Probabilistic Energy Flow Research Articles

Multi-Dimensional Holomorphic Embedding Method for Probabilistic Energy Flow Calculation in Integrated Distribution Power and Heating System

Multi-Dimensional Holomorphic Embedding Method for Probabilistic Energy Flow Calculation in Integrated Distribution Power and Heating System

Over-limit risk assessment method of integrated energy system considering source-load correlation

In an integrated energy system, source-load multiple uncertainties and correlations lead to an over-limit risk in operating state, including voltage, temperature, and pressure over-limit. Therefore, efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial. This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system. The method accurately and efficiently calculated the over-limit probability of the system state variables, considering the coupling conditions of electricity, heat, and gas, as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads. To further evaluate and quantify the impact of uncertainty factors on the over-limit risk, a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed. This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation. Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time. It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties.

Probabilistic energy flow analysis of the integrated electricity and gas system considering multiform uncertainties

The coupling of electricity and gas systems has been ever-augmented with the wide deployment of gas-fired generators, which facilitates the conception of the integrated electricity and gas system (IEGS). Probabilistic energy flow (PEF) analysis is usually conducted to assess the operating status of the IEGS by calculating the probability distribution of state variables (e.g., gas pressure, gas flow, voltage, and power flow). However, multiform time-variant uncertainties can simultaneously reside in the IEGS, including discrete (e.g., the component failure or functioning) and continuous ones (e.g., renewable energy outputs). Existing PEF analysis works cannot completely deal with time-variant multiform uncertainties featured with different mathematical characteristics. This limitation hinders the estimation of potential operating risks of the IEGS. To address this, this paper proposes a generalized framework for analyzing the probabilistic energy flow of the IEGS considering multiform uncertainties. Firstly, both time-varying random working states and variable outputs of components are represented as probabilistic models utilizing the Lz- transform technique. The probabilistic model is composed of some representative states depicting possible realizations of the component's performance and corresponding probabilities. On this basis, the optimal energy flow (OEF) operator is defined to aggregate probabilistic models of different components to determine probabilistic models of energy flows in the IEGS. Furthermore, multidimensional indices are constructed to comprehensively explore the probabilistic features of energy flows and the impact of probabilistic energy flows on the system performance. In this paper, the system performance mainly refers to the energy-serving capability of the IEGS. Specifically, probabilistic distribution characteristics of energy flows are explicitly displayed by relevant expectations, standard deviations as well as skewnesses. Indices such as the nodal expected gas and electricity not supplied are adopted to evaluate the influence of the probabilistic energy flow on the system performance. Numerical studies reveal that energy flows through different pipelines or power lines present diversified statistical characteristics, which indicates that they are influenced by multiform uncertainties to different extents.

An Interval Probabilistic Energy Flow Calculation Method for CCHP Campus Microgrids

Probabilistic models are used to describe the uncertainty of injected power in traditional probabilistic energy flow (PEF) calculations. Owing to the difficulty of obtaining the accurate distribution parameters of the probability models, the uncertainty of the distribution parameters needs to be considered in the PEF calculation of a combined cooling, heating, and power campus microgrid (CCHP-CMG). In this article, the parameterized probability box (p-box) model is applied for describing the uncertain distribution parameters of the probability model of injected power, and the interval PEF (IPEF) calculation model of a CCHP-CMG is developed. An affine arithmetic-based interval point estimation method (AIPEM) is put forward to solve the IPEF calculation model. The location intervals of injected powers are calculated from their p-boxes, and the interval energy flow calculation is performed with each location interval to estimate the moment intervals of the output variables. Affine arithmetic is used to reduce the interval expansion in the interval calculation, and to further reduce the interval expansion caused by affine multiplication/division operation, a method of removing the accumulation of extra noise terms is proposed. In addition, based on the Nataf inverse transformation and the p-box model, an AIPEM considering the correlation of injected powers is proposed that considers the uncertain correlation coefficients. A case study on a CCHP-CMG shows that the proposed AIPEM has higher computational accuracy than the existing interval cumulant method, especially when the uncertain fluctuation range of the injected power is large, although it consumes more CPU time.

A gradient descent direction based-cumulants method for probabilistic energy flow analysis of individual-based integrated energy systems

This study presents an individual-based model of heterogeneous integrated energy systems, which is composed of sub-energy system individuals and coupling individuals. Different evolution rules are depicted within one individual to form an accurate model of a complex system. Based on the model, a gradient descent direction iterative method (GDDM) is proposed for energy flow calculation of integrated energy systems to improve its convergence performance. Furthermore, a GDDM-based cumulants method (GDDM-CM) is presented to analyse the probabilistic energy flow distribution throughout the system. To verify the effectiveness of the proposed method, simulation studies have been undertaken in a modified complex integrated energy system. The results show that GDDM performs better than non-gradient descent method (NGDM). It converges faster and is insensitive to the initial point. Besides, GDDM-CM can obtain similar results as monte-carlo sampling method does and its calculation accuracy is better than that of the point estimate method. Moreover, the proposed GDDM-CM can greatly reduce the calculation time compared with the other two methods. Utilizing the GDDM-CM, the coupling effects of different individuals are investigated and the weak points in each individual can be identified.

A Stochastic Response Surface Method Based Probabilistic Energy Flow Analysis Method for Integrated Electricity and Gas Systems

An efficient probabilistic energy flow (PEF) analysis method is proposed for integrated electricity and gas systems (IEGSs) in this letter. A network decoupling based deterministic energy flow (DEF) calculation method is developed. A PEF analysis method is presented based on stochastic response surface method (SRSM). The computational efficiency and convergence performance of DEF and PEF are largely improved. The correctness and effectiveness of the proposed method are verified by using three test systems.

Calculation of Probabilistic Energy Flow of Colding-heating-electricity Interconnection System Considering Wind Power

In order to effectively evaluate the impact of wind power connection on the operation of the colding-heating-electricity interconnection system, a probabilistic energy flow calculation method based on the semi-invariant method and improved K-means is proposed. First, a colding-heating-electricity joint system model was established. Secondly, based on the semi-invariant method, an improved K-means clustering technique was proposed to solve the problems caused by different input random sample types. Finally, the efficiency of the proposed method is verified by simulation.

Probabilistic energy flow calculation for regional integrated energy system considering cross-system failures

With the increasing coupling among multiple energy systems, component failure of one system may spread to other systems, which results in cross-system failures that will remarkably affect the operation state of RIES. In this regard, this paper proposes a probabilistic energy flow calculation approach for regional integrated energy system (RIES), taking into account those cross-system failures (CSF), as well as fluctuations of renewable generation and multi-energy loads. An iterative screening method based on failure identification matrix is developed to find failed components cross different systems caused by CSF. With this method, all CSFs can be divided into three categories to simplify the interaction analysis among systems. Finally, a Monte Carlo based probabilistic energy flow calculation framework is proposed considering those three types of CSF. Numerical studies are performed on a RIES test system with 101 nodes and a practical RIES system with 119 nodes, which verifies the effectiveness of the proposed method. The results also show that the approach can provide more comprehensive insight of the interaction among different energy systems in various failure situations.

Probabilistic Energy Flow Analysis for Integrated Energy Systems Considering Correlated Uncertainties

The integrated energy systems (IES) contain various energy sub-systems, thus uncertainties of one sub-system will not only threaten its own safety operation but also be likely to have a significant impact on the operation of other sub-systems. Therefore, it is of great significance to study the variation of state variables when random fluctuations emerge in the IES. This paper proposes a unified steady state probabilistic energy flow (PEF) analysis for the IES considering the correlated uncertainties. Firstly, a Latin hypercube sampling with inverse Nataf transformation is developed to deal with correlated uncertainties of the IES. Furthermore, in order to solve the PEF problem, a unified method, Newton-Raphson embedded with Newton Downhill technique, is proposed to accelerate the iteration and improve the computation efficiency. The effectiveness of the proposed PEF analysis method is verified by a set of test results conducted on an IES composed of the IEEE 118-bus system coupled with 15-natural gas system and 32-district heating system.

Probabilistic Dynamic Optimal Energy Flow Calculation for the gas-electric coupling system

Due to pollution and energy shortage, the multi-energy interconnected system formed by natural gas and electricity coupling has attracted a lot of attention because of its cleanness and high efficiency. The operation of natural gas system and electrical power system belong to different time scales. At the same time, more and more new energy sources such as wind and photovoltaic power generation in the electrical power system increase the uncertainty of power supply. The operation of gas-electric coupling system must consider the impact of these factors. In response to these problems, this paper considers the impact of timing on the operating state of gas-electric coupling system based on the multi-period optimization model, and proposes a probabilistic energy flow calculation method, namely the Latin Hypercube sampling Markov Chain simulation (LHMCS). This method can quickly obtain the influence of random factors on the operating state of gas-electric coupling system. Finally, a simulation example verifies the rationality and effectiveness of the method proposed.

Convex Optimization of Integrated Power-Gas Energy Flow Model With Applications to Probabilistic Energy Flow

Energy flow calculation is a fundamental problem of the integrated power and gas system (IPGS) operation and planning. However, the nonlinear gas flow model introduces major challenges to the energy flow calculation. In this paper, we propose a tractably convex optimization model to solve the energy flow problem in IPGSs. It is demonstrated that the proposed optimization model has the same optimal solution as the original nonlinear steady energy flow model. Also, piecewise linearization is adopted to tightly linearize the nonlinear objective function of the model, which transforms the formulated convex optimization into a linear program one. Thus, the computation complexity of the proposed energy flow model is significantly reduced as compared with the existing methods. In addition, the proposed model can be extended to probabilistic energy flow estimation. Extensive case studies are conducted to validate the effectiveness of the proposed energy flow model using three IPGSs.

Interval Probabilistic Energy Flow Calculation of CCHP Campus Microgrid Considering Interval Uncertainties of Distribution Parameters

Due to the absence of historical data and the errors of measurement instruments, there may be uncertainties in the distribution parameters of the random variables describing the uncertain fluctuations of node power including renewable energy station output and load power in the combined cooling heating and power (CCHP) campus microgrid. In this paper, intervals are used to describe the uncertainties of distribution parameters of the random variables, and an interval probabilistic energy flow (IPEF) calculation model of the CCHP campus microgrid is established. Introducing the interval arithmetic (IA) into the cumulant method, an IA-based IPEF algorithm is proposed to obtain the analytical expressions of probability density function or cumulative distribution function intervals of the state variables. Moreover, affine arithmetic (AA) is introduced to address the interval extension problem in the calculation, and an AA&IA-based IPEF algorithm is proposed. By constructing the correlation transformation matrixes, the correlation among different node power is considered in the IPEF calculation. A case study on a CCHP campus microgrid demonstrates that the results of the AA&IA-based IPEF algorithm are more accurate than those of the IA-based IPEF algorithm by using the results of the double-layer Monte Carlo method as a reference. Moreover, the proposed algorithms are more efficient than the double-layer Monte Carlo method.

Probabilistic energy flow and risk assessment of Electricity–Gas systems considering the thermodynamic process

The existing analyses of the electricity–gas systems generally do not regard the nodal gas temperature as a state variable. However, the temperature variations caused by the thermodynamic process are important because of the following two reasons: 1) gas temperature influences the pressure and gas flow; 2) gas temperature is an important indicator for hydrate formation in gas networks, which greatly jeopardizes the gas transmission. We herein propose a probabilistic energy flow method considering the thermodynamic process in gas networks. To build the energy flow model, the algebraic pipeline flow model under the non-isothermal condition is derived according to a set of partial differential equations reflecting the thermodynamic process. The models of pressure-regulating stations and compression stations considering the thermodynamic processes are presented in an electricity–gas analysis for the first time. Risk indices are proposed to quantify the risks of hydrate formation and state variables that exceed the limits. The simulation results of two test cases demonstrate the effectiveness of the proposed method.

Probabilistic Energy Flow Calculation through the Nataf Transformation and Point Estimation

With the increasing capacity of renewable energy sources, uncertainties regarding renewable energy and other dynamic loads in integrated energy systems (IESs) are increasing. Thus, it is necessary to study the probabilistic energy flow (PEF) of IESs. However, existing PEF calculation methods such as the point estimate method (PEM) are computationally inefficient when there are many random variables and estimated points; moreover, relatively large errors can occur when the estimated points are outside their limits. Hence, this paper presents a calculation method that addresses these problems. Because there are correlations among the variables, the Nataf transformation is employed to control the correlation quickly and effectively. A model for an IES that is interconnected with natural gas and electricity systems and accounts for the uncertainties of wind plants, photovoltaic power plants, and dynamic gas loads is presented. Correlations between wind plants and photovoltaic power plants are handled using the Nataf transformation. Finally, a modified PEM is developed to solve the PEF. For situations in which the estimated points exceed their boundaries, the power transformation and equal constraint transformation methods are used. The results of time-domain simulations demonstrate the effectiveness of the proposed approach.

Unified probabilistic energy flow analysis for electricity–gas coupled systems with integrated demand response

Under a multi-energy environment, the conventional demand response programmes in smart-grid will be evolving to the concept of integrated demand response (IDR). In order to examine such an effect, this study presents a new methodological framework to conduct the probabilistic energy flow (PEF) analysis of integrated energy systems, while considering the potential effects of IDR. For this aim, a new price-elasticity-based IDR model is developed. As distinct from existing works, this study explicitly focuses on the stochastic characteristics of IDR programme and a novel Z-number-based technique is introduced to handle the relevant uncertainties that involved. The proposed Z-number model can not only capture the inherent randomness in demand-side performance but also account for the potential impact of information availability/reliability on the representation of IDR. Since the presence of Z-numbers makes the authors model include different types of uncertain representations, the fuzzy expectation technique and centroid method are jointly used to make them manageable under the same framework. A comprehensive algorithm based on the cumulant method and an optimisation scheme is employed to solve this PEF problem with consideration of IDR. The proposed framework is illustrated through numerical studies and simulation results confirm its effectiveness in actual implementations.

Probabilistic Energy Flow Analysis for Urban Energy Systems Considering Correlated Uncertainties

Abstract The urban energy system (UES) with the distribution electricity network (DEN) and distribution gas network can achieve a multiple-energy utilization. However, massive uncertainties in an UES will pose potential risks to the secure operation of both networks. This paper proposes a probabilistic energy flow (PEF) framework considering correlated uncertainties in photovoltaic outputs and loads, operation modes, as well as asymmetrically three-phase characteristic of DEN. The Monte Carlo Simulation method based on Latin Hypercube Sampling and Nataf transformation is then adopted to solve the PEF problem. Numerical results obtained from a sample UES demonstrate the effectiveness of the proposed PEF framework and disclose the effects of uncertainties, correlations and operation modes on the operation of the UES.

Probabilistic Energy Flow Analysis of MCE System Considering Various Coupling Units and the Uncertainty of Distribution Generators

The deep interdependence between electrical and natural gas system has great influence on the security of multi-carrier energy (MCE) systems. The large-scale access to distribution generation (DG) of clean energy makes it necessary to investigate the effects of uncertainties on the safe and stable operation of MCE systems. However, the existing probabilistic energy flow analysis for MCE systems pays little attention to the difference between the correlation of various DG during calculating the energy flow. Therefore, this paper investigates the probabilistic energy flow considering the uncertainties and correlations of various DG. A framework of MCE systems considering various coupling units including power to gas (P2G) and energy hub is initially proposed. For this, the Monte Carlo simulation (MCS) method based on Latin hypercube sampling (LHS) and Nataf transformation is specially designed to calculate the energy flow for each sample. The energy flow calculation method is composed of three-phase AC/DC alternating iteration method and improved Newton node mesh method considering the control mode of compressors. Finally, the test system containing improved IEEE123-node system and NGS48-node system have verified the accuracy of the proposed probabilistic energy flow calculation method.

Probabilistic-Interval Energy Flow Analysis of Regional Integrated Electricity and Gas System Considering Multiple Uncertainties and Correlations

This paper studies the uncertainty analysis of the regional integrated electricity and gas system (IEGS) composed of a three-phase unbalanced power distribution system and a gas distribution network. The well-known probabilistic energy flow (PEF) analysis of IEGS is extended to a comprehensive probabilistic-interval energy flow (PIEF) problem by modeling uncertainties in pipeline parameters as interval variables and spatial dependency between them as correlation angles. The polynomial chaos expansion (PCE) method is introduced to IEGS analysis and an improved PCE-based method combining the technique of dimension-reduction and the sparse-PCE is proposed to address the numerous uncertainties involved in a regional IEGS. A methodological framework considering probabilistic/interval uncertainties, linear/rank correlations among probabilistic variables, and dependency between interval variables is then developed for both PEF and PIEF analysis. Numerical simulations of a sample regional IEGS are used to verify the accuracy and efficiency of the proposed method, as well as the superiority of the proposed PIEF analysis over the existing PEF analysis.

Investigation of Carrier Demand Response Uncertainty on Energy Flow of Renewable-Based Integrated Electricity–Gas–Heat Systems

Since there are heavy interdependencies among the electrical, heat, and gas systems to supply various load types worldwide, operation of multi-energy carrier (MEC) systems faces critical challenges. Moreover, any uncertainty rising in one carrier would directly influence the energy flow and the secure operation of the whole MEC system. This issue intensifies when an MEC system is integrated with industrial energy carrier demand response (ECDR) consumers and renewable energy sources (RESs). Large industrial ECDR consumers can increase the system uncertainty by randomly participating in the demand response programs and utilizing various carriers. Accordingly, this paper presents a powerful probabilistic tool named 2 m + 1 point estimate strategy for energy flow analysis of an integrated MEC system considering ECDR, RES, and various load types uncertainties to control the risks associated with the uncertainties. In addition, a new decomposition technique is presented to accelerate the energy flow solving of the integrated MEC systems. This technique has been promoted by adding a novel noniterative method named holomorphic embedding and the less-computational graph methods for solving the energy flow of the MEC. The proposed probabilistic energy flow is tested on an integrated MEC system employing various incentives for industrial ECDR consumers.

Probabilistic Steady-State Operation and Interaction Analysis of Integrated Electricity, Gas and Heating Systems

The existing studies on probabilistic steady-state analysis of integrated energy systems (IES) are limited to integrated electricity and gas networks or integrated electricity and heating networks. This paper proposes a probabilistic steady-state analysis of integrated electricity, gas and heating networks (EGH-IES). Four typical operation modes of an EGH-IES are presented at first. The probabilistic energy flow problem of the EGS-IES considering its operation modes and correlated uncertainties in wind/solar power and electricity/gas/heat loads is then formulated and solved by the Monte Carlo method based on Latin hypercube sampling and Nataf transformation. Numerical simulations are conducted on a sample EGH-IES working in the “electricity/gas following heat” mode to verify the probabilistic analysis proposed in this paper and to study the effects of uncertainties and correlations on the operation of the EGH-IES, especially uncertainty transmissions among the subnetworks.