages that maximize resilience indicators. Moreover, we analyze a multiobjective scenario to maximize the DSR and minimize the number of utilized sectionalizing switches, which provides an efficient reconfiguration model preventing additional costs associated with closing unutilized sectionalizing switches. We tested our model on a virtually generated 33-bus distribution network and a real 234-bus power distribution network, demonstrating how using the sectionalizing switches can increase power accessibility in outage conditions.

Fast QC Relaxation of the Optimal Power Flow Using the Line-Wise Model for Representing Meshed Transmission Networks

In this paper, we investigate the recently introduced McCormick-based Quadratic Convex (QC) relaxation of the Optimal Power Flow (OPF) where Line-Wise Model (LWM) is used for representing meshed transmission systems (QC-LW OPF) for the sake of decisively determining its relationship to other available convex relaxations in the literature. We also extend the recently introduced convex envelope of the tangent function so it would be suitable for test cases with any voltage angle difference range. A computational study where the recently proposed QC-LW OPF formulation is compared to an equivalent McCormick based QC relaxation of the OPF where Bus Injection Model (BIM) is used for representing meshed transmission systems (QC-BI OPF) is presented in this paper. This computational study was conducted using test cases that belong to different operational categories using 123 test cases from the PGLib-OPF library with a bus size range between 3 up to 6515 buses for the sake of understanding the effect of the change of operating conditions on the quality of solutions obtained using the QC-LW OPF and QC-BI OPF formulations. Results are compared using several metrics that testify to the obtained solution’s quality and the problem’s computational complexity. Comparison of results shows that the QC-LW relaxation neither dominates nor is dominated by the QC-BI relaxation in terms of solution quality. Therefore, it dominates the Second Order Cone (SOC) relaxation and neither dominates nor is dominated by the Semidefinite Programming (SDP) relaxation. Furthermore, it is shown that the QC-LW OPF has reduced the number of relaxed trigonometric functions and McCormick envelopes needed when compared to the QC-BI OPF, leading to a faster solution time for more than 84% of the test cases in the range of 2% up to 67%.

Linear Optimal Power Flow Formulation Derived from Optimality Condition

Linear Optimal Power Flow Formulation Derived from Optimality Condition