Linearized Power Flow Model Research Articles

An Accelerated Stackelberg Game Approach for Distributed Energy Resource Aggregator Participating in Energy and Reserve Markets Considering Security Check

With increasing distributed energy resoures (DERs) integration, the strategic behavior of a DER aggregator in electricity markets will significantly affect the secure operation of the distribution system. In this paper, the interactions among the DER aggregator, energy and reserve markets, and distribution system are investigated through a single-leader-multi-follower Stackelberg game model with the DER aggregator as the leader and the independent system operator and distribution system operator as the followers. To guarantee the operation security of the distribution system, security check problems under three different scenarios are involved in the follower level, which is linearized using a mixed-integer linearized power flow model. Then, using the strong duality theorem, the proposed model is converted into a bi-level mixed-integer linear (BMILP) programming model with only mixed-integer linear follower-level problems. Next, an accelerated relaxation-based bi-level reformulation and decomposition algorithm is proposed to solve the BMILP problem. Finally, case studies are carried out on a constructed integrated transmission and distribution (T&D) system and a practical integrated T&D system to verify the effectiveness of the proposed model and algorithm. The simulation results indicate that the available downward reserve of the DER aggregator will decrease with the security limitation of the distribution system.

An Accelerated Stackelberg Game Approach for Distributed Energy Resource Aggregator Participating in Energy and Reserve Markets Considering Security Check

With increasing distributed energy resoures (DERs) integration, the strategic behavior of a DER aggregator in electricity markets will significantly affect the secure operation of the distribution system. In this paper, the interactions among the DER aggregator, energy and reserve markets, and distribution system are investigated through a single-leader-multi-follower Stackelberg game model with the DER aggregator as the leader and the independent system operator and distribution system operator as the followers. To guarantee the operation security of the distribution system, security check problems under three different scenarios are involved in the follower level, which is linearized using a mixed-integer linearized power flow model. Then, using the strong duality theorem, the proposed model is converted into a bi-level mixed-integer linear (BMILP) programming model with only mixed-integer linear follower-level problems. Next, an accelerated relaxation-based bi-level reformulation and decomposition algorithm is proposed to solve the BMILP problem. Finally, case studies are carried out on a constructed integrated transmission and distribution (T&D) system and a practical integrated T&D system to verify the effectiveness of the proposed model and algorithm. The simulation results indicate that the available downward reserve of the DER aggregator will decrease with the security limitation of the distribution system.

MicroGrid Resilience-Oriented Scheduling: A Robust MISOCP Model

This article introduces a Robust Mixed-Integer Second Order Cone Programming (R-MISOCP) model for the resilience-oriented optimal scheduling of microgrids (MGs). This is developed for MGs that are islanded due to a scheduled interruption from the main grid, where minimizing both operational costs and load shedding is critical. The model introduced presents two main benefits. Firstly, an accurate second order cone power flow model (SOC-PF) is used, which ensures global optimality. Through a comparison with a piecewise linear power flow model on a modified IEEE 33 bus network, it is demonstrated that failure to accurately model power flow equations, can result in a significant underestimation of the operational cost of almost 12%. Secondly, uncertainty is modelled using a robust approach which allows trade-offs between the uncertainty that a MG operator is willing to tolerate, and performance. In this article, performance criteria considered are operational cost and load shedding. Market price, demand, renewable generation and islanding duration are considered as uncertain variables. Results show that by controlling the budget of uncertainty, the MG operator can achieve an almost 20% reduction in the operating cost, compared to a fully robust schedule, while achieving 0% probability of shedding more demand than expected.

A Relaxed PV Bus Model in Linear Power Flow

The emerging multimodal active energy resource units make it necessary to introduce constant voltage amplitude buses (PV bus) in the operation of modern distribution systems. To enable the analysis of PV bus using linear power flow models, this letter proposes a relaxed PV bus model for solving power flow related problems using an indirect modeling approach. The developed PV bus model can be applied to linear power flow models in both rectangular and polar coordinates. Besides, the approximation error of the relaxed PV bus model can also be controlled and quantified. Lastly, the proposed relaxed PV bus model is validated using different test systems. According to the numerical studies, the maximum error on the PV bus approximation in all case studies is only 0.61%, and the maximum error of the supporting power at the PV buses is 6.5%.

Exploiting the Operational Flexibility of Wind Integrated Hybrid AC/DC Power Systems

Due to inherent uncertainty and volatility, large-scale wind power integration brings new challenges to the flexible operation of power systems. It calls for more flexibility not only from the generation side but also from the grid side, especially the hybrid HVAC/HVDC transmission grid. This paper proposes to exploit potential flexibility by controlling power flows through HVDC by the merits of its flexible regulation capability, as well as HVAC transmission switching (TS). To synergistically schedule them with generator units, a security-constrained economic dispatch (SCED) model for wind integrated hybrid AC/DC power system is presented to realize the co-optimization of generation and TS. Wherein, multiple uncertainties (wind power fluctuation and generator failure) are considered, and the linear power flow model of the DC grid is adopted. Then the model is transformed into a two-stage (normal state optimization and corrective dispatch verification) robust optimization (RO) model for iterative solution by the aid of the column-and-constraint generation (C&CG) algorithm. Case studies on the modified IEEE 14-bus and IEEE 118-bus systems show that TS can assist HVDC to optimize power flow distribution and provide more flexibility, meanwhile optimizing HVDC transmission power and TS can greatly reduce the operating cost of the whole system.

On the minimal set of controllers and sensors for linear power flow

We consider a linear power flow model with interval-bounded nodal power injections and limited line power flows. We determine the minimal number of power injections to control based on a minimal set of measurements, such that the overall system is feasible for all assignments of the non-controlled power injections. For the important case where the possible measurements are the nodal power injections, we show that the problem can be solved efficiently as a mixed-integer linear program (MILP). When also line power flows are considered as potential measurements, we derive an iterative, greedy algorithm that provides a feasible, but potentially conservative solution. We apply the developed algorithms to both a simple microgrid and a modified version of the IEEE 118 bus test power system. We show that in both cases a sparse solution in terms of the number of required controllers and measurements can be obtained. Moreover, the number of required measurements can be reduced significantly if line flow measurements are considered additionally to nodal power injections.

Minimize Linearization Error of Power Flow Model Based on Optimal Selection of Variable Space

Linear power flow models are widely used in power system analysis because it brings huge computational benefits, especially for power system optimization problems. The improvement of the linearization accuracy is greatly beneficial to the power system operation considering the large amount of power scheduled based on linear power flow models. We find that the linearization error substantially changes with different selections of the variable space. In this paper, we formulate a model to find the linear power flow model with the minimized linearization error based on the optimal selection of the variable space. The expression of the variable space is generalized as a polynomial function. A simplified model of power losses is proposed. The difference between the selection of the variable space and the common hot-start approaches is illustrated. The effectiveness of the proposed linear power flow model is verified by power flow calculation and optimal power flow (OPF) calculation in the IEEE and Polish test systems.

Preventive and Curative Actions by Meshed Bipolar HVDC-Overlay-Systems

With the introduction of high voltage direct current (HVDC)-system, especially the voltage source converter (VSC)-based implementation, new degrees of freedom have been made accessible for system operation and for ensuring system security where they can be utilized either in the form of preventive or curative actions. This work presents an optimization model meant to determine a set of preventive and curative relief measures. Thereby, both methods are not calculated in a successive approach, but the optimization of the curative actions is nested into the preventive optimization. By doing so, a feedback can be given in cases the preventive optimization is evolving in regions where no feasible curative actions can be provided. A numerical case study is conducted, where the calculation of the curative set-points is performed with a linearized power flow model. Thereby, an effect is shown, where preventive insecure contingencies appear as curative feasible as the linearized model does not show any violated constraints.

A loop-analysis theory based power flow method and its linear formulation for low-voltage DC grid

As in alternative-current (AC) networks, the power flow (PF) task is the backbone of direct-current (DC) networks with the development of DC power system. In this paper, it proposes a loop-analysis theory based iterative power flow method for the low-voltage DC (LVDC) power grids. And further the linear power flow model via Taylor's series expansion has also been developed. Based on the node-branch incidence matrix, the relationship between the branch powers and node injection powers has been established. The formation of the proposed iterative method and its linear approximation are derived based on the branch powers rather than the injected currents. ZIP loads and V buses (multi-slack buses) have been taken into account. Simulation results verified the accuracy and speed of the proposed methods in comparison with the classical Newton-Raphson method and two other existing linear methods for both radial and meshed networks.

Gradient-Based Multi-Area Distribution System State Estimation

The increasing distributed and renewable energy resources and controllable devices in distribution systems make fast distribution system state estimation (DSSE) crucial in system monitoring and control. We consider a large multi-phase distribution system and formulate DSSE as a weighted least squares (WLS) problem. We divide the large distribution system into smaller areas of subtree structure, and by jointly exploring the linearized power flow model and the network topology, we propose a gradient-based multi-area algorithm to exactly and efficiently solve the WLS problem. The proposed algorithm enables distributed and parallel computation of the state estimation problem without compromising any performance. Numerical results on a 4,521-node test feeder show that the designed algorithm features fast convergence and accurate estimation results. Comparison with traditional Gauss-Newton method shows that the proposed method has much better performance in distribution systems with a limited amount of reliable measurement. The real-time implementation of the algorithm tracks time-varying system states with high accuracy.

Linearizing Power Flow Model: A Hybrid Physical Model-Driven and Data-Driven Approach

Linear power flow model is advantageous for the fast operational analysis and the efficient optimization of the power systems. In this letter, we propose a hybrid physical model-driven and data-driven approach for linearizing power flow model. In this proposed approach, the linear power flow model contains two parts, i.e., the existing physical-equation-based linear power flow model and the linearized error model. The linearized errors are obtained by the partial least squares regression based data-driven approach. The proposed linear power flow model can retain the useful inherent information from the physical model and utilize the ability of data analysis to extract the inexplicit linear relationship. Simulations on the four test systems have validated that the proposed hybrid linear model exhibits a much better performance on the branch power flow calculation than other linear power flow models.

Conic Optimization for Electric Vehicle Station Smart Charging With Battery Voltage Constraints

This article proposes a new convex optimization strategy for coordinating electric vehicle charging, which accounts for battery voltage rise and the associated limits on maximum charging power. Optimization strategies for coordinating electric vehicle charging commonly neglect the increase in battery voltage, which occurs as the battery is charged. However, battery voltage rise is an important consideration since it imposes limits on the maximum charging power. This is particularly relevant for dc fast charging, where the maximum charging power may be severely limited, even at the moderate state of charge levels. First, a reduced-order battery circuit model is developed, which retains the nonlinear relationship between the state of charge and maximum charging power. Using this model, limits on the battery output voltage and battery charging power are formulated as the second-order cone constraints. These constraints are integrated with a linearized power flow model for three-phase unbalanced distribution networks. This provides a new multiperiod optimization strategy for electric vehicle smart charging. The resulting optimization is a second-order cone program and, thus, can be solved in polynomial time by standard solvers. A receding horizon implementation allows the charging schedule to be updated online, without requiring prior information about when vehicles will arrive.

Estimating DLMP confidence intervals in distribution networks with AC power flow model and uncertain renewable generation

This study proposes a systematic approach for estimating intervals of distribution locational marginal price (DLMP) and corresponding confidence levels without requiring an exact probability distribution of renewable generation. The DLMP is evaluated based on a variant of the alternating current (AC) power flow model, namely, the branch flow model, in which network losses, bus voltage, and reactive power are taken into account. Based on the exactness of convex relaxation of optimal power flow, the authors developed a linear AC power flow model by applying second-order cone relaxation and global polyhedral approximation. Given a set of renewable power volatility, interval prediction of the DLMP is formulated as a bilevel linear program, which is solved via a mixed-integer linear program (MILP). Considering variances and unimodality of renewable power forecast error, a conservative estimation of the confidence level is expressed by a generalised Gauss inequality, which comes down to semidefinite programming. The proposed method is a natural extension of existing interval and probabilistic forecast methods, leveraging the proposed linear AC power flow model and inexact probability distributions of renewable power output, and could be promising in future distribution power markets. Case studies corroborate the effectiveness of the proposed method.

Transmission-Line Switching for Load Shed Prevention via an Accelerated Linear Programming Approximation of AC Power Flows

Topology control has garnered recent consideration for its value in leveraging the flexiblity of the power system. However, using even the latest technology, the AC optimal transmission switching (ACOTS) problem remains intractable. As such, most research addressing optimal topology control is based on the DCOPF, which has been shown to be particularly inaccurate for transmission switching decisions in the AC power system. This paper accelerates an existing mixed-integer linear power flow model which has been shown to be very accurate in approximating AC power flows. The aim of the new model is to identify transmission switching candidates to prevent load shed caused by contingency events. The methodology was tested on the IEEE 118-bus test case system and presents promising results. The model proposed herein was found to almost always identify AC optimal switching actions while significantly reducing the necessary solution time in comparison with the base model.

Linear three-phase power flow model for autonomous microgrid systems

In this work, the authors propose a linear three-phase power flow model for droop-controlled autonomous microgrids. The proposed linear power flow model adopts a set of linear algebraic equations and can, with small errors, be compared to the detailed nonlinear model, account for the lack of a slack bus, the variability of autonomous microgrid frequency and the operation of distributed generation units in droop control mode. The proposed linear model can as such provide a fast and accurate power flow solution and has great potential in the stochastic planning and online operational applications of autonomous microgrids, as well as in linear or convex optimal power flow and probabilistic load flow analysis. The accuracy and effectiveness of the proposed linear power flow model are validated through different case studies considering balanced and unbalanced test systems.

Optimal scheduling of distribution network incorporating topology reconfiguration, BES and load response: A MILP model

Distributed generation (DG) is becoming increasingly important due to the serious environmental pollution caused by conventional fossil-energy-based generation and the depletion of non-renewable energy. As the flexible resources in the active distribution network (ADN), battery energy system (BES) and responsive load (RL) are all able to assist renewable DG integration in day-ahead dispatch. In addition, the security and economic level can be well improved by adjusting network topology. Therefore, in this paper, a coordinated day-ahead scheduling method incorporating topology reconfiguration, BES optimization and load response is presented to minimize the total day-ahead operation cost in the ADN. Linearized current injection models are presented for renewable DG, RL and BES based on the linear power flow model, and an extensible linear switching operations calculation (ELSOC) method is proposed to address the network reconfiguration. Thus, a mixed integer linear programming (MILP) model is proposed to optimal coordinated operation of ADN. The correctness and effectiveness of the proposed method are demonstrated by the simulations on the modified test system. In addition, the combined scenario and Monte-Carlo method is used to handle the uncertainties of loads and DGs, and the results of different uncertainties can further verify the feasibility of the proposed model.

Distributionally Robust Optimal Reactive Power Dispatch with Wasserstein Distance in Active Distribution Network

The uncertainties from renewable energy sources (RESs) will not only introduce significant influences to active power dispatch, but also bring great challenges to the analysis of optimal reactive power dispatch (ORPD). To address the influence of high penetration of RES integrated into active distribution networks, a distributionally robust chance constraint (DRCC)-based ORPD model considering discrete reactive power compensators is proposed in this paper. The proposed ORPD model combines a second-order cone programming (SOCP)-based model at the nominal operation mode and a linear power flow (LPF) model to reflect the system response under certainties. Then, a distributionally robust optimization (WDRO) method with Wasserstein distance is utilized to solve the proposed DRCC-based ORPD model. The WDRO method is data-driven due to the reason that the ambiguity set is constructed by the available historical data without any assumption on the specific probability distribution of the uncertainties. And the more data is available, the smaller the ambiguity would be. Numerical results on IEEE 30-bus and 123-bus systems and comparisons with the other three-benchmark approaches demonstrate the accuracy and effectiveness of the proposed model and method.

Decoupled piecewise linear power flow and its application to under voltage load shedding

Various optimizations in power systems based on the AC power flow model are inherently mixed-integer nonlinear programming (MINLP) problems. Piecewise linear power flow models can handle nonlinearities and meanwhile ensure a high accuracy. Then, the MINLP problem can be turned into a tractable mixed-integer linear programming (MILP) problem. However, piecewise linearization also introduces a heavy computational burden because of the incorporation of a large number of binary variables especially for large systems. To achieve a better tradeoff between approximation accuracy and computational efficiency, this paper proposes a model named decoupled piecewise linear power flow (DPWLPF) for transmission systems. The P-Q decoupling characteristic is used to ease the evaluation of the piecewise cosine functions in the power flow equations. Therefore, in optimizations, the coupling between variables is reduced. Moreover, an under voltage load shedding (UVLS) approach based on DPWLPF is presented. Case studies are conducted for benchmark systems. The results show that the DPWLPF facilitates the solution of optimal power flow (OPF) and UVLS problems than conventional piece-wise models. And DPWLPF still enhances the approximation accuracy by the decoupled piecewise modeling.

On Linear Analysis of the Power Flow Equations for DC and AC Grids With CPLs

This express brief presents an approximation of the power flow problem for alternating-current (ac) and directcurrent (dc) distribution networks by using a linear representation of the hyperbolic constraints i = p/v ↔ II* = ℤ/V related to the power balance at each constant power load node. Taylor's or Laurent's series expansion methods are not required to obtain an equivalent linear power flow model. The proposed linear method allows us to achieve a high quality approximation of the power flow modeling without iterative procedures. Our simulation results show the accurate estimation of the voltage profile in distribution networks by the proposed linear approach in comparison to existing methods in specialized literature for ac and dc networks, including linear estimators or classical numerical methods, such as Gauss-Seidel and Newton-Raphson approaches. Numerical implementation of those approaches is carried out in the MATLAB 2017a programming environment.

A linear power flow model for balanced distribution network with droop-controlled DSTATCOM and voltage controlled DG

Fast and accurate voltage profile estimation is of great importance for secure operation of distribution systems. In this paper, we propose a linear power flow model for calculating the voltage profile of balanced distribution networks in presence of the droop-controlled ideal DSTATCOM, the voltage-controlled distributed generation, and the voltage-dependent loads. The advantage of our proposed linear model is that both the voltage controlled units and the loads with voltage-dependent characteristic can be well considered. The good performance of the proposed model is validated on the modified 33-bus distribution network and the modified 874-bus distribution network.