Interval Power Flow Research Articles

A support vector regression-based interval power flow prediction method for distribution networks with DGs integration

In distribution networks with distributed generators (DGs), power generation and load demand exhibit increased randomness and volatility, and the line parameters also suffer more frequent fluctuations, which may result in significant state shifts. Existing model-driven methods face challenges in efficiently solving uncertain power flow, especially as the size of the system increases, making it difficult to meet the demand for rapid power flow analysis. To address these issues, this paper proposes an SVR-based interval power flow (IPF) prediction method for distribution networks with DGs integration. The method utilizes intervals to describe system uncertainty and employs Support Vector Regression (SVR) for model training. The input feature vector consists of the intervals of active power generation, load demand, and line parameters, while the output feature vector represents the intervals of voltage or line transmission power. Ultimately, the SVR-based IPF prediction model is established, capturing the linear mapping relationship between input data and output IPF variables. Simulation results demonstrate that the proposed method exhibits high prediction accuracy, strong adaptability, and optimal computation efficiency, meeting the requirements for rapid and real-time power flow analysis while considering the uncertainty in distribution networks with DGs integration.

A Data-Driven Approach for Reactive Power Optimization Incorporating Interval Values for Renewable Power Generation

The increasing integration of renewable energy sources into modern electric grids has led to a rise in uncertain factors that must be managed to maintain voltage security during reactive power optimization (RPO). Traditional deterministic RPO methods fail to account for these uncertainties, which can result in power grid security issues such as voltage violations. To address these challenges, this paper proposes a data-driven interval-based reactive power optimization method (IRPOM). The IRPOM represents the uncertainties associated with renewable power generation and load demands as intervals within the RPO problem formulation. The proposed method uses an improved particle swarm optimization algorithm to solve the RPO problem. In each iteration, the uncertain power flow is solved using the optimizing-scenarios method- (OSM-) based interval power flow (IPF) algorithm. This approach calculates the real power losses and checks whether state quantities, including voltage, power flow, and generator output, exceed their limits. Furthermore, a data-driven modeling approach is introduced to reduce the conservativeness of the IRPOM solutions. The effectiveness of the proposed method is demonstrated through detailed computational analysis on a modified IEEE 30-bus system. The results show that the proposed approach ensures economic efficiency while maintaining a low bus voltage threshold crossing probability close to zero.

An interval power flow method for radial distribution systems based on hybrid second-order cone and linear programming

Interval power flow (IPF) analysis is a promising approach to handling the uncertainty of renewable energy sources and loads in power systems. However, most existing studies are based on standard interval arithmetic or affine arithmetic. Little work can be found that employs traditional optimization methods to obtain the interval results, especially based on convexified power flow formulation. Therefore, this paper proposes an IPF method based on a hybrid second-order cone and linear programming for radial distribution systems. First, the optimization process of IPF based on the standard power flow model is introduced. Given the non-convexity of the standard power flow model, an IPF framework based on second-order cone programming (SOCP) for radial distribution systems is proposed. Moreover, considering that the SOCP formulation can only achieve half of the whole IPF process, a revised linear DistFlow formulation is adopted, and the IPF method based on a hybrid second-order cone and linear programming is devised. Finally, the proposed IPF method is compared with the standard power flow equation-based IPF method and the Monte Carlo method. The simulation results on modified IEEE 33-node and 69-node distribution systems demonstrate the effectiveness and prospects of the proposed IPF method.

Lifting Model-Based Interval Analysis for DC Power Flow in Presence of Uncertain Network Reactance and Power Injections

This paper develops a novel interval power flow model in the presence of both uncertain network reactance and power injections. To reduce the interval growth of the interval operators, the traditional DC power flow equations are lifted into a higher dimensional space, where the admittance matrix is split into three parts and interval linear equations are with a higher dimension. Furthermore, an iteration method is designed to solve the proposed model and the convergence is strictly proved and guaranteed. Numerical studies on several test systems show the effectiveness of the proposed model.

Study on Dynamic Interval Power Flow Calculation of Microgrid Based on Monte Carlo Algorithm

In order to effectively monitor the stability of the microgrid, based on the advantages of the Monte Carlo algorithm, a dynamic interval power flow calculation method for microgrid is designed. First, based on the multilayer complex structure of the microgrid, the hierarchical topology of its interval structure is analyzed. Then, the micropower flow model is designed, the affine algorithm is used to accurately describe the relationship between the variables in the microgrid structure, and the dynamic interval affine calculation is completed. Therefore, in the dynamic interval of the microgrid, the Monte Carlo probability method is applied to obtain more random data. Based on this, a model is established for the probability of wind power generation and photovoltaic power generation to simulate the output characteristics of the microgrid. Finally, the voltage variable, active power variable, and reactive power variable are calculated and embedded in the iterative algorithm to realize the stochastic power flow calculation of the microgrid. After experimental verification, the voltage amplitude calculated using the method proposed in this article has a small error value compared to the actual voltage, with a minimum error value of 0.2, close to 0. Under the condition of convergence accuracy of 10−6, the minimum convergence frequency is 3 times, and the power flow calculation process time does not exceed 38 seconds. This proves that the algorithm has high calculation accuracy, good convergence performance and timeliness, and can provide certain technical support for the stable operation of microgrids.

Optimal Allocation and Sizing of Distributed Generation Using Interval Power Flow

Modern distribution systems and microgrids must deal with high levels of uncertainty in their planning and operation. These uncertainties are mainly due to variations in loads and distributed generation (DG) introduced by new technologies. This scenario brings new challenges to planners and system operators that need new tools to perform more assertive analyses of the grid state. This paper presents an optimization methodology capable of considering uncertainties in the optimal allocation and sizing problem of DG in distribution networks. The proposed methodology uses an interval power flow (IPF) that adds uncertainties to the combinatorial optimization problem in charge of sizing and allocating DG units in the network. Two metaheuristics were implemented for comparative purposes, namely, symbiotic organism search (SOS) and particle swarm optimization (PSO). The proposed methodology was implemented in Python® using as benchmark distribution systems the IEEE 33-bus and IEEE 69-bus test distribution networks. The objective function consists of minimizing technical losses and regulating network voltage levels. The results obtained from the proposed IPF on the tested networks are compatible with those obtained by the PPF, thus evidencing the robustness and applicability of the proposed method. For the solution of the optimization problem, the SOS metaheuristic proved to be robust, since it was able to find the best solutions (lowest losses) while keeping voltage levels within the predetermined range. On the other hand, the PSO metaheuristic showed less satisfactory results, since for all test systems, the solutions found were of lower quality than the ones found by the SOS.

A Central Limit Theorem-Based Method for DC and AC Power Flow Analysis Under Interval Uncertainty of Renewable Power Generation

This paper proposes a central limit theorem-based method (CLTM) to overcome the conservatism of interval DC and AC power flow analysis under uncertainty of renewable power generation. Interval DC power flow (IDCPF) models are solved by expressing the bus angle and active transmission power as linear combinations of interval nodal power injections, and then the central limit theorem is applied to obtain high-confidence intervals of DC power flow variables. Interval AC power flow (IACPF) models are solved by first applying the optimizing-scenarios method to acquire more accurate affine arithmetic forms of the power flow variables defined according to linear combinations of nodal power injections, and then high-confidence intervals of AC power flow variables are obtained via the central limit theorem. In addition, a criterion formulation is established to evaluate the errors of interval power flow methods. The results of simulations validate the effectiveness and superiority of the proposed method relative to the performances of previously established methods, including the Monte Carlo simulation, affine arithmetic-based method and optimizing-scenarios method.

A surrogate-assisted point estimate method for hybrid probabilistic and interval power flow in distribution networks

The uncertain power flow analysis is essential to assessing the operating states of distribution networks under various uncertainties. The power flow calculations considering the single uncertainty have been well developed. However, different types of uncertainties may coexist in power systems. In order to comprehensively assess the impacts of mixed uncertainties, this paper proposes a surrogate-assisted point estimate method to solve the hybrid probabilistic and interval power flow. By taking advantage of the point estimate method and high-fidelity surrogate model, the proposed method can accurately obtain the relevant results of power flow calculation with high efficiency. The numerical studies in IEEE 33-bus and 141-bus systems verify the accuracy and efficiency of the proposed method by comparing it with other well-known methods.

Fully Decentralized Robust Modelling and Optimization of Radial Distribution Networks Considering Uncertainties

Fully decentralized optimization of multi-agent distribution networks considering uncertainties is essential to improving user-side energy utilization efficiency and flexibility, whereas a contradictory centralized coordinator aware of system-level information is inevitably introduced in existing researches. To address the problem, this paper introduces the flexibility boundaries of nodes to express their adjustability under uncertainties and constructs a flexibility transition model to express their neighbourhood relationship. Besides, a robust interval power flow model is established to consider the stochastic impact of generation on nodal voltage through neighbourhood information exchange. Based on the above two models, the robust optimization problem is established to minimize the baseline operation cost and maximize the allowable generation limits of non-dispatchable renewable energy sources. The model involves no global information and is solved with alternating direction method of multipliers(ADMM) in a fully decentralized way. Case study on a modified IEEE 33-bus and a 118-bus system is presented and the proposed method is compared with conventional multi-level robust formulations. The results suggest the effectiveness and correctness of the newly proposed method.

An interval power flow for unbalanced distribution systems based on the Three-Phase Current Injection Method

Interval arithmetic has been widely used for power systems analysis considering uncertainties associated with load and generation data. However, little work can be found within the context of three-phase distribution systems due to their peculiarities such as radial topology and unbalanced loads. In this paper, a novel methodology is proposed for determining interval results of a power system load flow for three-phase unbalanced distribution networks based on the Three-Phase Current Injection Method (TPCIM) in which state variables are considered in rectangular coordinates. In the proposed approach, active and reactive powers at each load bus are modelled as interval values to represent their inherent uncertainties and the Krawczyk operator is applied into the power flow equations in order to provide reliable interval three-phase results. Additionally, the use of interval extensions and an angular rotation technique are proposed to overcome overestimation problems associated with the interval solutions. Computational simulations are carried out using IEEE 13, 33 and 69-bus test systems. The main contribution of this work is the proposition of two methods based on traditional interval arithmetic to reduce the diameter of the solution and achieve similar results when compared with Affine Arithmetic and Monte Carlo Simulations. Additionally, the computational time associated with the algorithm is extremely advantageous. The method is useful for power distribution systems operation and planning studies, representing a viable and useful tool for calculating the impact of uncertain input data on the power flow results.

A novel affine power flow method for improving accuracy of interval power flow data in cyber physical systems of active distribution network

A novel affine power flow method for improving accuracy of interval power flow data in cyber physical systems of active distribution network

Interval power flow analysis of microgrids with uncertainties: an approach using the second-order Taylor series expansion

Interval power flow analysis of microgrids with uncertainties: an approach using the second-order Taylor series expansion

Intending Island Service Restoration Method With Topology-Powered Directional Traversal Considering the Uncertainty of Distributed Generations

The intending island service restoration method is one of the core technologies of self-healing control for smart distribution systems, which aims to maximize the restoration of the out-of-service loads in the out-of-service area without faults quickly. For this reason, a topology-weighted directional traversal intending island recovery method considering the uncertainty of distributed generation sources is proposed. First, divide the network level of the power-loss feeder and calculate the interval power flow of the feeder before the fault, and obtain the power flow direction when the active and reactive power of the faulted branch is the smallest so as to determine whether the distributed generation supply in the non-faulty power-loss area can restore all load power supply. If not, to determine the island recovery plan, continue to compare the distributed generation supply and the load capacity at all levels, and give priority to recovering loads with higher importance levels and smaller network levels. The traversal of the topological authority and direction effectively reduces the island recovery time and can make full use of the distributed generation output to maximize the recovery of the non-faulty power-loss area. Taking the PG&E 69-node system as an example and using the BFGS trust region algorithm to calculate the island power flow without unbalanced nodes for verification, the results show that this method consumes less time and can restore more load power supply than the existing island recovery method, which verifies the method’s effectiveness and reasonability.

Distribution Network Dynamic Reconfiguration Based on Improved FuzzyC‐MeansClustering with Time Series Analysis

The rapid growth of distributed energy resources integrated in distribution systems leads to an increasing need of continuously and automatically changing the system topology to realize the economic operation of distribution networks. This paper proposes an optimization model of dynamic reconfiguration for distribution networks based on a new method of time series analysis. Equivalent daily curve considering time‐varying nature of distributed generator and load demands is divided by an improved fuzzy C‐means clustering algorithm, where the indicator of section function is set to find the optimal reconfiguration time intervals. The uncertainty of distributed generator outputs and load demands is described by the interval algorithm. Then the affine Taylor expansion is adopted to solve the interval power flow equations. The reconfiguration optimization model is solved with decimal particle swarm optimization algorithm based on loop search. The optimal dynamic reconfiguration of a modified 70‐bus test system with distributed generators is carried out and the simulation results demonstrate the effectiveness and superiority of the proposed method. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Reactive power optimization under interval uncertainty of renewable power generation based on a security limits method

With the integration of high penetration renewable power generation, the voltage security of power systems is threatened. The reactive power optimization including interval uncertainty (RPOIU) model is applied for developing a voltage control strategy to ensure that the state variables of a power grid reside within their safe operating limits under uncertainty of renewable power generation and load demand input data. However, the RPOIU model is not solved properly in terms of solution efficiency and optimization effect at present. This paper proposes a novel method for solving the RPOIU model, which is denoted as the security limits method (SLM). The proposed SLM first computes the security limits of the state variables of the RPOIU model using an interval power flow method, i.e., the optimizing-scenarios method, and then transforms the RPOIU model into a deterministic model whose constraints conform to the established security limits. The deterministic model is then efficiently solved using the interior point method. We verify that the voltage control strategy obtained by solving the deterministic model is able to ensure that the values of the state control variables satisfy the constraints of the original RPOIU model, but with more conservative security limits. This issue is addressed by modifying the security limits of the SLM through defining an interval ratio, and applying an iterative predictor-corrector procedure. The result obtained by the proposed SLM and modified SLM based on numerical simulations are compared with those obtained by a previously proposed method for effectively solving the RPOIU model, and analysis of the results demonstrates the advantages, effectiveness, and good applicability of the proposed SLM.

Probability box theory-based uncertain power flow calculation for power system with wind power

Abstract The uncertainty of wind speed may lead to the deviation and change of wind power output, which influences the stability of wind farm. Therefore, in this paper, a probability box (p-box) based uncertain power flow model for wind power is proposed, which initially introduces p-box to power flow calculation. A probabilistic interval power flow model with both probability and interval is established. Firstly, the drift interval of wind speed is obtained and its p-box model is established by analyzing the distribution of wind speed. Secondly, the wind power output p-box is derived from the wind speed p-box based on the relationship between wind power output and wind speed, then the p-box of wind power output is discretized and introduced into the power flow equation to obtain the power flow p-box model. Finally, Newton–Raphson method is used to solve the power flow p-box model. Experiments on data collected from a wind farm (running standard IEEE30-bus test system) in Inner Mongolia demonstrate that our method is more effective and accurate than the traditional Monte Carlo simulation (MCS) and classical interval power flow (IPF) method.

Research on interval multi-objective optimal power flow in AC-DC systems considering wind power fluctuation

Aiming at the influence of the uncertainty of power system operating parameters such as wind power fluctuation on AC-DC hybrid system, an interval optimal power flow calculation method based on interval and affine arithmetic is proposed in this paper. First, AC and DC interval power flow model is constructed based on the relationship between interval and affine arithmetic, and the uncertainties such as the new energy generation output of the system are expressed as interval variables; static security performance index (PI) is introduced in AC-DC multi-objective optimal power flow objective functions, which take the system’s power generation cost and network loss into account; the Pareto optimal solution set is distributed uniformly in space by using the particle swarm algorithm to solve the interval optimal power flow model. Finally, MATLAB simulation examples are used to verify that the method can optimize the system’s power generation cost, network loss and static safety index while considering wind power fluctuation.

Calculation and Analysis of the Interval Power Flow for Distributed Energy System Based on Affine Algorithm

Due to the coupling of different energy systems, optimization of different energy complementarities, and the realization of the highest overall energy utilization rate and environmental friendliness of the energy system, distributed energy system has become an important way to build a clean and low-carbon energy system. However, the complex topological structure of the system and too many coupling devices bring more uncertain factors to the system which the calculation of the interval power flow of distributed energy system becomes the key problem to be solved urgently. Affine power flow calculation is considered as an important solution to solve uncertain steady power flow problems. In this paper, the distributed energy system coupled with cold, heat, and electricity is taken as the research object, the influence of different uncertain factors such as photovoltaic and wind power output is comprehensively considered, and affine algorithm is adopted to calculate the system power flow of the distributed energy system under high and low load conditions. The results show that the system has larger operating space, more stable bus voltage and more flexible pipeline flow under low load condition than under high load condition. The calculation results of the interval power flow of distributed energy systems can provide theoretical basis and data support for the stability analysis and optimal operation of distributed energy systems.

A Solution of Interval Power Flow Considering Correlation of Wind Power

Interval power flow (IPF) is an important tool for steady state analysis of wind power system. All the existing interval power flow calculation methods require that the input interval variables are independent of each other, so the interval correlation of wind power in wind farms cannot be reasonably taken into account. In response to this question, in this article, firstly, the correlation Angle is used to describe the interval correlation of wind power output, and constructed an interval power flow model considering wind power correlation. Then, affine transformation technique is used to realize the de-correlation of wind power output and convert the wind power output into independent interval variables. Finally, monte carlo method or optimal scenario algorithm is used to solve interval power flow, and the maximum and minimum values of the power flow state quantity which are the interval distribution are obtained. Simulating in the modified ieee-14 and 118 systems, and the results verified the effectiveness and feasibility of the proposed methods.

Interval power flow calculation algorithm for multi‐terminal dc distribution networks considering distributed generation output uncertainties

Interval power flow calculation algorithm for multi‐terminal dc distribution networks considering distributed generation output uncertainties