Linear Flow Model Research Articles

Modeling and optimal scheduling of battery energy storage systems in electric power distribution networks

Thanks to the unique features, deployment of battery energy storage systems in distribution systems is ever-increased. Therefore, new models are needed to capture the real-life characteristics. Beside active power, the battery energy storage system can exchange reactive power with the grid due to the inverter-based connection. Although some previous works have considered this issue, a detailed linear model suitable for the realistic large scale distribution systems is not addressed adequately. In this context, this paper proposes a mixed integer linear programming model for optimal battery energy storage system operation in distribution networks. The proposed model considers various parts of the battery energy storage system including battery pack, inverter, and transformer in addition to linear modeling of the reactive power and apparent power flow limit. Moreover, a linear power flow model is used to calculate voltage magnitudes and power losses with high accuracy. The proposed model is applied to the IEEE 33-bus test case and the results prove the accuracy and efficiency of the proposed model. The results demonstrate that considering reactive capability of the batteries offers new benefits including voltage profile improvement, decreasing reactive power flow in the network, reducing network losses, and releasing network and substation capacity.

Modeling of heterogeneous reservoirs with damaged hydraulic fractures

The significance of unconventional low-permeability reservoir development has been well recognized in recent years. Pressure- and rate-transient behavior analyses are crucial methods of finding key factors that affect well production in these formations. However, previous analytically or semi-analytically based gridless models of pressure- and rate-transient response analyses for multi-stage fractured horizontal wells (MFHWs) mainly deal with homogeneous reservoirs and merely use skin factors to describe the fracture damage. Fracture damage normally refers to the conductivity reduction within hydraulic fractures and the matrix permeability or porosity impairment near the fracture face. The widely applied skin factors are derived under steady-state conditions and are accurate only when the damage length is small. They also show inability to simulate partially propped fractures and multiple fracture damage scenarios.In this article, a general linear flow model is established to simulate heterogeneous box-shaped unconventional low-permeability reservoirs with closed outer boundaries and various hydraulic fracture damage, including (1) fracture near-wellbore damage caused by proppant flow-back, over-displacement, embedment, crush or ineffective placement, (2) partially propped fractures with undamaged near-wellbore regions, (3) fracturing fluid leak-off damage in the adjacent matrix, (4) dual or multiple damage effects under unsteady-flow conditions. The fracture interference in heterogeneous reservoirs is modeled by our previous method (Zeng et al., 2018).The basic model has been verified against well-testing software KAPPA, and satisfactory agreements are obtained. New type curves have been generated with distinct flow regimes for specific fracture damage mechanisms. Sensitivity analyses have been conducted to find out how the damage properties affect the shape and duration of these distinct regimes. Reservoir heterogeneity combined with fracture damage makes the pressure and rate behavior deviate significantly from the undamaged homogeneous one and controls the fracture flux distribution in certain regimes during production for both constant-rate and constant-pressure production cases. In addition, field data from the Cardium unconventional light oil formation in western Canada are analyzed through type curve matching, which demonstrates the benefits of our model. The findings of this model are potentially helpful for field data analyses and fracture damage diagnostic.

Temporal Fluctuations and Poroelasticity Can Generate Chaotic Advection in Natural Groundwater Systems

AbstractAlthough steady, isotropic Darcy flows are inherently laminar and nonmixing in the absence of diffusion, it is well understood that transient forcing via engineered pumping schemes can induce rapid, chaotic mixing flows in groundwater. In this study we explore the propensity for such mixing to arise in natural groundwater systems subject to cyclical forcings, for example, tidal or seasonal influences. Using a conventional linear groundwater flow model subject to tidal forcing, we show that under certain conditions these flows generate Lagrangian transport and mixing phenomena (chaotic advection) near the tidal boundary. We show that aquifer heterogeneity, storativity, and forcing magnitude cause reversals in flow direction over the forcing cycle which, in turn, generate coherent Lagrangian structures and chaos. These features significantly augment fluid mixing and transport, leading to anomalous residence time distributions, flow segregation, and the potential for profoundly altered reaction kinetics. We define the dimensionless parameter groups which govern this phenomenon and explore these groups in connection with a set of well‐characterized tidal systems. The potential for Lagrangian chaos to be present near discharge boundaries must be recognized and assessed in field studies.

A Nonlinear Analytical Algorithm for Predicting the Probabilistic Mass Flow of a Radial District Heating Network

This paper develops a nonlinear analytical algorithm for predicting the probabilistic mass flow of radial district heating networks based on the principle of heat transfer and basic pipe network theory. The use of a nonlinear mass flow model provides more accurate probabilistic operation information for district heating networks with stochastic heat demands than existing probabilistic power flow analytical algorithms based on a linear mass flow model. Moreover, the computation is efficient because our approach does not require repeated nonlinear mass flow calculations. Test results on a 23-node district heating network case indicate that the proposed approach provides an accurate and efficient estimation of probabilistic operation conditions.

A General Formulation of Linear Power Flow Models: Basic Theory and Error Analysis

Linear power flow models are widely used in power systems to simplify the nonlinear power flow equations. The DC power flow model is one of the representatives. There are many other linear power flow models that improve the DC power flow model with the inclusion of $\boldsymbol{Q}$ and $\boldsymbol{v}$ . However, existing linear models are derived based on empirical mathematical approximation without a general methodology guidance. In this paper, we found that the fundamental difference among different linear power flow models lies in the formulation of “independent variables.” Based on this finding, a general formulation of linear power flow models is proposed. The linearization error is theoretically analyzed. In particular, the case when $\boldsymbol{ \theta }$ and $\boldsymbol{v^{\scriptscriptstyle k}}$ are regarded as independent variables is thoroughly investigated. Method for finding the linear power flow with the minimum error is presented. The formulation of the independent variables associated with the minimum linearization error is determined by the distribution of state variables $\boldsymbol{v}$ and $\boldsymbol{\theta }$ . It is shown that the linearization error when $\boldsymbol{v^{\scriptscriptstyle 2}}$ is regarded as an independent variable is normally smaller than that for $\boldsymbol{v}$ because of the special properties of the distribution of $\boldsymbol{v}$ in power grids.

Boundary-integral representations for ship motions in regular waves

Alternative boundary-integral representations of linear potential flow around a ship that travels at a constant speed in regular (time-harmonic) waves in deep water are considered. In particular, the Neumann–Michell (NM) boundary-integral representation of steady flow around a ship traveling in calm water is extended to ship motions in regular waves. This extension is straightforward, but points to a complicated and unresolved fundamental issue that is essentially related to the formulation of a linear flow model. Indeed, the NM boundary-integral flow representation for ship motions in regular waves given in the study suggests that the classical Neumann–Kelvin (NK) flow model may be inadequate at a ship waterline. Alternative NM boundary-integral representations of time-harmonic ship waves that are consistent with the previously given NM representation of ship waves in calm water, and yield Fourier–Kochin representations of ship waves that are well suited for numerical computations, are given.

Real-time dispatchable region of renewable generation constrained by the reactive power and voltage profiles in AC power networks

A large amount of renewable energy generation (REG) has been integrated into power systems, challenging the operational security of power networks. In a real-time dispatch, system operators need to estimate the ability of the power network to accommodate REG with a limited reserve capacity. The real-time dispatchable region (RTDR) is defined as the largest range of a power injection that the power network can accommodate in a certain dispatch interval for a given dispatch base point. State-of-the-art research on the RTDR adopts a DC power flow model regardless of the voltage profiles and reactive power, which can overlook potentially insecure operational states of the system. To address this issue, this paper proposes an AC power flow based RTDR model simultaneously considering the reactive power and voltage profiles constraints. The nonlinear constraints in our model are approximated using a linear power flow model together with a polytope approximation technique for quadratic constraints. An adaptive constraint generation algorithm is used to calculate the RTDR. Simulation results using the IEEE 5-bus and 30-bus systems illustrate the advantages of the proposed model.

A Loop-Analysis Theory Based Linear Power Flow Method for Three-Phase Distribution Power System

The linear power flow (LPF) models are particularly important in the context of optimization algorithms for three-phase distribution systems with the high penetration of distributed renewable generations. With several approximations on network modelling, voltage drop calculation, link power calculation, voltage controlled bus and ZIP loads, this paper proposed a three-phase LPF model for three-phase unbalanced distribution networks based on loop-analysis theory. The method can deal with voltage-controlled (PV) & Load (PQ) buses and ZIP loads. The approximations of the star and delta connections ZIP loads are also analyzed. Test results based on several standard IEEE test feeders and an improved 615bus test system proved the effectiveness and accuracy of the proposed algorithm. The proposed LPF solution gives a simple, more robust, and potentially faster solution for modern distribution power system.

Energy management optimization of a smart wind power plant comparing heuristic and linear programming methods

This paper aims at optimizing the energy management of a smart power plant composed of wind turbines coupled with a Lithium Ion storage device in order to fulfill a power production commitment to the utility grid. The application of this case study is typically related to islanded electric grids. Our work particularly investigates and compares two classes of energy management strategies for design purpose: a first capable of providing the global optimum of the power flow planning from a Linear Programming (LP) approach thanks to a priori knowledge of future events in the environment; a second, based on a classical control heuristic without any a priori knowledge on the future, applicable in real time. Beyond the future objectives in terms of system design (techno-economical sizing optimization), the comparison of both approaches also aims at improving the predefined heuristic from the analysis of the ideal reference provided by the global LP optimizer. In this scope, a linear power flow model of the power plant is developed in compliance with a LP solver (Cplex). A particular attention is paid to the techno-economic optimization including storage cost evaluation, commitment failure penalties and exploitation gains. Simulations and optimizations are carried out over one year in order to take variability and seasonal features of the wind potential into account.

Pressure-rate convolution and deconvolution response for fractured conventional and unconventional reservoirs using new decline rate model

Abstract This paper introduces new approach for pressure-rate convolution and deconvolution analysis of multi-stages hydraulically fractured conventional and unconventional reservoirs. This approach demonstrates the impact of variable Sand face flow rate on reservoir performance. A new model for P/R deconvolution is used to convert pressure pulse from variable flow rate to single and constant rate response. The target of this study is fractal reservoirs with and without stimulated and unstimulated reservoir volume. Multi-linear flow regimes approach is used to describe pressure behavior in the reservoirs while decline flow rate behavior is described by newly proposed model in this study. This model depicts, instead of van Everdingen model, indirectly the declining rate with time by using pressure responses with production time. Decline flow rate behavior simulated by linear and bi-linear flow models are also studied and compared with the one obtained by the new model. Several analytical models are used in this study by applying P/R convolution and deconvolution technique and solved for constant and variable flow rate considering different reservoir configurations and operating conditions. The results are interpreted and analyzed for better understanding pressure behaviors, flow regime types, and productivity index trends for continuously changing flow rate especially at early production time. Estimating stimulated reservoir volume ( V s r v ) is considered one of the applications of convolved pressure since it is calculated from pseudo-steady state flow when late time boundary dominated flow regime is reached. The outcomes of this study can be summarized as: 1) Introducing new approach for pressure-rate convolution and deconvolution technique for multi-stages hydraulically fractured reservoirs by applying new decline flow rate model that indirectly simulates variable flow rate with time. 2) Generating analytical models for dimensionless pressure and flow rate for constant and variable flow rate using the concept of P/R convolution and deconvolution. 3) Comparing the result of newly proposed models with the results obtained by applying van Everdingen model for decline rate behavior. 4) Studying the applicability of linear and bi-linear flow models in converting variable flow rate pressure response to single and constant flow rate pressure response. 5) Applying the deconvolution technique to simulate pressure response at late production time to estimate stimulated reservoir volume. The most interesting points are: 1) The main difference in wellbore pressure behavior between variable and constant flow rate can be seen at early production time, however intermediate production time could also show very limited changes for the case of variable rate wellbore pressure. 2) A unit slope line flow regime could be developed for varied flow rate pressure response at very early production time similar to the wellbore storage dominated flow regime. 3) Productivity index calculated by the proposed models for variable flow rate is greater than the index for constant flow rate. 4) The impact of petrophysical properties of porous media and hydraulic fracture characteristics on pressure response are similar in the two cases of variable and constant flow rate. 5) The decline rate models for linear and bi-linear flow are not applicable in pressure deconvolution technique.

Evaluation of the Vertical Producing Degree of Commingled Production via Waterflooding for Multilayer Offshore Heavy Oil Reservoirs

Recently, commingling production has been widely used for the development of offshore heavy oil reservoirs with multilayers. However, the differences between layers in terms of reservoir physical properties, oil properties and pressure have always resulted in interlayer interference, which makes it more difficult to evaluate the producing degree of commingled production. Based on the Buckley–Leverett theory, this paper presents two theoretical models, a one-dimensional linear flow model and a planar radial flow model, for water-flooded multilayer reservoirs. Through the models, this paper establishes a dynamic method to evaluate seepage resistance, sweep efficiency and recovery percent and then conducts an analysis with field data. The result indicates the following: (1) the dynamic difference in seepage resistance is an important form of interlayer interference during the commingled production of an offshore multilayer reservoir; (2) the difference between commingled production and separated production is small within a certain range of permeability ratio or viscosity ratio, but separated production should be adopted when the ratio exceeds a certain value.

Optimal Power and Gas Flow With a Limited Number of Control Actions

Current research regarding integrated optimal power and gas flow (OPGF) requires the adoption of a relatively large number of control actions, which is undesirable in practice owing to time and communication constraints. The present study therefore proposes OPGF with a limited number of control actions (denoted as OPGF-LNC). Imposing a bound on the number of control actions inevitably introduces binary variables, and the resulting OPGF-LNC with nonlinear power/gas flow models is a mixed integer nonlinear programming (MINLP) problem. To avoid the poor numerical performance of MINLP, dc power flow model and a two-stage piecewise linear gas flow model are employed. Consequently, the OPGF-LNC problem is simplified to a mixed-integer linear programming (MILP) problem. Furthermore, the tradeoff between control costs and the allowed number of control actions is analyzed, and soft constraint relaxation is deployed when satisfactory results are unattainable using all possible control actions. Simulation results using an IEEE 39-node system and a 48-node gas network verify that the MILP approach always produces reliable optimal solutions while MINLP is divergent in some cases. In addition, substantial benefits are demonstrated with the implementation of reasonable soft constraint relaxation.

Distributionally Robust Chance-Constrained Approximate AC-OPF With Wasserstein Metric

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In the presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate ac-OPF. The power flow model employed in the proposed OPF formulation combines an exact ac power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on the IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate ac model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties.

Effective Dynamic Scheduling of Reconfigurable Microgrids

This paper develops an effective model for microgrid optimal scheduling with dynamic network reconfiguration. Network reconfiguration can effectively alter local power flow and thus provide an opportunity to reduce microgrid distribution network losses during grid-connected operation (supporting microgrid economic objectives) and to reduce potential load curtailments during the islanded operation (supporting microgrid reliability objectives). The proposed optimal scheduling model is decomposed into a grid-connected operation master problem and an islanded operation subproblem. A novel and highly accurate dynamic linear power flow model, with the ability of line switching, is developed and included in both problems. The optimal schedule determined in the master problem is assessed to meet the microgrid islanding feasibility in the subproblem. If infeasible, the decision variables are amended using the islanding cuts, which will accordingly revise the network reconfiguration, as well as the schedule of dispatchable units, energy storage, and adjustable loads. The simulation results on a test microgrid demonstrate the effectiveness and satisfying performance of the proposed model.

Mathematical model of fractured horizontal well in shale gas reservoir with rectangular stimulated reservoir volume

Production performance analysis of shale gas development has posed a great challenge for scientific researchers owing to the complex percolation mechanisms and massive hydraulic fracturing. The unique accumulation and transporting mechanisms in multiscaled shale pores, such as adsorption, desorption, diffusion, and slippage, lead to gas seepage behaviors deviating from Darcy flow. Compared to previous studies, this paper presents a comprehensive mathematical model incorporating all the above percolation mechanisms for multistage fractured horizontal wells, in which rectangular stimulated reservoir volume (SRV) is taken into account. A numerical solution for such a complex model is obtained by extending the boundary element method to application. The verification of our method is confirmed by comparison to semi-analytical results of a simplified model. From the well testing type curves, nine flow regimes were identified, most of which are not observed in conventional linear flow models. The effects of sensitive parameters, such as fracture number, size and permeability of the SRV, and formation permeability, on pressure type curves were investigated. The comprehensive model in this paper provides another method to guide the development of shale gas reservoirs, including interpretation of formation properties, guiding fracturing design, and conducting post-fracture evaluation.

Boundary integral representations of steady flow around a ship

The classical Neumann–Kelvin (NK) linear potential flow theory of ship waves in calm water and the related Neumann–Michell (NM) theory are considered. Five alternative boundary integral representations are given: (1) the classical NK integro-differential representation, called “classical NK formulation”, which corresponds to an inconsistent linear flow model, (2) a modification of the classical NK flow formulation that corresponds to a consistent linear flow model and is called “consistent NK formulation”, (3) a flow representation, called “NM potential and velocity formulation”, that involves the flow potential ϕ and the velocity components ϕd and ϕt along two unit vectors d and t tangent to the ship hull surface, and yields an integro-differential equation for determining ϕ, (4) a flow representation, called “NM velocity formulation”, that only involves ϕd and ϕt and yields a pair of coupled integral equations for determining (ϕd,ϕt), and (5) a flow representation called “NM potential formulation” that only involves the flow potential ϕ and yields an integral equation for determining ϕ. The two NK formulations involve both a surface integral over the ship hull surface and a line integral around the ship waterline, whereas the three NM formulations do not involve a waterline integral. All flow representations other than the classical NK representation are based on a consistent linear flow model.

Real-time energy-efficient traffic control via convex optimization

This article proposes a macroscopic traffic control strategy to reduce fuel consumption of vehicles on highways. By implementing Greenshields fundamental diagram, the solution to Moskowitz equations is expressed as linear functions with respect to vehicle inflow and outflow, which leads to generation of a linear traffic flow model. In addition, we build a quadratic cost function in terms of vehicle volume to estimate fuel consumption rate based on COPERT model. A convex quadratic optimization problem is then formulated to generate energy-efficient traffic control decisions in real-time. Simulation results demonstrate significant reduction of fuel consumption on testing highway sections under peak traffic demands of busy hours.

Market Clearing and Settlement Using Participant Based Distributed Slack Optimal Power Flow Model for a Double Sided Electricity Auction Market – Part II

Even though non-linear Optimal Power Flow model proposed in part I of this paper provide accurate results there are difficulties in solving a full non-linear Optimal Power Flow model since it is a very time consuming. Due to speed and robustness, Linear Programming model is preferred by the system operators for nodal price calculations. It is a challenge to incorporate transmission losses into the losses linear DC power model. In this paper to model line flow operating limit inequality constraint a lossless Participant Based Distributed Slack DC Power Flow model is derived from Newton-Raphson state correction scheme used in part I of this work. An equivalent lumped linear model for the Participant Based Distributed Slack Lumped Nonlinear Optimal Power Flow model called Participant Based Distributed Slack Lumped Linear Optimal Power Flow model is developed in part II of this paper. The results for market clearing and settlement of double sided electricity market by the proposed lumped linear model are compared with the nonlinear model proposed in part I of this work using case studies on PJM system, IEEE 30 bus system and IEEE 118 bus system. The results obtained indicate the speed and robustness of the proposed linear model.

Market Clearing and Settlement Using Participant Based Distributed Slack Optimal Power Flow Model for a Double Sided Electricity Auction Market –Part I

AC and DC OPF models are widely used in the industry for market clearing and settlement of primary electricity market. Research works available in literature have used locational marginal price to price electricity. But this research work by using Participant Based Distributed Slack Power Flow and a unique delivery/withdrawal point for energy called “Market Center” provides a new framework for pricing the electricity. Using the Participant Based Distributed Slack Power Flow Model a Participant based Distributed Slack Non Linear Optimal Power Flow model for market clearing and settlement of primary transaction in a double sided electricity auction market is proposed in Part 1 of this paper. Part II of this paper proposes an equivalent lumped linear model for faster convergence. The optimal solution gives the optimal schedules of the individual participants. The formulae to decompose the amount payable to GENCO participant and marginal price to be received from the DISCO participant into its components are also derived which is different from all available techniques in literature which calculate bus wise nodal prices. The proposed model is tested on Radial Five bus system and PJM system for market clearing and settlement of primary electricity market.

Numerical simulation of wave resonance in the narrow gap between two non-identical boxes

Wave resonance in the narrow gap between two side-by-side non-identical boxes is investigated by employing a two-dimensional numerical wave flume based on the OpenFOAM® package. The focus of this study is to examine the influence of the energy transformation and the energy dissipation on the hydrodynamic behavior of wave response around resonant conditions. Numerical simulations show that the unrealistic wave resonant responses in the narrow gap by the linear potential flow model are due to not only the energy dissipation induced by the fluid rotational motion, but also the energy transformation associated with the free surface. With the increase of incident wave amplitude, relatively more energy is reflected, leading to the decrease of wave resonant response and energy dissipation in the narrow gap at the resonant frequency. When slightly away from the resonant frequency, the energy dissipation becomes the dominant factor for the decrease of wave response in the narrow gap with increasing the incident wave amplitude. As for the influence of gap configuration, on one hand, energy dissipation has the dominant effect for the typical case of small upstream and large downstream box drafts. On the other hand, the reflected energy is more important for the typical large upstream and small downstream box drafts. More resonant fluid exists in the gap with the increase of gap breadth, leading to the decrease of reflection coefficient and the increase of transmission coefficient.