Direct Current Optimal Power Flow Research Articles

Optimal sizing and placement of battery energy storage system for maximum variable renewable energy penetration considering demand response flexibility: A case in Lombok power system, Indonesia

Indonesia is a tropical climate country with considerable renewable electrical energy source prospects, including photovoltaic (PV) and wind energies. Nevertheless, several variable renewable energy sources (VREs) have exhibited uncertain attributes and substantial reliance on natural conditions, leading to unstable load-related power supply risks. Hence, integrating battery energy storage systems (BESSs) with VRE generators is a dependable approach to bolster renewable energy generator applications on a large-scale grid while providing load demand flexibility. This study determined adequate sizing and placement of the BESS to achieve maximum VRE generator penetration while considering the demand response flexibility. Key indicators, including technical minimum load and system ramp capacity, were identified to achieve maximum penetration of thermal generators. This study also combined a unit commitment procedure with direct current optimal power flow (DC-OPF) as a novel approach to determine the maximum VRE penetration level. An optimization problem model concerning mixed integer linear programming (MILP) was subsequently employed in this study using the CPLEX solver in the general algebraic modeling system (GAMS). The study is based on the IEEE RTS-24 system modified and a real-life case study of the Lombok energy system in Indonesia. Results from the simulated Lombok power system highlighted that optimal sizing and placement of the BESS could lower system costs by 37.66%, 33.63%, and 22.26% compared to the current system conditions during the weekday, weekend, and the lowest day scenarios, respectively. The VRE penetrations of the generators were also higher than the current system conditions by 83%, 51%, and 39% during the weekday, weekend, and the lowest day scenarios, respectively.

A data-aided robust approach for bottleneck identification in power transmission grids for achieving transportation electrification ambition: a case study in New York state

As the enthusiasm for electric vehicles passes the range anxiety and other tests, large-scale transportation electrification becomes a prominent topic in research and policy discussions. In consequence, the public attention has shifted upstream and holistically towards the integration of large-scale transportation electrification to power systems. This paper proposes a method to identify bottlenecks in power transmission systems to accommodate large-scale and stochastic electric vehicles charging demands. First, a distributionally robust chance-constrained direct current optimal power flow model is developed to quantify the impacts of stochastic electric vehicles charging demands. Subsequently, an agent-based model with the trip-chain method is applied to obtain the spatiotemporal distributions of electric vehicles charging demands for both light-duty electric vehicles and medium and heavy-duty electric vehicles. The first two moments of those distributions are extracted to build an ambiguity set of electric vehicles charging demands. Finally, a 121-bus synthetic transmission network for New York State power systems is used to validate the future transportation electrification in New York State from 2025 to 2035. Results show that the large-scale transportation electrification in New York State will account for approximately 13.33 % to 16.79 % of the total load demand by 2035. The transmission capacity is the major bottleneck in supporting New York State to achieve its transportation electrification. To resolve the bottlenecks, we explore some possible solutions, such as transmission capacity expansion and distributed energy resources investment. Wind power shows an advantage over solar energy in terms of total operational costs due to better peak alignment between wind power and electric vehicles charging demand.

High-resolution, open-source modeling of inland flooding impacts on the North Carolina bulk electric power grid

Although damages to local distribution systems from wind and fallen trees are typically responsible for the largest fraction of electricity outages during hurricanes, outages caused by flooding of electrical substations pose a unique risk. Electrical substations are a key component of electric power systems, and in some areas, the loss of a single substation can cause widespread power outages. Before repairing damaged substations, utilities must first allow floodwaters to recede, potentially leaving some customers without power for weeks following storms. As economic losses from flooding continue to increase in the U.S., there has been increasing attention paid to the potential impacts of flooding on power systems. Yet, this attention has mostly been limited to geospatial risk assessments that identify what assets are in the path of flooding. Here, we present the first major attempt to understand how flooding from hurricanes and other extreme precipitation events affects the dynamic behavior of power networks, including losses of demand and generation, and altered power flows through transmission lines. We use North Carolina, hit by major hurricanes in three of the past seven years, as a test case. Using open-source data of grid infrastructure, we develop a high-resolution direct current optimal power flow model that simulates electricity production and generators and power flows through a network consisting of 662 nodes and 790 lines. We then simulate grid operations during the historical (2018) storm Hurricane Florence. Time series of flooding depth at a discrete set of ‘high water’ mark points from the storm are used to spatially interpolate flooding depth across the footprint area of the storms on an hourly basis. Outages of substations and solar farms due to flooding are translated to location-specific losses of demand and solar power production throughout the network. We perform sensitivity analysis to explore grid impacts as a function of the height of sensitive equipment at substations. Results shed light on the potential for localized impacts from flooding to have wider impacts throughout the grid (including in areas not affected by flooding), with performance tracked in terms of transmission line flows/congestion, generation outputs, and customer outages.

Getting prices right on electricity spot markets: On the economic impact of advanced power flow models

As the share of variable renewable energy increases, adequate prices on electricity spot markets become increasingly important as they set signals for scarcity, investment, or demand response. Market prices are derived from the underlying welfare maximization problem. On electricity spot markets, this optimization problem is based on the non-convex and non-linear Alternating Current Optimal Power Flow (ACOPF) model. Since the ACOPF is intractable, electricity markets around the world use a linear approximation, the Direct Current Optimal Power Flow (DCOPF) model. Recent research has led to better non-linear relaxations of the ACOPF. We show that these non-linear relaxations increase welfare and imply significantly lower redispatch costs and side-payments. Most importantly, we show that the price signals obtained from non-linear relaxations are much improved. The DCOPF often yields high price differences between nodes when there is no line congestion in the AC-feasible solution or vice versa. Such biased price signals pose a significant problem in practice as they lead to inefficient demand response, distorted investment signals, and incorrect congestion incomes. The use of non-linear relaxations mitigates this problem and provides an important advantage of the resulting prices over prices based on the DCOPF.

Stochastic multi‐stage joint expansion planning of transmission system and energy hubs in the presence of correlated uncertainties

AbstractDeployment of Energy Hubs (EHs) across the power grid can alleviate the Transmission System (TS) capacity and substitute the conventional fossil fuel‐based thermal units. Therefore, this paper presents a tri‐level multi‐stage Joint Expansion Planning of the Transmission system and EHs (JEPT&EHs). In this approach, the Cholesky decomposition technique combined with the Nataf transformation is applied to make the uncertain input parameters correlated. Then, the k‐means data‐clustering method is employed to reduce the initial correlated samples. In the first level, the Transmission System Operator (TSO) optimizes the planning and scheduling strategies associated with the TS capacity requirements and operation costs of the conventional generators. In the second level, the financers specify the expansion of the EHs based on the Locational Marginal Prices (LMPs). In the third level, the Direct Current Optimal Power Flow (DCOPF) is determined to update the LMPs by the Independent System Operator (ISO). The optimization problem is an Equilibrium Problem with Equilibrium Constraints (EPEC) since there are multiple financers across the TS. The proposed model is implemented on the IEEE standard 30 bus TS to present the effectiveness of the EHs' deployment and the impact of the correlations in the total costs of the TSO and financers.

Dynamic Optimal Power Flow for Microgrids Considering Efficiency and Lifetime of Battery Using MINLP

This paper proposes an approach to operate optimally microgrids considering battery efficiency and lifetime. The microgrid system consists of several generators such as diesel generators, microturbines, photovoltaics, wind turbines, and battery. It is assumed that the battery belongs to an ancillary service provider. Therefore, a different price for battery charge and discharge transactions is applied. In addition, battery efficiency and lifetime are also incorporated into the operation problem. The problem is formulated as Dynamic Direct Current Optimal Power Flow (DDCOPF). The objective function is to minimize generation cost and line losses, as well as to maximize battery lifetime. However, unlike other standard DDCOPF, network losses modeled as the quadratic equation are included in the problem formulation. As formulated problem comprises discrete and continuous variables, Mixed Integer Non-Linear Programming (MINLP) is used to solve it. In order to show the effectiveness of the proposed approach, a modified IEEE 30 bus is used as a test system.

Best Response Intersection: An Optimal Algorithm for Interdiction Defense

We define the interdiction defense problem as a game over a set of targets with three stages: a first stage where the defender protects a subset of targets, a second stage where the attacker observes the defense decision and attacks a subset of targets, and a third stage where the defender optimizes a system using only the surviving targets. We present a novel algorithm for optimally solving such problems that uses repeated calls to an attacker’s best response oracle. For cases where the defender can defend at most k targets and the attacker can attack at most z targets, we prove that the algorithm makes at most [Formula: see text] calls to the oracle. In application to the direct current optimal power flow problem, we present a new mixed integer programming formulation with bounded big-M values to function as a best response oracle. We use this oracle along with the algorithm to solve a defender-attacker-defender version of the optimal power flow problem. On standard test instances, we find solutions with larger values of k and z than shown in previous studies and with runtimes that are an order of magnitude faster than column and constraint generation. Funding: This work was sponsored in part by the U.S. Department of Energy Office of Electricity’s Advanced Grid Modeling (AGM) program. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory [Contract DE-AC52-07NA27344].

Data-Driven Electricity Price Risk Assessment for Spot Market

Electricity price risk assessment (EPRA) is essential for spot market analysis and operation. The statistical moments (i.e., the mean and standard deviation) of the price need to be assessed to support market risk control. This paper proposes a data-driven approach for EPRA based on the Gaussian process (GP) framework. Compared with the deep learning algorithms, GP has two merits: (1) the scale of training sample required is small and (2) the time-consuming hyperparameter tuning process is avoided. However, the direct application of GP for EPRA is not tractable due to the complicated discrete relationship between the system operating status and the electricity price. To deal with that, a data-driven EPRA framework is developed that contains a GP surrogate model for the direct current optimal power flow (DC-OPF) problem and a hybrid model-data-based hybrid electricity price calculation method. To guarantee the accuracy of EPRA, an adaptability criterion and a second verification process based on the Karush–Kuhn–Tucker (KKT) condition are developed to distinguish the samples with GP learning errors. Numerical results carried out on IEEE benchmark systems demonstrate that the proposed method can achieve exactly the same EPRA results as Monte Carlo (MC) simulation, which significantly improved the computational efficiency.

Power and Traffic Nexus: From Perspective of Power Transmission Network and Electrified Highway Network

The increasing prevalence of electric vehicles (EVs) and emerging dynamic wireless charging techniques have increased the interdependence between the power and transportation systems. This article investigates a new nexus scenario-the electrified highway network (EHN) and power transmission network (PTN). The independent but interrelated operation of the two networks is analyzed. On the EHN side, a combined charging-driving navigation (CCDN) model, which rigorously considers EV travel speed and charge/discharge behaviors, is proposed and formulated as an irregular dynamic programming problem. A chronological search algorithm is designed to derive the optimal charging-driving decision sequences. The economic operation of the PTN is formulated as a direct current optimal power flow problem. Hourly location marginal prices (LMPs) and charge/discharge demands are exchanged between the two networks. The abovementioned LMP-based interaction forms a Nash-type game, in which both networks aim to minimize their own operation costs. A best-response decomposition algorithm combined with a continuous LMP method is developed to identify the equilibrium state. Numerical results demonstrate the effectiveness of the CCDN model in modeling EV charging-driving behaviors in the EHN. Moreover, the results show that, under certain conditions, mobility of EVs along the EHN can economically relieve the transmission congestion and reduce the operation costs of both networks.

Optimization of Economic Efficiency in Distribution Grids Using Distribution Locational Marginal Pricing

Distribution locational marginal pricing (DLMP) is an increasingly popular pricing signal that can be used to incentivize grid-friendly behavior of distributed energy resources (DER) to optimize economic efficiency in distribution grids. In this paper, a lossy direct-current optimal power flow (DCOPF) is utilized to obtain the iterative calculation framework for DLMPs. The two-stage algorithm iterates between the transmission system optimal power flow (OPF) and the distribution system OPF until no significant changes in DLMPs are observed. Real power losses are estimated using a static piecewise linear approximation technique. A sampling algorithm is proposed to minimize the possible convergence issues associated with the proposed mathematical model. DLMPs are calculated for the i) contemporary, ii) enhanced, and iii) meshed distribution grids using an IEEE 34-bus test feeder. The test transmission system is modeled using an IEEE 30-bus system. Finally, the calculated DLMPs in the enhanced grid are compared against three existing pricing mechanisms via three case studies to prove its validity and superiority, especially in congested systems with high penetration of price-responsive loads (PRLs).

Security constrained DC OPF considering generator responses

Considering generator responses after contingencies provides a more practical solution to security constrained direct current optimal power flow (DCOPF) problems. The major difficulty of solving such OPF includes the large number of contingencies and non-convexity of the generator response constraints. In the literature, mixed-integer linear programming (MILP) formulation relying on big-M technique has been applied to deal with the non-convexity of generator response constraints. In this paper, we further improve the solving speed by formulating generator response constraints via bilinear expressions and adopting Benders’ decomposition technique to decompose the problem into a master problem and multiple subproblems, with each subproblem associated with a contingency. Benders’ decomposition strategies were investigated in this research to seek an efficient decomposition approach. Through preserving bilinear expressions related to the base case power while relaxing the rest bilinear expressions via McCormick envelops, we designed an efficient Benders’ decomposition strategy. Case study results demonstrate the efficiency of the proposed formulation compared to the state-of-the-art formulations.

DeepOPF: A Deep Neural Network Approach for Security-Constrained DC Optimal Power Flow

We develop <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepOPF</monospace> as a Deep Neural Network (DNN) approach for solving security-constrained direct current optimal power flow (SC-DCOPF) problems, which are critical for reliable and cost-effective power system operation. <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepOPF</monospace> is inspired by the observation that solving SC-DCOPF problems for a given power network is equivalent to depicting a high-dimensional mapping from the load inputs to the generation and phase angle outputs. We first train a DNN to learn the mapping and predict the generations from the load inputs. We then directly reconstruct the phase angles from the generations and loads by using the power flow equations. Such a predict-and-reconstruct approach reduces the dimension of the mapping to learn, subsequently cutting down the size of the DNN and the amount of training data needed. We further derive a condition for tuning the size of the DNN according to the desired approximation accuracy of the load-generation mapping. We develop a post-processing procedure based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> -projection to ensure the feasibility of the obtained solution, which can be of independent interest. Simulation results for IEEE test cases show that <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepOPF</monospace> generates feasible solutions with less than 0.2% optimality loss, while speeding up the computation time by up to two orders of magnitude as compared to a state-of-the-art solver.

Unit commitment direct current optimal power flow using mixed-integer linear programming

This paper presents unit commitment direct current optimal power flow (UCDCOPF) using mixed-integer linear programming (MILP) for daily power system generation planning. This method is used to solve optimum operation in an electric power system by scheduling generator status and calculating the optimal power output of each generator in a power system. The calculation of this method considers several constraints in the power system. A modified IEEE 24 bus system is used as a test system to show the ability of the proposed approach.

Data quality aware chance‐constrained DC‐OPF: a variational Bayesian Gaussian mixture approach

The contamination of outliers severely damages the data quality, resulting in the inaccurate data-driven optimisation model. This study proposes a data quality aware chance-constrained model for the direct current optimal power flow (DC-OPF) problem under uncertainties. Under the framework of Bayesian statistics, the variational Bayesian Gaussian mixture model (VBGMM) is employed to extract the probabilistic information from the available historical data, i.e. realisations of random variables. VBGMM can identify the outliers by capturing their probability characteristics, in which way improving the data quality. Notably, VBGMM automatically determines the number of components, which is a remarkable difference from the conventional Gaussian mixture model. In addition, based on the affine policy, a method integrating VBGMM with chance-constrained programming is proposed to make VBGMM scalable. The proposed method is firstly tested on a 6-bus system for an illustrative purpose, and then on a 118-bus system for validating the potential practical application. Comparative studies verify the effectiveness of the proposed method.

Accelerated and Robust Analytical Target Cascading for Distributed Optimal Power Flow

Augmented Lagrangian-based distributed algorithms, such as analytical target cascading (ATC), may converge slowly, oscillate around the optimal point, or diverge if penalty multipliers are not selected appropriately or the level of importance of objective terms is not balanced. This article presents accelerated, robust ATC (AR-ATC) to solve the optimal power flow (OPF) distributedly. A function is designed to determine a balancing coefficient whose incorporation into ATC makes a balance between the important level of terms of objective functions. If penalty multipliers are initialized to large values, the proposed function creates a balancing coefficient to avoid the premature convergence or divergence. A balancing coefficient will be created to reduce the number of iterations if penalty multipliers are set to small values. Mathematical justifications and simulation studies are performed to analyze the effectiveness of AR-ATC. Potential applications of the proposed method on auxiliary problem principle, alternating direction method of multipliers (ADMM), and Nesterov-based ADMM are also studied numerically. Simulations are performed for direct current optimal power flow and alternating current optimal power flow problems.

Optimal planning of renewable energy systems for power loss reduction in transmission expansion planning

Purpose The purpose of this study is to address the efficiency of power losses representation while still reducing the computational burden of an optimal power flow (OPF) model in transmission expansion planning (TEP) studies. Design/methodology/approach A modified TEP model is formulated with inclusions of linearized approximation of power losses for a large-scale power system with renewable energy sources. The multi-objectives function determines the effect of transmission line losses on the optimal power generation dispatch in the power system with and without inclusion of renewable energy sources with emphasis on minimizing the investment and operation costs, emission and the power losses. Findings This study investigates the impact of renewable energy sources on system operating characteristics such as transmission power losses and voltage profile. Sensitivity analysis of the performance for the developed deterministic quadratic programming models was analyzed based on optimal generated power and losses on the system. Research limitations/implications In the future, a comparison of the alternating current OPF and direct current (DC) OPF models based on the proposed mathematical formulations can be carried out to determine the efficiency and reduction of computation process of the two models. Practical implications This paper proposed an accurate way of computing transmission losses in DC OPF for a TEP context with a view of achieving a minimal computation time. Originality/value This paper addresses the following objectives: develop a modified DC OPF with a linearized approximation of power losses in TEP problem with large integration of RES. Investigate the impact of RES on system operating characteristics such as transmission power losses and voltage profile.

A potential security threat and its solution in coupled urban power-traffic networks with high penetration of electric vehicles

The growing penetration of electric vehicles (EVs) and the popularity of fast charging stations (FCSs) have greatly strengthened the coupling of the urban power network (PN) and traffic network (TN). In this paper, a potential security threat of PN-TN coupling is revealed. Different from traditional loads, a regional FCS outage can lead to both the spatial and temporal redistribution of EV charging loads due to EV mobility, which further leads to a power flow redistribution. To assess the resulting potential threats, an integrated PN-TN modeling framework is developed, where the PN is described by a direct current optimal power flow model, and the TN is depicted by an energy-constraint traffic assignment problem. To protect the privacy of the two networks, an FCS outage distribution factor is proposed to describe the spatial-temporal redistribution ratio of the charging load among the remaining FCSs. Moreover, to protect the security and efficiency of the coupled networks, a price-based preventive regulation method based on the spatial demand elasticity of the EV charging load is developed to reallocate the charging load as a solution to insecure situations. Numerical simulation results validate the existence of the PN-TN coupling threat and demonstrate the effectiveness of the regulation method to exploit the spatial flexibility of EV loads.

A Distributed Dual Consensus ADMM Based on Partition for DC-DOPF With Carbon Emission Trading

This article presents a distributed alternating direction method of multipliers (ADMMs) approach for solving the direct current dynamic optimal power flow with carbon emission trading (dc-DOPF-CET) problem. Generally, the ADMM-based distributed approaches disclose boundary buses and branches information among adjacent subsystems. As opposed to these methods, the proposed method (dc-ADMM-P) adopts a novel strategy which uses consensus ADMM to solve the dual of dc-DOPF-CET while only discloses boundary branches information among adjacent subsystems. Moreover, the convergence performance of dc-ADMM-P is improved by reducing the number of dual multipliers and employing an improved update step of the multiplier. DC-ADMM-P is tested on cases ranging from 6 to 1062 buses, with comparison with other distributed/decentralized methods. The simulation results verify the high efficiency of dc-ADMM-P in solving the dc-DOPF problem with complex (nonlinear) factors which can be formulated as convex separable functions. Meanwhile, it also shows the improvement of convergence performance by reducing the number of dual multipliers and employing a new update strategy for the multiplier.

Exploring the Profitability and Efficiency of Variable Renewable Energy in Spot Electricity Market: Uncovering the Locational Price Disadvantages

While variable renewable energy (VRE) has been developed for decades, VRE market participation is developing relatively slowly, despite the potential economic efficiency it may bring. This paper tries to specify the efficiency of VRE in a deregulated pool-based electricity market. Based on standard pool-based market design, this paper built a direct current optimal power flow (DC-OPF) based simplified 2-settlement spot electricity market model conjugating electricity and ancillary service clearing. To address the outcomes of the imperfect market in the real world, this paper studied the consequences brought by agents’ learning and strategic behaviors. Simulations under different ancillary service levels and reliability cost levels are carried out. The results show that VRE may be unprofitable in the market, especially when learning and strategic behavior is considered. Learning and strategic market behavior will also hamper the role of VRE as a “better” energy source. This paper shows and proves a locational marginal price (LMP) disadvantage phenomenon, which will lead to low profitability of VRE. Three major suggestions are given based on the results.

Distributed Optimal Power Flow in Hybrid AC–DC Grids

Distributed or multi-area optimal power flow (OPF) in alternating current (AC) grids is currently a subject undergoing intense study to cope with computational burdens in large-scale grids and to maintain self-control of a regional system operator. However, future power grids will most likely be hybrid grids consisting of the conventional AC transmission system combined with high voltage direct current (DC) technology. Thus, we reformulate the full AC–DC OPF problem such that it becomes separable and, therefore, accessible to distributed algorithms. Then, we show in detail two different approaches on the decomposition of a hybrid AC–DC grid. Finally, we implement an improved alternating direction of multipliers method (ADMM) as well as the most recently proposed augmented Lagrangian based alternating direction inexact Newton (ALADIN) method. Simulation results show that optimality gaps below 0.01% are reached with both decomposition approaches and algorithms for two different test systems (5-bus and 66-bus). Furthermore, convergence rates and wall clock times are reduced by around one order of magnitude from ADMM to ALADIN.