Current Flow Equations Research Articles

Probabilistic power flow based on physics-guided graph neural networks

Under the high-dimensional and nonlinear stochastic power system environment, artificial intelligence (AI) is becoming a promising alternative to the urgent demand for probabilistic power flow (PPF). However, traditional AI only learns from experiences in data and is unable to satisfy physical constraints. This may result in an undesired infeasible inference for PPF. To solve this issue, a physics-guided graph neural network (PG-GNN)-based PPF analysis method is proposed. At the outset, alternating current power flow equation is analyzed to produce gradients for connecting physical component and AI. The proposed PG-GNN then incorporates the sensitivities into the training tensor graph such that it enables in-depth physical learning. At the same time, the GNN serves to enhance generalizability in varying topology. The well-trained PG-GNN is finally leveraged to surrogate traditional power flow solver in Monte-Carlo method, such that real-time PPF is realized. Numerical tests on benchmarks demonstrate that, in unseen scenarios, the proposed method enables only little gap versus the exact baseline, but beats other rivals with over tenfold computing efficiency. Moreover, comparisons with traditional AIs manifest that our method holds better generalizability.

Robust Allocation of FACTS Devices in Coordinated Transmission and Generation Expansion Planning considering Renewable Resources and Demand Response Programs

Nowadays, the use of demand response programs (DRPs) in a variety of long-term and short-term planning problems has been explored. In this paper, a generation and transmission expansion planning (GTEP) model along with FACTS device allocation is presented. Furthermore, demand response programs are taken into account for more load flexibility. The proposed model is presented as a multi-objective minimizing problem considering emission, cost, and voltage security index. Furthermore, the conventional Pareto optimization is adopted using fuzzy weighted sum method (FWSM) to achieve a single-objective model. The final problem is constrained by equations of alternative current power flow, operation and voltage security limits, planning model of shunt FACTS devices, and operation of the DRP. Adaptive robust optimization (ARO) is used to reach suitable models for the active power of renewable resources and power consumption. As a main search algorithm, a hybrid combination of water cycle algorithm (WCA) and ant lion optimization (ALO) is proposed to find the optimum solution with a small standard deviation. The problem is tested on different standard IEEE systems, and the results validate the operation and network security improvement due to optimal location of FACTS devices. According to the results, the economic and environmental status of the network has also improved.

An MISOCP-Based Decomposition Approach for the Unit Commitment Problem with AC Power Flows

Unit Commitment (UC) and Optimal Power Flow (OPF) are two fundamental problems in short-term electric power systems planning that are traditionally solved sequentially. The state-of-the-art mostly uses a direct current (DC) approximation of the power flow equations. However, utilizing the DC approach in the UC-level may lead to infeasible or suboptimal generator commitment schedules for the OPF problem. In this paper, we aim to simultaneously solve the UC Problem with alternating current (AC) power flow equations, which combines the challenging nature of both UC and OPF Problems. Due to the highly nonconvex nature of the AC flow equations, we utilize the mixed-integer second-order cone programming (MISOCP) relaxation of the UC Problem as the basis of our solution approach. The MISOCP relaxation is utilized for finding both a lower bound and a candidate generator commitment schedule. Once this schedule is obtained, we solve a multi-period OPF problem to obtain feasible solutions for the UC problem with AC power flows. For smaller instances, we develop two different algorithms that exploit the recent advances in the OPF literature and obtain high-quality feasible solutions with provably small optimality gaps. For solving larger instances, we develop a Lagrangian decomposition based approach that yields promising results.

Green Micro-Grid Operation Constrained to Reliability and Flexibility Indices in the Presence of Distributed Generations and Energy Storage Systems

This paper presents microgrid (MG) operation constrained to the reliability, flexibility, and environment indices in the presence of distributed generations (DGs) and energy storage systems (ESSs). The proposed scheme minimizes the total expected operating cost of MGs and DGs. It is also subject to alternating current (AC) power flow equations of MGs, constraints of operation, reliability, and flexibility in MG, and operation model of power sources and storage devices. Stochastic programming is incorporated to model uncertainties of load, energy price, the active power of renewable energy generation, availability of MG equipment, sources, and storage devices. Following on, a hybrid solver formed by combining artificial bee colony (ABC) and sine-cosine algorithm (SCA) is adopted to achieve the optimal solution with approximate conditions of unique ultimate response. Eventually, the suggested scheme is implemented on a 69-bus radial MG, where the numerical results confirm the capability of the scheme in improving the operation, reliability, flexibility, and environment status of the MG.

Analysis of a bulk-surface thermistor model for large-area organic LEDs

The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the threedimensional bulk glass substrate and two semilinear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder’s fixed-point theorem. Moreover, since the heat sources are a priori only in L^1 , the concept of entropy solutions is used.

Optimal Planning of Charging Stations in Coupled Transportation and Power Networks Based on User Equilibrium Conditions

In this article, we propose a new bi-level approach for the location and sizing of charging stations (CSs) based on the integration of power distribution and traffic networks. At the lower level, the user choices and the energy demands are modeled through the user equilibrium (UE) traffic assignment conditions that are derived for the case of electric vehicles (EVs). At the higher level, an optimization problem is formalized for the optimal planning of locations, sizes, and unit prices of a set of new charging stations in a territory, which embeds the UE conditions and the power flow model as constraints. The main novelties lie in the integration of traffic and electric networks, the optimization problem formalization, and the inclusion of detailed electrical issues [such as the complete alternating current (AC) power flow equations and the modeling of CSs with different phases]. The developed model has been applied to a case study in the Genoa Municipality. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In the past decade, particular attention has been posed on EVs and renewable resources to reduce pollutants’ emissions. Critical issues regard the setting of adequate recharging infrastructure and the integration of huge distributed and intermittent loads and production into the electrical grid. This article aims to provide a bi-level optimization model that can be used by companies or public administrations that want to install charging stations in a territory. In particular, the transportation and the electrical networks are integrated: at a lower level, the UE traffic assignment conditions are derived to represent users’ choices and to assess the energy demand. At the upper level, an optimization problem is formalized, to minimize costs of installation and power losses and to maximize benefits, which embeds as constraints the UE conditions for the case of EVs and the model of the electrical grid (load flow equations, bounds over-voltage, apparent power, etc.).

Ensemble Learning Based Convex Approximation of Three-Phase Power Flow

Though the convex optimization has been widely used in power systems, it still cannot guarantee to yield a tight (accurate) solution to some problems. To mitigate this issue, this paper proposes an ensemble learning based convex approximation for alternating current (AC) power flow equations that differs from the existing convex relaxations. The proposed approach is based on three-phase quadratic power flow equations in rectangular coordinates. To develop this data-driven convex approximation of power flows, the polynomial regression (PR) is first deployed as a basic learner to fit convex relationships between the independent and dependent variables. Then, ensemble learning algorithms such as gradient boosting (GB) and bagging are introduced to combine learners to boost model performance. Based on the learned convex approximation of power flow, optimal power flow (OPF) is formulated as a convex quadratic programming problem. The simulation results on IEEE standard cases of both balanced and unbalanced systems show that, in the context of solving OPF, the proposed data-driven convex approximation outperforms the conventional semi-definite programming (SDP) relaxation in both accuracy and computational efficiency, especially in the cases that the conventional SDP relaxation fails.

Design and optimization of the shape of electrostatic precipitator system

The electro hydrodynamic (EHD) flows in “Optimization of The Shape of Electrostatic Precipitator System” which also known as “Cup shape W type electrostatic precipitator system” is simulated on Ansys Fluent 6.1. The “Cup shape” and W type electrostatic precipitator System is comparing with old flat plate Electrostatic precipitator system. According to theoretical analysis model particle flow is analyze by time-averaged conservation equations of mass and momentum, the electric field and current flow equations. The W-type collecting surface space charge density is more uniform rather than the flat plate. It is seen that on W- type circular wire plate number of trap particle is more than the flat plate type, whereas number of escapes is more in flat type. Which state that increasing the voltage of this design increase the efficiency of the system. In Cup design electrode release greater number of ion or electric field than the circular type electrode. Together using of w type and cup type electrode increase the ionization and collection area, which improve the efficiency of the ESP design. Separate work on practical design already done in previous work for Cup shape electrode with the flat collecting plate design and w type collecting plate with circular wire which give much better results.

Dimension reduction of thermistor models for large-area organic light-emitting diodes

&lt;p style='text-indent:20px;'&gt;An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ p(x) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \varepsilon&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.&lt;/p&gt;

Identifying critical risks of cascading failures in power systems

Potential critical risks of cascading failures in power systems can be identified by exposing those critical electrical elements on which certain initial disturbances may cause maximum disruption to power transmission networks. In this work, we investigate cascading failures in power systems described by the direct current (DC) power flow equations, while initial disturbances take the form of altering admittance of elements. The disruption is quantified with the remaining transmission power at the end of cascading process. In particular, identifying the critical elements and the corresponding initial disturbances causing the worst-case cascading blackout is formulated as a dynamic optimization problem (DOP) in the framework of optimal control theory, where the entire propagation process of cascading failures is put under consideration. An Identifying Critical Risk Algorithm (ICRA) based on the maximum principle is proposed to solve the DOP. Simulation results on the IEEE 9-Bus and the IEEE 14-Bus test systems are presented to demonstrate the effectiveness of the algorithm.

Linear optimal power flow using cycle flows

Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe a new formulation that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, in settings that include computationally challenging applications such as multi-period LOPF with storage dispatch and generation capacity expansion. We show that the new formulation of the LOPF solves up to 7 times faster than the angle formulation using a commercial linear programming solver, while another existing cycle-base formulation solves up to 20 times faster, with an average speed-up of factor 3 for the standard networks considered here. If generation capacities are also optimized, the average speed-up rises to a factor of 12, reaching up to factor 213 in a particular instance. The speed-up is largest for networks with many buses and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation and the computational challenge of operation and planning in such networks.

Impact of optimised distributed energy resources on local grid constraints

Abstract Optimisation models have been extensively used for finding optimal configuration and operation of distributed energy technologies. The main objective in most of these models is to find the optimal configuration of distributed energy technologies that will meet a certain energy demand with the least cost and emissions. Local grid constraints are not considered in the optimisation of distributed energy resources in most of these models. This implies that some optimal solutions from these models may not be possible to integrate due to a violation of steady state voltage and thermal limits which are important to Distribution Network Operators (DNO). In some cases, where a joint optimisation approach is utilised and local grid constraints are considered, it becomes computationally complex due to the nonlinear nature of Alternating Current (AC) power flow equations for electricity networks. In this paper, the impact of optimised Distributed Energy Resources (DER) on a modelled microgrid was evaluated with an AC time series power flow using a soft-linking method. The soft-linking method avoids the computationally complex nature of joint optimisation methods. Different scenarios of the optimised DER were simulated and evaluated based on voltage excursions and energy losses. The results provide insights into the impact of local grid constraints on the adoption of different scenarios of optimised DER.

Results of natural gas parameters influence on processes that occur in turbine flow meters

In the article criteria of possible turbine flow meters calibrations with natural gas parameters fluctuations are determined Article presents experimentation results of natural gas parameters influence on processes that occur in turbine flow meters. In the article criteria of possible turbine flow meters calibrations with natural gas parameters fluctuations are determined. It is known that metrological parameters of the turbine flow meter in operating systems will deter from certificated values. With pressure, temperature or other fluctuations of a flow physical parameters the flow meter's results will occur within of a error space determined by Reynolds number equation for current flow passing through turbine grille. With Reynolds numbers constant aspect ratio the relative error of a flow meter will stay irrelevant of type of the environment. It is suggested to use value as a criterion that is not affected by thermodynamic parameters and physical characteristics of a environment but of turbine grille model and mechanical state of a flow meter . Turbine flow meter SM-RI-X-KG1000, DN200 was used for experiment with flow volume varies of 80 m 3 /h to 1600 m 3 /h and pressure varies of 100 kPa to 700 kPa. Results of theoretical calculations and experimental research data for Reynolds number ratio is shown on a graph of a turbine flow meter speed on pressure dependency. It is determined that flow meter designed for low-pressure environment should be calibrated for actual range of operating environment pressure and temperature values. ДСТУ EN 12261:2006. «Лічильники газу турбінні. Загальні технічні умови», that allows usage of gas flow meter designed for 100mbar environment in set up with 4bar of absolute gas pressure should be corrected. For improving of quality and precision of gas flow measurement, calibration of gas flow meters, designed for 5bar of absolute gas pressure, should be performed in an actual working environment keeping in mind range of realistic pressure and temperature fluctuations. Calibrations of gas flow meters that designed for gas pressure environment exceeding 5bar could be performed with environment that keeps defined Reynolds numbers ratio constant.

Thermistor systems of &lt;i&gt;p&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;)-Laplace-type with discontinuous exponents via entropy solutions

We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (ⅰ) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ⅱ) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

Analysis of [formula omitted]-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current–voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a p(x)-Laplace-type problem with piecewise constant exponent.We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)-Laplacian. Finally, Schauder’s fixed-point theorem is used to show the existence of solutions.

Appearance of effective surface conductivity: An experimental and analytic study

Surface conductance measurements on p-type doped germanium show a small but systematic change to the surface conductivity at different length scales. This effect is independent of the structure of the surface states. We interpret this phenomenon as a manifestation of conductivity changes beneath the surface. This hypothesis is confirmed by an analysis of the classical current flow equation. We derive an integral formula for calculating of the effective surface conductivity as a function of the distance from a point source. Furthermore we derive asymptotic values of the surface conductivity at small and large distances. The actual surface conductivity can only be sampled close to the current source. At large distances, the conductivity measured on the surface corresponds to the bulk value.

Linear Current Flow Equations With Application to Distribution Systems Reconfiguration

Conventionally, power flow equations are used for distribution systems (DS) analysis to find the nodal voltages. For the particular form of the DS reconfiguration problem, however, a direct formulation in terms of branch flows allows a substantial increase in solution efficiency from an optimization point of view. In this paper, a set of linear current flow (LCF) equations are derived for DS. This formulation is then used within the network reconfiguration problem for loss minimization. A mixed-integer quadratically constrained programming (MIQCP) formulation, together with a mixed-integer linear programming (MILP) formulation, are proposed in this paper and assessed through simulations. In these comparisons, the MILP formulation shows computational advantages over the MIQCP version and the preceding literature. The proposed methods are evaluated on several test systems.

Particulate removal via electrostatic precipitators — CFD simulation

In this study, a CFD model for the description of operation of electrostatic precipitators has been developed. The model is based on a theoretical analysis focused on the gas flow by time-averaged conservation equations of mass and momentum, on particulate flow and particle charging, and on the electric field and current flow equations. The model takes into consideration all the processes involved inside an industrial ESP, namely the fully coupled three coexisting fields of gas flow, particle dynamics and electrostatics. The model was extensively applied to a 2D geometry provided with detail: (a) the gas flow field (pressure, velocities, turbulence), (b) the particle history (trajectories, charge, velocities, residence time, fate, etc.), as well as particle-associated Eulerian variables (concentration, charge density, etc.), and (c) electrostatic parameters (potential, strength, ion current density).

Time-dependent Onsager equations and perturbation propagation

The diffusion of perturbations is investigated, beginning with Onsager-type equations for particle current and energy flow and conservation equations. This method takes into consideration the interplay of density and temperature, and the phenomena produced by this interplay—often neglected with other approaches—are fully accounted for. Time evolution of density and temperature profiles are presented. The speed of propagation of perturbation is calculated.

Constriction resistance of microcone-based contacts

Based on a recently developed semiconductor processing technique for field emitters, molybdenum cones are made on the surface of contact members to achieve low constriction resistance. A model was set up to calculate the constriction resistance of such a cone-based contact interface. The constriction resistance is obtained from the solution to a linear system of current flow equations for the contact structure. The effects of surface features such as number, size and distribution of microcontacts, and cluster area on the constriction resistance were evaluated by a computer simulation. The least-square optimization of numerical results led to an empirical expression which relates the constriction resistance to the cone radius, cone density and cluster area. The model predicts the reduction of constriction resistance as a function of cone density, and it agrees well with the experimental results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>