Probabilistic analysis of load flow equations

Contents The paper presents a new method of load flow calculation in which the known node powers are treated as random variables. The unknown node voltages are assumed to be normally distributed and therefore only the average values and the second order moments should be calculated. These are calculated from the average values, variances and covariances of the node powers. The probabilistic load flow equations have been led out from the quadratic deterministic load flow equations and they are quartic. The deterministic equations have 2 ^w–1 solutions forw-nodes and therefore the probabilistic equations should have at least the same number of solutions. In hitherto literature only a single solution of probabilistic load flow equations is studied. In this paper the original method for finding all solutions is presented. The analysis of load flow equations for 2-node power system is performed.

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Anwendung der Wahrscheinlichkeitsrechnung auf die Analysis von Lastflußgleichungen

Übersicht Dieser Aufsatz behandelt eine neue Methode zur Lastflußberechnung, in der Knotenleistungen nicht mehr als bekannte Werte, sondern als Wahrscheinlichkeitsverteilungen behandelt werden. Die unbekannten Knotenspannungen werden als normalverteilt vorausgesetzt und nur die Mittelwerte und zweiten Momente sind zu bestimmen. Diese werden aus den Mittelwerten, Varianzen und Kovarianzen der Knotenleistungen berechnet. Die Lastflußgleichungen nach der Wahrscheinlichkeitsrechnung werden aus den bekannten quadratischen Lastflußgleichungen hergeleitet. Es sind Gleichungen 4. Grades, die fürw Netzknoten 2 ^w–1 Lösungen zulassen, so daß auch die Gleichungen nach der Wahrscheinlichkeitsrechnung mindestens genau so viele Lösungen haben. Die bisherigen auf der Wahrscheinlichkeitsrechnung basierenden Verfahren berücksichtigen nur einzelne Lösungen, während in der vorliegenden Abhandlung eine neue Methode entworfen wird, die alle Lösungen berücksichtigt. Die Analyse der Lastflußgleichungen wird an einem Zweiknotennetz demonstriert.

     

A COMPARISON OF PROBABILISTIC POWER FLOW WITH DETERMINISTIC BASED SOLUTIONS

ABSTRACT

A probabilistic formulation of power flow problem in which the number of equations is less than the number of the unknowns is discussed. The structure of the traditional Jacobian matrix encountered in conventional power flow is altered, and the resulting problem is one of a minimal least squares. A minimization technique using Crout's factorization and the Newton's method of successive approximation are proposed to obtain solutions. The probabilistic algorithm is applied to IEEE 14 and 37 bus systems, and computational results were obtained. The probabilistic power flow results arc compared with a deterministic based stochastic power flow solution for the 14 bus lest system. The probabilistic approach produces better results in terms of accuracy and computational time when compared with the corresponding deterministic based solutions.     

基于广义多项式混沌法的电力系统随机潮流

近年来,随着风、光电源的大量接入,系统运行的不确定性增大,考虑了系统运行随机因素的随机潮流受到更广泛的关注。提出了一种基于广义多项式混沌法的电力系统随机潮流计算方法。该方法利用广义多项式混沌法的正交多项式逼近思想,将系统的随机性分离至正交多项式基,并利用直角坐标潮流方程的二次性避免非线性潮流方程展开的高阶截断误差,进而利用随机Galerkin法,将随机潮流方程转换为一组确定性方程,通过此方程的求解获得随机潮流状态变量的正交多项式逼近系数,由此系数可获得相关变量的期望和方差,并可结合蒙特卡洛仿真,获得变量的概率密度。IEEE 9节点系统的算例表明,该方法的计算误差大致随多项式逼近阶数的上升而指数下降,通常条件下三阶逼近即可获得较高的精度,具有比蒙特卡洛仿真法更高的计算效率。     

基于广义多项式混沌法的含风电电力系统随机潮流

风电的大量接入加剧了电力系统运行的不确定性,随机潮流计算是分析电力系统不确定性的重要工具。针对风电接入,提出了基于广义多项式混沌理论的方法求解随机潮流。该算法选取最优广义多项式混沌基函数构成级数展开式来近似表示随机输入变量,再根据基函数的正交性,构造确定性方程组,进而将求解系统状态变量概率分布的问题转化为其广义多项式混沌逼近系数的求解问题。针对风电出力概率分布复杂且需计及风速相关性的情况,采用等价变换方法处理随机输入变量。对IEEE 30节点系统的随机潮流计算结果表明:所提算法的计算精度较高;与蒙特卡洛模拟法相比,计算量较小。     

HYDRO-THERMAL POWER SYSTEM LOSS MINIMIZATION: A PROBABILISTIC PERTURBATION APPROACH

ABSTRACT

We present a formulation of the power system loss minimization problem that includes power demand uncertainty. We assume that the perturbations in the system power demand and its resultant effects on the system voltages as random variables with zero mean and some variance.

The equality constraints associated with the formulation are characterized by the power flow equations in polar form utilizing the nodal admittance matrix approach. The algorithm is applied to 4 different IEEE hypothesized test system, using Newton's method and Powell's penalty function approach, and computational results are obtained. A comparison of the probabilistic approach with deterministic based solutions for three systems are offered. The difference in voltage profiles and other variables may be minimal for smaller systems. However, these differences appear to be substantial as the system increases both in size and capacity.     

Probabilistic power flow with non-conforming electric loads

A comprehensive model for probabilistic power flow is proposed for the purpose of estimating the statistics of bus voltage magnitudes and circuit currents, given the statistics of the loads. The proposed method is based on the quadratized power flow model and a non-conforming electric load model. The statistics of the electric load are accurately represented with the statistics of a few independent random variables and the statistics of bus voltage magnitudes and circuit current flows are computed from the linearized model of bus voltage and circuit current flow with respect to these independent load random variables. The method is validated via Monte Carlo simulations in which the problem is fully solved for each random sample, thus incorporating non-linearities resulting from the AC power flow equations. Both the linearization solution and the Monte Carlo approach are based on the Single Phase Quadratic Power Flow (SPQPF) model of the power system. The proposed method has been implemented and applied to the IEEE 24-bus Reliability Test System (RTS). The results are presented in the paper.     

点估计法在电压稳定性分析中的应用

针对电压稳定概率评估中难以考虑支路故障随机性的问题,提出了基于点估计法的电压稳定分析方法。该方法不仅可以方便地统一处理线路故障及节点注入功率的不确定性,还克服了线性化方法难以计及发电机无功出力约束的不足;避免了潮流方程线性化,通过少量确定性计算即可求出系统负荷裕度以及电压崩溃临界点处节点电压幅值与相角的前几阶矩,从而获得各待求随机量的概率分布,确定某一负荷水平下的电压失稳概率。与蒙特卡罗方法相比,点估计法在具有较高计算精度的同时,体现出求解速度快的特点。数值仿真分析了随机变量标准差和系统规模对点估计法计算精度的影响,并指出在综合考虑计算精度和计算时间的情况下,2点估计法的结果令人满意。     

Load supplying capability of interconnected power systems: A probabilistic approach

Prudent planning of a power system always requires determination of its load supplying capability (LSC). However, the assessment of the ability of a power system to supply loads can become very difficult when the problem data are uncertain. In this case the use of current deterministic approaches is not adequate and the application of probabilistic techniques is the most feasible alternative. This paper presents a probabilistic approach for the evaluation of the LSC of interconnected power systems in order to capture the unavoidable random nature of loads due to bus load forecasting. The method treats loads as random correlated variables with normal distribution and it can determine the probability distribution of the system LSC and of all the quantities which characterize the state of the network in this particular condition. A numerical example is included to illustrate the usefulness of the proposed technique.     

Existence and stability conditions of nonlinear probabilistic load flow solutions

Contents The paper presents a new method of nonlinear probabilistic load flow calculations forPU-bus. The probabilistic load flow equations have been led out from the quadratic load flow equations. The analysis of load flow equations is performed for 2-bus power system consisting ofPU-bus and slack bus. The nonlinear probabilistic load flow equations may have multiple or no solutions. Some of the solutions are steady state stable and some-unstable.

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Existenz- und Stabilitätsbedingungen von Lösungen nichtlinearer Lastflußgleichungen nach der Wahrscheinlichkeitsrechnung

Übersicht Dieser Aufsatz behandelt eine neue Methode zur nichtlinearen Lastflußberechnung nach der Wahrscheinlichkeitsrechnung. Die Lastflußgleichungen nach der Wahrscheinlichkeitsrechnung werden aus quadratischen Lastflußgleichungen hergeleitet. Die Analyse der Lastflußgleichungen wird an einem Zweiknotennetz mitPU- und Bilanzknoten demonstriert. Die nichtlinearen Lastflußgleichungen nach der Wahrscheinlichkeitsrechnung können viele oder keine Lösung haben. Einige Lösungen besitzen statische Stabilität, andere wiederum nicht.

     

Point estimate method for probabilistic load flow of an unbalanced power distribution system with correlated wind and solar sources

Context: The electric parameters of the power networks are usually analysed through deterministic power flows; however, the variation in load demands and power fluctuation of renewable generators cannot be considered with the deterministic power flows because it uses specific power values. The probabilistic power flow methods are better for this purpose since they apply techniques to include and reflect the uncertainty of input variables on the results obtained.Objective: This paper extends the Point Estimate Method (PEM) applied to the probabilistic power flow of an unbalanced power distribution system with dispersed generation and variable power factors. This method is applied to include uncertainties of loads and power sources such as wind and solar. As PEM requires independent input random variables, but usually there is spatial correlation between loads or power sources; therefore, Cholesky decomposition is applied to deal with this situation.Method: In this paper are combined the scheme 2m+1 of the Point Estimate Method with the Cholesky decomposition and some approximation methodologies to estimate the cumulative distribution function of some electrical parameters.Results: The results obtained are the moments about the mean of the output variables, which are used in conjunction with some approximation methodologies to obtain an estimation of the Cumulative Distribution Function for nodes or branch parameters. The proposed methodology is tested on the three-phase unbalanced IEEE 123-node test system, and results are compared with those obtained from the benchmark Monte Carlo simulation.Conclusions: There are comments on some pertinent information about Point Estimate Method performance on this kind of power systems.     

Point Estimate Schemes to Solve the Probabilistic Power Flow

This paper analyzes the behavior of Hong's point estimate methods to account for uncertainties on the probabilistic power flow problem. This uncertainty may arise from different sources as load demand or generation unit outages. Point estimate methods constitute a remarkable tool to handle stochastic power system problems because good results can be achieved by using the same routines as those corresponding to deterministic problems, while keeping low the computational burden. In previous works related to power systems, only the two-point estimate method has been considered. In this paper, four different Hong's point estimate schemes are presented and tested on the probabilistic power flow problem. Binomial and normal distributions are used to model input random variables. Results for two different case studies, based on the IEEE 14-bus and IEEE 118-bus test systems, respectively, are presented and compared against those obtained from the Monte Carlo simulation. Particularly, this paper shows that the use of the scheme provides the best performance when a high number of random variables, both continuous and discrete, are considered.     

概率潮流分析中节点电流和PV节点无功功率的均值和协方差计算

传统的概率潮流分析通常基于节点功率和节点电压的概率分布特性,在此基础上形成了概率最优潮流、概率负荷分析、概率稳定等。文章以节点注入功率和PV节点电压运行曲线为基础,依据随机变量数字特征的基本性质和概率潮流计算的相关方程,分别采用不同的方法推导了节点电流、PV节点的无功功率及平衡节点功率的均值和协方差计算公式,比较和分析了各种算法的计算精度。理论分析和算例结果表明了这些算法的有效性,算法的精确性依赖于节点电压的均值和协方差的计算精度。     

交直流电力系统概率潮流计算

基于概率理论的解析法,提出了包含直流系统的概率潮流模型。以节点注入功率和PV节点电压运行曲线构成系统的多运行方式,并计及节点注入之间的相关性,对交直流电力系统进行概率潮流计算,获得了节点电压及直流系统变量的均值和协方差。由随机变量数字特征的基本性质和交流系统概率潮流计算的原理,在直角坐标系下推导了不同直流控制方式的交直流概率潮流算式。以整流器定电流控制、逆变器定电压控制为例,在改造后的新英格兰39节点系统上计算了节点电压及换流器各角度的均值和标准差,分析结果表明了所提算法的有效性。     

Monte Carlo simulation-based probabilistic assessment of DG penetration in medium voltage distribution networks

With the growing use of renewable energy sources, Distributed Generation (DG) systems are rapidly spreading. Embedding DG to the distribution network may be costly due to the grid reinforcements and control adjustments required in order to maintain the electrical network reliability. Deterministic load flow calculations are usually employed to assess the allowed DG penetration in a distribution network in order to ensure that current or voltage limits are not exceeded. However, these calculations may overlook the risk of limit violations due to uncertainties in the operating conditions of the networks. To overcome this limitation, related to both injection and demand profiles, the present paper addresses the problem of DG penetration with a Monte Carlo technique that accounts for the intrinsic variability of electric power consumption. The power absorbed by each load of a medium voltage network is characterized by a load variation curve; a probabilistic load flow is then used for computing the maximum DG power that can be connected to each bus without determining a violation of electric constraints. A distribution network is studied and a comparison is provided between the results of the deterministic load flow and probabilistic load flow analyses.     

基于两点估计法的交直流混合系统电压稳定概率评估

传统的交直流系统电压稳定分析大多是建立在确定性方法上的,没有考虑系统中各种随机因素的影响,不利于交直流系统运行的经济性和合理性。为了弥补确定性方法的不足,提出了一种基于两点估计法的交直流混合系统电压稳定分析方法。该方法考虑负荷水平和负荷增长系数的随机性,建立了交直流混合系统的负荷裕度模型。通过少量确定性计算求出负荷裕度和临界点处节点电压的前3阶矩,从而得到待求变量的近似概率分布,确定某一负荷水平下交直流系统电压失稳的概率。与蒙特卡罗法进行比较,计算精度满足要求,并且大大地缩短了计算时间。     

基于随机配置点法的概率潮流算法

可再生能源的大量接入使得电力系统的不确定性增加,对电力系统的运行和控制提出了新的挑战。从不确定性量化理论出发,提出了一种基于随机配置点法的概率潮流算法。该算法将不确定性输入变量的概率分布表述为广义多项式混沌的谱系数,通过构建一个规模可控的确定性非线性方程组,将待求变量的概率分布函数求解转换为广义多项式混沌的谱系数求解问题,可较好地解决概率潮流计算中求解精度和计算复杂度之间的矛盾。在IEEE 14节点和IEEE 118节点系统的仿真计算中,该算法的有效性、实用性和准确性得到了验证,对于含新能源并网的概率潮流等不确定性问题具有较好的工程应用前景。     

Investigating distributed generation systems performance using Monte Carlo simulation

A novel algorithm to evaluate the performance of electric distribution systems, including distributed generation (DG) is proposed. This algorithm addresses the deterministic and the stochastic natures of these electrical systems. Monte Carlo simulation is employed to solve the system operation randomness problem, taking into consideration the system operation constraints. The uncertainties in the locations, exported penetration level, and the states (on or off) of the DG units constitute the random parameters of the studied systems. The introduced algorithm incorporates these parameters with the traditional Newton-Raphson solution of the power flow equations. Monte Carlo simulation is implemented to perform the analysis of all the possible operation scenarios of the system under study and thus ensure the validity of the results. The proposed algorithm is employed to obtain the hourly power flow solution for a typical DG connected system. The system loading follows several typical load curves based on load bus types. Furthermore, new hourly steady-state operating system parameters are evaluated to describe the system behavior under the DG random operation. The results obtained are presented and discussed.     

计及双馈风力发电机内部等值电路的电力系统随机潮流计算

研究了含变速恒频双馈式发电机的风电场接入系统后对电压质量的影响.在双馈式发电机简化等值电路的基础上建立了风电场的确定性潮流模型,建立了风力发电机的随机分析模型,并在这二者的基础上运用基于半不变量法的随机潮流进行计算.算例结果表明,风电场接入系统后,对系统中各节点的电压质量均产生影响,其中风电场节点的电压越限概率最高,对非风电场节点,其电压越限概率与其基准电压值及距风电场节点电气距离远近有关.     

考虑线路故障的随机潮流

该文给出了一种考虑线路随机故障的随机潮流算法。文中将线路随机故障等效为线路端节点注入功率的扰动，将负荷波动和发电机故障都当作节点注入功率的变化，以线性化的潮流方程计算它们引起节点电压和支路功率的变化量。为减小线性化引起的误差，对系统影响较大的离散扰动以确定性潮流计算系统状态的变化。在求节点电压和支路功率分布的各阶半不变量时对连续正态分布和离散分布2部分分别计算。用Von Mises提出的方法由各阶矩求离散分布，与正态分布卷积后获得电压和支路功率的分布函数。对RTS24节点系统的算例分析了线路故障对随机潮流结果的影响。测试表明，文中的随机潮流算法和蒙特卡罗法计算结果是一致的，计算速度则要快得多。对某实际系统的测试则给出了随机潮流应用的可能性。     

Stochastic Optimal Load Flow Using Newton-Raphson Iterative Technique

ABSTRACT

In this paper we consider the problem of stochastic optimal power flow for an all thermal electric power system. The formulation includes the effects of variables uncertainty modeled as stochastic and normally distributed. The system performance is represented by the load flow equations in polar form, reformulated to include the uncertainty aspect. Inequality constraints are handled using Powell's penalty function method. We employ a variational approach to derive the optimality conditions resulting in a set of nonlinear equations. Newton's method is employed, to actually implement the optimum strategy by solving the nonlinear set of optimality conditions iteratively. An application example involving the IEEE standard 14 bus system is presented and the stochastic solution is compared with a deterministic solution for the same system in order to demonstrate the applicability of the proposed method.