Probabilistic Power Flow Research Articles

GPU-accelerated sparse matrices parallel inversion algorithm for large-scale power systems

State-of-the-art Graphics Processing Unit (GPU) has superior performances on float-pointing calculation and memory bandwidth, and therefore has great potential in many computationally intensive power system applications, one of which is the inversion of large-scale sparse matrix. It is a fundamental component for many power system analyses which requires to solve massive number of forward and backward substitution (F&B) subtasks and seems to be a good GPU-accelerated candidate application. By means of solving multiple F&B subtasks concurrently and a serial of performance tunings in compliance with GPU’s architectures, we successfully develop a batch F&B algorithm on GPUs, which not only extracts the intra-level and intra-level parallelisms inside single F&B subtask but also explores a more regular parallelism among massive F&B subtasks, called inter-task parallelism. Case study on a 9241-dimension case shows that the proposed batch F&B solver consumes 2.92 μs per forward substitution (FS) subtask when the batch size is equal to 3072, achieving 65 times speedup relative to KLU library. And on the basis the complete design process of GPU-based inversion algorithm is proposed. By offloading the tremendous computational burden to GPU, the inversion of 9241-dimension case consumes only 97 ms, which can achieve 8.1 times speedup relative to the 12-core CPU inversion solver based on KLU library. The proposed batch F&B solver is practically very promising in many other power system applications requiring solving massive F&B subtasks, such as probabilistic power flow analysis.

Investigating Univariate Dimension Reduction Model for Probabilistic Power Flow Computation

In this paper, Johnson system and Gaussian copula are employed to model correlated random variables in power flow equations, whereby the probabilistic power flow (PPF) problem can be mapped to an independent and identically distributed probability space. The univariate dimension reduction (UDR) model is introduced to approximate the function relationship between PPF inputs and outputs. Using Beta distribution and symmetric Johnson SU distribution, two sets of orthogonal polynomials are derived to generate quadrature weights and points, and various point estimate methods (PEMs) based on the UDR model are developed to calculate the moments of PPF outputs. Finally, the proposed PEMs are verified by numerical simulations on the IEEE 57-bus system, IEEE 118-bus system, and IEEE 300-bus system.

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.

Resilient Transmission Hardening Planning in a High Renewable Penetration Era

Hardening components in transmission systems is a practice to improve system resilience against possible disturbances caused by natural disasters. In a power system with a very high penetration of renewable energy, the system hardening will be further complicated by the uncertainty and variability of renewable energy. In this paper, we study the transmission line hardening planing problem in the context of probabilistic power flows injected by the high penetration of renewable energy. We assume that the probabilistic information of renewable energy is incomplete and ambiguous and propose a data-driven approach to approximate the renewable uncertainty sets. We then extend the $N-1$ security criteria to multiple simultaneous contingencies and seek to prepare a hardening plan for the worst-case scenarios. A two-stage data-driven stochastic model is formulated by considering the joint worst-case wind output distribution and transmission line contingencies. Then, we apply the Column-and-Constraints generation method to solve the proposed model. To test the effectiveness of the proposed approach, we conduct experiments on 24-bus and 118-bus test systems. We numerically show that the data-driven approach can effectively address the uncertainty ambiguity and the proposed approach can produce effective hardening plans that improve the system resilience.

Optimal power flow considering predictability of power systems

Ever increasing use of renewable energies as uncertainty resources has caused the power system state to become highly unpredictable. Accurate prediction of the power system state is very important, especially in operational decisions, market contracts and risk management.This paper presents a probabilistic optimal power flow (P-OPF) in order to maximize the predictability of the system while minimizing the total cost of power generation. The well-known non-dominated sorting genetic algorithm (NSGA) is used to manage multiple objective functions considering operational constraints. In order to show the efficiency of the proposed method, the IEEE 30-bus standard test system is selected as a case study and the results are presented. The importance of this study and efficiency of the proposed method are discussed comprehensively.

Handling uncertainties with affine arithmetic and probabilistic OPF for increased utilisation of overhead transmission lines

Large-scale integration of variable and unpredictable renewable-based generation systems poses significant challenges to the secure and reliable operation of transmission networks. Application of dynamic thermal rating (DTR) allows for a higher utilisation of transmission lines and effectively avoids high-cost upgrading and/or reinforcing of transmission system infrastructure. In order to efficiently handle ranges of uncertainties introduced by the variations of both wind energy sources and system loads, this paper introduces a novel optimization model, which combines affine arithmetic (AA) and probabilistic optimal power flow (P-OPF) for DTR-based analysis of transmission networks. The proposed method allows for the improved analysis of underlying uncertainties on the supply, transmission and demand sides, which are expressed in the form of probability distributions (e.g. for wind speeds, wind directions, wind power generation and demand variations) and related interval values. The paper presents a combined AA-P-OPF method, which can provide important information to transmission system operators for evaluating the trade-off between security and costs at a planning stage, as well as for selecting optimal controls at operational stage. The AA-P-OPF methodology is illustrated for a day-ahead planning, using a case study of a real transmission network and a medium size test distribution network.

A probabilistic power flow calculation method considering the uncertainty of the static frequency characteristic

In a power system, power generation and load have frequency response characteristics, which randomly fluctuate with changes in operating status. This study investigates a probabilistic power flow method that considers the unit and load uncertainty of the static frequency characteristic. Firstly, a calculation model is established on the basis of the characteristics of the frequency modulation performance of the unit and load. Then a calculation method is developed using the concept of dynamic power flow in order to determine the probability distribution of the active power flow of each line under the occurrence of a fault in the system. In the method, Monte Carlo sampling with the semi-invariant method is applied for analysis and calculation. The IEEE-30-buses system is taken as an example to analyze the impact of different responses of units on the power flow distribution of various branches. The method discussed herein is compared with the Monte Carlo simulation method to verify its effectiveness.

Evaluación Probabilística y Gestión del Riesgo de la Cargabilidad de la Red por la Puesta en Servicio del Metro de Quito considerando el Movimiento Estocástico de los Trenes Eléctricos

El presente artículo propone una metodología para evaluar el impacto de las cargas móviles (trenes eléctricos) del sistema eléctrico del Metro de Quito en la red de la Empresa Eléctrica Quito. Esta metodología, basada en simulación de Montecarlo, permite la generación aleatoria de escenarios de operación de transporte masivo considerando: el desplazamiento, consumo de energía y condiciones de operación de los trenes eléctricos. La propuesta metodológica permite la evaluación probabilística de flujos de potencia mediante funciones de densidad de probabilidad (PDF por sus siglas en inglés); y adicionalmente, propone el cálculo de índices de riesgo para medir, dado un grado de confianza, los mínimos o máximos valores esperados de nivel de carga, voltaje, entre otros factores que caracterizan el nivel de cargabilidad de una red.

Combined Gaussian Mixture Model and cumulants for probabilistic power flow calculation of integrated wind power network

With the large-scale wind power penetration, probabilistic power flow plays an important role in power system uncertainty analysis. This paper proposes a novel Gaussian Mixture Model to fit the probability density distribution of short-term wind power forecasting errors with the multimodal and asymmetric characteristics. Cumulants are used to calculate mean value and deviation of state variables for each random combination result of Gaussian components. Probabilistic power flow is acquired by summing up all the Gaussian probability density functions with weights counted by the product of Gaussian components in each random combination. Parallel probabilistic power flow computation by use of the Gaussian Mixture Model and cumulants could simplify the calculation procedure in large scale of integrated wind power network. Case studies are carried out in modified IEEE 57-bus test system to verify advantages of the novel approach. Results show that the computational efficiency and accuracy are well improved in the proposed method.

Convex Relaxations of Probabilistic AC Optimal Power Flow for Interconnected AC and HVDC Grids

High Voltage Direct Current (HVDC) systems interconnect AC grids to increase reliability, connect offshore wind generation, and enable coupling of electricity markets. Considering the growing uncertainty in power infeed and the complexity introduced by additional controls, robust decision support tools are necessary. This paper proposes a chance constrained AC-OPF for AC and HVDC grids, which considers wind uncertainty, fully utilizes HVDC control capabilities, and uses the semidefinite relaxation of the AC-OPF. We consider a joint chance constraint for both AC and HVDC systems, we introduce a piecewise affine approximation to achieve tractability of the chance constraint, and we allow corrective control policies for HVDC converters and generators to be determined. An active loss penalty term in the objective function and a systematic procedure to choose the penalty weights allow us to obtain AC-feasible solutions. We introduce Benders decomposition to maintain scalability. Using realistic forecast data, we demonstrate our approach on a 53-bus and a 214-bus AC-DC system, obtaining tight near-global optimality guarantees. With a Monte Carlo analysis, we show that a chance constrained DC-OPF leads to violations, whereas our proposed approach complies with the joint chance constraint.

A frequency and duration analysis method for probabilistic optimal power flow with wind farms

Probabilistic optimal power flow (POPF) is an important tool in power system planning and operation. One limitation of conventional POPF is that only the probability information of random variables is obtained as a reference for related analyses. Frequency and duration information often plays an important role in power system assessment. In this study, a frequency and duration analysis method for POPF with wind farms (WFs) is proposed, which is based on Markov chains by improving the traditional probability‐frequency distribution function (PFDF) method. The main advantage of the proposed method is that highly accurate solutions can be obtained with less computation. Random input variables, including intermittent loads and WF power outputs associated with both wind speed uncertainties and wind turbine (WT) failures, are modeled using the corresponding PFDFs. With the proposed method, not only probability information but also frequency and duration information of random POPF outputs are efficiently and analytically computed through the operations of PFDFs of random inputs. Moreover, an optimization method for determining the clustering number of random states is proposed to improve the credibility of stochastic process modeling of Markov‐chain‐based random variables. The test on the modified IEEE‐RTS79 system with WFs demonstrates the rapidity and validity of the proposed method. © 2019 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Recommended Air Conditioner Temperature Based on Probabilistic Power Flow Considering High-Dimensional Stochastic Variables

Current power management systems face multiple reliability challenges associated with integrating a population of air conditioner (AiCo) loads into future grids. These challenges, such as the AiCo overload issue, result in a substantial discrepancy between supply and demand during the summer peak, and a few power consumers have to bear the pain of switch power brownouts. The main theme of this paper is the engineering application of AiCo control laws considering the flexible tradeoff between power economics and thermal comfort. The recommended AiCo operating temperature should be reproduced according to different micrometeorological environments. The novelty of the study is that the air conditioner temperature is recommended based on probabilistic power flow (PPF) considering high-dimensional stochastic variables. A fast calculation method is presented to solve PPF calculation problems based on singular value decomposition, principal component analysis and a polling method. This experiment aims to obtain different PPF results corresponding to different AiCo operating temperatures. The effectiveness and efficiency of the proposed method are verified by comparing the method with the Latin hypercube sampling algorithm in a 118-bus power system. Numerical tests verify the benefits of nonnegative matrix factorization and the stochastic response surface method (SRSM) for PPF calculations considering high-dimensional stochastic variables.

Probabilistic Power Flow Analysis for Hybrid HVAC and LCC-VSC HVDC System

Some innovative works have been done to probabilistic power flow (PPF) analysis for hybrid HVAC and LCC-VSC HVDC system in this paper. Firstly, a unified method considering precise model of converters is proposed to solve a general deterministic power flow (DPF) calculation including hybrid LCC (Line Current Converter) and VSC (Voltage Source Converter), the pure VSC-MTDC (Voltage Source Converter-Multiple Terminal Direct Current) and pure LCC system. Meanwhile, with a large amount of renewable energy sources integrated to the main grid through DC grids, it will impose a stochastic impact on the secure operation of such hybrid AC/DC grids. Therefore, it becomes necessary to model the probabilistic uncertainties and analyze their effects on the operation of hybrid AC/DC systems under different control modes. Nevertheless, most power flow analysis methods for hybrid AC/DC system are still deterministic in nature. Therefore, a probabilistic method based on the combination of Nataf transformation and Latin hypercube sampling (LHS) is developed and proposed to solve this complex PPF problem in an efficient manner, which considers correlated various probabilistic uncertainties, e.g. wind speeds, solar radiations and loads following different types of probability distribution. Finally, the effectiveness of the unified DPF method is validated in a modified IEEE 14-bus system, while the proposed PPF is verified in a modified IEEE 118-bus system and the effects of uncertainties on the diverse operation modes of hybrid AC/DC grids are discussed as well.

Static Security Risk Assessment for Islanded Hybrid AC/DC Microgrid

Due to the lack of the support from the main grid, the islanded hybrid ac/dc microgrid will more likely cause the power flow severe fluctuation and even the microgrid to be split once a serious contingency occurs. To maintain the security and stability of the system under such a condition, this paper proposes a new static security risk assessment of the islanded hybrid ac/dc microgrid considering that the hybrid ac/dc microgrid was split. First, this paper presents a static security analysis criterion to determine the post-contingencies splitting scheme and a new load-shedding strategy to achieve power balance of subsystems. On this basis, a probabilistic power flow calculation method based on Levenberg-Marquardt with a nonmonotone line search (LMNL) algorithm is applied to calculate the probabilistic power flow of the islanded subsystems. Finally, the probability distribution of static security risk assessment indexes, including the dc loss load risk, ac loss load risk, and interlinking converter (ILC) active power limit violation risk, are obtained. The simulation results on the test system demonstrate the necessity of considering the hybrid ac/dc microgrid was split, and prove the feasibility and effectiveness of the proposed method.

An Efficient Probabilistic Approach Based on Area Grey Incidence Decision Making for Optimal Distributed Generation Planning

The increase in the scale of distribution networks significantly reduces the efficiency of intelligent planning for distributed generation (DG). To improve the efficiency of intelligent DG planning and avoid the impact of uncertainty concerning renewable energy on it, this paper proposes a sensitivity index for the bus-embedded multi-objective estimation of distribution algorithm (MEDA) based on the semi-invariant probabilistic power flow approach to achieve an optimal solution. The sensitivity indices of the buses are comprehensively enabled to obtain a new index and determine their sensitivity sequences based on the area grey incidence decision-making method. Subsequently, according to the uncertainty of wind turbine generators and photovoltaics, a probability model is established, and the semi-invariant method is used to solve for the probabilistic power flow according to a correlation model. Finally, the sensitivity of the proposed bus-embedded MEDA to enhancing the efficiency of the solution is examined. The optimal DG allocation scheme is obtained with the goal of achieving the lowest total cost in the planning year. Finally, the feasibility and effectiveness of the proposed model and method are verified using simulations of the IEEE 33-bus, IEEE 69-bus, and IEEE 118-bus test systems.

Probabilistic Power Flow Analysis of Bulk Power System for Practical Grid Planning Application

The sizes of PV power plants have grown in such a way that their effects on the power system can no longer be neglected. In order to address these issues, grid operators are forced to expand grid connection points, and a power flow analysis considering uncertain renewable generation is required. Thus, a modified probabilistic power flow (PPF) analysis for practical grid planning is suggested in this paper. The regularity and randomness of PV power are modeled by a Monte Carlo-based probabilistic model combining both k-means clustering and the kernel density estimation method. The certain cluster group is selected so as to reflect the severe PV generation scenario, and the chi-square test to represent the $n$ th conservative network planning was suggested. In order to provide the power flow result more effectively, a mapping function of graphic representation based on a significant grid code violation is provided in an automatic PPF tool written by Python scripts. Following this procedure yields a reasonable network design for various renewable energy penetration levels.

Data-Driven Probabilistic Power Flow Analysis for a Distribution System With Renewable Energy Sources Using Monte Carlo Simulation

This paper investigates the effect of uncertainty in the allocation of photovoltaic (PV) generation, solar irradiance, and its impact on the power flow in a distribution network. The solar irradiance available in the National Renewable Energy Laboratory Resource Data Center is clustered into two states: high and low irradiance defined by a threshold. The uncertainty is modeled based on Non-Gaussian distribution, obtained using kernel density estimation. This estimation aids in achieving the probability density function and cumulative distribution functions of the solar irradiance. Moreover, the load demand, wind speed, and generator location are modeled according to Gaussian, Weibull, and discrete uniform distribution functions, respectively. As a part of probabilistic power flow, the backward/forward sweep method is used to solve each scenario of the Monte Carlo simulation. The proposed framework is applied to the 33-node test system considering three different test cases. The first case considers deployment of PV systems in three microgrids of the electric grid, and the other two test cases analyze different levels of penetration of randomly allocated PV and wind power systems. At the end, the results indicate potential reverse power flow through certain branches of the grid, and the renewables have a major impact on the system.

Chance-constrained stochastic congestion management of power systems considering uncertainty of wind power and demand side response

This study proposes a new stochastic model based on chance constraints for network congestion management in the day-ahead power market. By jointly considering the uncertainty of wind power and demand side response, the proposed optimization model can help determine the optimal daily dispatch of generators and demand responsive loads to minimize the risk of transmission congestion. To this end, transmission line congestion probability (TLCP) is constructed in chance constraints to ensure system congestion less than a certain level. The indexes of load loss probability (LLP) and wind curtailment probability (WCP) are proposed and incorporated as chance constraints to achieve a high reliability of power supply and high utilization of wind generation, respectively. In the optimization process, the proposed stochastic optimization model is transformed to an equivalent deterministic model by using the probability distribution of random variables, and the influence of transmission system loss on transmission congestion management is considered. To calculate the satisfaction degree of chance constraints under the determined dispatch of generators and demand responsive loads, the probabilistic power flow based on the cumulant method is used, and the risk of transmission congestion level is quantified by the index, congestion risk value (CRV). Simulation results of a modified PJM 5-bus system and an IEEE 118-bus test system demonstrate the effectiveness of the proposed approach for congestion management in the day-ahead power market.

Impact analysis of traction loads on power grid based on probabilistic three‐phases load flow

Electrified railway traction power supply system is directly powered by 110 kV or 220 kV (some 330 kV) high-voltage power grid. The uncertainty, non-linearity, and asymmetry of traction loads like the electric locomotives and electric multiple units (EMUs) exert the negative sequence and harmonics impact on the high-voltage transmission network. In order to evaluate the impact of the uncertainty and asymmetry of traction loads on the high-voltage transmission grid, a probabilistic three-phase power flow model of transmission network considering the probability model of traction loads is proposed. First, the power probability model of traction loads is established according to the running characteristics of traction loads. Second, the Monte Carlo simulation method is applied for the probabilistic three-phase power flow model calculation considering the traction loads. Finally, the simulation calculation is carried out on the balanced IEEE-14 bus three-phase test system. The results show that the asymmetric traction loads cause the unbalance of the grid nearby to deteriorate. The further away from the traction substation, the smaller impact of the traction loads.

Data clustering based probabilistic optimal power flow in power systems

Ever increasing use of renewable energies beside other uncertain parameters in power systems makes it necessary to evaluate power system issues, probabilistically. One of the important studies in power systems operating is optimal power flow (OPF), which should be considered as probabilistic OPF (POPF). However, the Monte Carlo simulation (MCS) method can be efficiently used for handling any type of uncertainty but this method suffers from large calculation requirements and cannot be implemented in various studies. Especially, in evolutionary based optimization problems the use of MCS method is completely restricted. Data clustering is an alternative method which keeps the accuracy of the MCS method and requires very less calculation burden. In this study, data clustering is used for probabilistic assessment of power system in POPF problem. The proposed method is very fast, accurate and can easily handle any type of correlation between stochastic input variables. The correlated input variables are generated by the Cholesky decomposition method. The Genetic Algorithm (GA) is used as optimization tool. In order to demonstrate and validate the performance of the proposed method, IEEE 30- and 118-bus standard test systems are studied and the results are compared with a modified version of the two-point estimate method.