摘要
中国正处在新型电力系统建设的加速转型期。新型电力系统在源网荷侧呈现的强不确定性发展趋势以及其高维度、强相关、多样化特点给传统的不确定性建模及分析方法带来挑战。文中围绕新型电力系统不确定性静态建模及量化分析方法展开评述。首先,梳理不确定性静态建模和量化分析的基本概念和框架;然后,以近几年文献调研为基础,重点从先验不确定性和后验不确定性的视角对不确定性静态建模方法进行整理和评述,以及从模型驱动和数据驱动两个层面对不确定性量化分析方法及其应用场景加以归纳和总结;最后,结合调研结果展望了未来新型电力系统不确定性静态建模和量化分析方法,指出应着力提高建模和分析方法的广度、计算效率、计算准确度并推动实际应用。
当前,人类社会正在加速迈入能源转型的新阶段。中国提出要积极稳妥推进碳达峰、碳中和,深入推进能源革命,加快规划建设新型能源体
新型电力系统呈现强不确定性发展趋
以概率论和统计学为基础的不确定性建模及分析方法是应对电力系统不确定性的有效手段。旨在对电力系统规划和运行中面临的各种不确定性进行定量刻画,化不确定为确定,为规避风险、降低不确定因素影响提供可靠的决策依据。然而,新型电力系统的不确定性具有新特征,主要体现为高维度、强相关、多样化,亟须提出适用于新型电力系统的准确、高效的不确定性建模和分析方法。
因此,本文面向新型电力系统,对近年来不确定性静态建模与量化分析的最新研究进展和应用情况进行系统性评述。首先,介绍电力系统不确定性的基本概念以及其静态建模和量化分析的基本框架,明确新型电力系统不确定性的特点和面临的挑战;其次,从先验不确定性和后验不确定性两个方面对相应的静态建模方法进行梳理;然后,基于不确定性概率二次模型,分别阐述模型驱动和数据驱动两类不确定性量化分析方法,并归纳其典型应用场景;最后,对未来的研究重点进行展望。
美国国家科学研究委员会将不确定性定义为信息的缺乏或不完
电力系统中存在着各种各样的不确定性,可通过不同维度(如属性、来源、时间
与传统不确定性分类维度不同,本文将从先验不确定性和后验不确定性的新视角出发,阐述两类不确定性静态建模的差异性和关联性,以及在新型电力系统规划运行中的应用。这里给出基于贝叶斯定理的先验不确定性与后验不确定性的定
1)先验不确定性是指在考虑观测数据前,就能对不确定因素进行概率性描述的不确定性。
2)后验不确定性是指在考虑相关观测数据后,根据条件概率(贝叶斯定理)更新得到的不确定性。
可以看出,先验不确定性与后验不确定性的根本差别在于是否考虑观测数据以及是否基于条件概率。先验不确定性来源于历史资料或经验,不随观测数据的变化而变化;而后验不确定性是考虑补充资料后对先验不确定性的修正,会随着观测数据的变化而更新。两者在电力系统规划及运行中均广泛存在,且无法被消除,故需要对其进行有效的建模及分析,以减少不确定性对系统的影响。
在电力系统实际应用中,不确定性建模的范畴非常广。例如,可对
图1 不确定性静态建模及量化分析框架
Fig.1 Framework of static modeling and quantitative analysis for uncertainty
输入不确定性静态建模部分基于新能源出力、负荷波动等连续型随机变量和线路故障等离散型随机变量的原始数据,采用基于历史数据的先验不确定性和考虑观测数据的后验不确定性静态建模方法,分别建立不确定因素的先验和后验概率分布。
输出不确定性量化分析部分先将输入不确定性的概率分布转化为概率二次模型,即场景或数字特征;然后,采用模型驱动或数据驱动的不确定性量化分析方法,以电力系统分析中的方程问题、优化问题、微分方程问题为计算模型,求取输出随机变量的概率分布;最后,根据问题的时间尺度,应用于系统规划或运行评估。
新型电力系统不确定性呈现以下新的特点:
1)高维度:随着新能源渗透率的不断提高,尤其是分布式新能源的广泛接入,新型电力系统新能源发电单体数量将急剧增
2)强相关:由于新能源出力与气象因素密切相关,地理位置相近的发电系统间将存在强相关
3)多样化:除了新能源外,新型电力系统发展还带来了众多其他不确定性,如电动汽车充放电功率、电力电子设备故障等,其特性相差巨大。
新型电力系统的新特点给不确定性建模和分析带来了挑战。多样化决定了建模和分析的对象需要进行扩展;而高维度、强相关决定了相应的模型和算法需要改进,以保持或提升在复杂不确定性环境下的计算准确性和计算速度。
本章主要讨论基于概率分布的不确定性静态建模方法。2.1节和2.2节将分别阐述先验不确定性和后验不确定性的常用建模方法。
先验不确定性又分为基于历史资料的客观先验不确定性和基于经验的主观先验不确定性。本节主要针对前者的建模方法展开论述。
对于单维连续随机变量的概率分布建模,目前学者广泛采用的模型有非参数概率分布和参数概率分布两种。非参数累积分布函数(cumulative distribution function,CDF)建模常用经验CDF,它是与样本的经验测度相关的分布函数,由多个阶跃函数组成,在n个数据点处阶跃1/n,且满足函数在任意指定值处的值是样本历史数据观测值小于或等于指定值的比例。由于其阶跃特点,无法通过求导得到概率密度函数(probability density function,PDF),故一般采用核密度估计进行非参数PDF建模。文献[
参数概率分布则具有特定的显式表达式。一般对于不同类型的不确定因素,需要采用不同类型的概率分布进行建模。文献[
对于单维离散随机变量的概率分布建模,常用伯努利分布描述两状态。文献[
单维变量概率分布建模受新型电力系统多样化不确定性特点的影响,在参数建模方法的通用化方面存在局限性,需要针对特定的变量类型和历史数据,选取特定的参数概率分布。文献[
对于多维随机变量(包括连续型和离散型),常用相关系数矩阵对其相关性建模,包括Pearson相关系数、Spearman秩相关系数和Kendall秩相关系数。文献[
基于相关系数矩阵的多维变量相关性建模受新型电力系统强相关不确定性特点的影响,在复杂相关性刻画的准确性上存在短板。文献[
2.1.1节和2.1.2节所调研的方法是分别对随机变量的概率分布和相关性进行建模,另外还有同时建模的方法,即直接建立多维变量的联合概率分布,主要包括Copula函数、高斯混合模型和非参数联合概率分布。Copula函数建模是通过特定的函数形式将随机变量的边缘分布连接为一个联合概率分布的方法,常用的Copula函数包括高斯Copula、t Copula和阿基米德Copula等。文献[
高斯混合模型是采用多个多维高斯正态分布分量逼近联合概率分布的一种建模方法。文献[
此外,2.1.1节中的单维离散变量或离散化的连续变量的建模方法也能扩展到多维情形。文献[
多维变量联合概率建模同时受新型电力系统高维度、强相关、多样化不确定性的影响,主要表现在解析建模的准确度会下
先验不确定性的概率建模方法归纳于
图2 先验不确定性静态建模方法
Fig.2 Static modeling methods for prior uncertainty
后验不确定性的本质是在给定观测数据条件下的不确定因素的条件概率分布,本节着重从应用层面阐述其静态建模方法,以及与先验不确定性的联系。根据观测数据的不同,电力系统中常见的后验不确定性可进一步细分为3类:基于不确定因素点预测值、基于不确定因素短期数据和基于不确定因素之外的其他观测数据。
若不确定因素能获取到点预测值,可采用点预测+恒定预测误差的方法对后验不确定性进行静态建模,其中,点预测方法不属于本文范畴,可参考专门的文
为了弥补恒定预测误差的不足,部分学者采用了时变的条件预测误差。而更新方法又可分为区间更新和点更新两类。区间更新是将预测值取值范围划分为有限个区间,对每一个区间所对应的预测误差的条件历史数据分别建模,每次更新时只需要根据预测值所处的区间,取出对应预测误差的条件概率分布即可。例如,可对每一个区间的风电出力历史数据采用Beta分布进行拟
点更新方法不对条件历史数据进行建模,而是直接建立预测值与预测误差的联合概率分布,再根据预测值更新预测误差的条件概率分布。文献[
当无法得到点预测值时(通常因没有配备预测工具导
当不确定因素为离散随机变量时,可基于不确定因素之外的其他数据对后验不确定性进行静态建模。离散型变量的后验不确定性静态建模等价于对其分布参数进行后验估计,故条件概率分布退化为函数映射,在伯努利分布且单一输入信息下,最常见的即为脆弱性曲线。文献[
后验不确定性静态建模方法及其与先验不确定性之间的联系如
图3 后验不确定性静态建模方法及其与先验不确定性关系
Fig.3 Static modeling method for posterior uncertainty and its relationship with prior uncertainty
后验不确定性静态建模是先验不确定性模型、观测数据、条件概率的结合。新型电力系统高维度、强相关、多样化不确定性下先验不确定性模型准确度下降,而观测数据的缺失、不全、壁垒等问题也会更加凸显,将进一步降低后验不确定性模型的准确
不确定性量化或传播分析是通过输入随机变量的概率模型计算输出随机变量的概率模型的过程。3.1节介绍输入随机变量模型的处理方法,3.2节评述从输入到输出的计算方法,3.3节总结3类应用场景。第2章中建模得到的先验不确定性和后验不确定性都可以采用本章提到的方法。
第2章中基于概率分布建立的不确定性静态模型通常不直接用于不确定性量化分析,而是先将其转化为可以直接分析计算的“概率二次模型”。本节主要讨论后续不确定性量化分析方法会用到的场景和数字特征两种概率二次模型,其余模型如不确定窗、模糊数、区间等可参考相应文
场景是从给定概率分布采样得到的样本数据。根据第2章建立模型的不同,其采样方法也不同。对于单个或者多个独立随机变量的连续型/离散型概率分布,只需将[0,1]区间上的均匀采样数据通过逆CDF映射到样本空间。若多个随机变量既满足各自的概率分布,还满足一定相关系数矩阵,则可使用Nataf变换将其转换到多维正态分布空间进行采
先验和后验不确定性模型精度的下降会使得采样场景的精度也随之下降,而在新型电力系统高维度不确定性影响下,采样效率也将下降。尤其MCMC采样的序列相关性将变得更强,所需采样次数更多,进一步降低采样效
数字特征是指能够刻画随机变量某些方面性质特征的量,包括均值、方差、协方差、分位数、矩、半不变量等。其中,前3种数字特征包含在矩或半不变量中,分位数可视为逆CDF在特定点处的值;矩和半不变量是概率分布的完整描述,即一组矩值或一组半不变量值与一个概率分布(包括单维和多维)一一对应,矩和半不变量之间也能够互相转化。文献[
数字特征也与不确定性模型精度密切相关,而在新型电力系统高维度不确定性下,数字特征的数量也将急剧增加,尤其是高阶数字特
此外,2.1.1节提到的连续概率分布的离散化也可以视为概率二次模型转化,此处不再赘述。
模型驱动的输出不确定性计算方法可分为3类:模拟法、解析法和近似
模拟法或蒙特卡洛模拟法是通过对不确定因素采样,对每个样本点进行计算,最后根据统计结果得到输出变量概率分布特征的一种方法。对模拟法的改进主要是从采样源头进行,在保证模拟精度的同时减少采样次数,包括拉丁超立方采样和准蒙特卡洛模拟,两种方法都属于伪随机采样,前者基于分层采样,后者基于Sobol、Halton等低差异序列。文献[
解析法的理论基础是概率论中采用卷积求取随机变量函数的概率分布。其在理论上可以完全准确地求得输出随机变量的概率分布,但输出变量与输入变量之间必须满足显示线性函数关系,故无法直接用于非线性、优化、微分方程等复杂模型。对解析法的改进可以归为两类:一类是对卷积算法的改进,包括快速傅里叶变换、序列运算理
近似法是模拟法和解析法在计算精度和计算效率之间的折衷方法,其目的是近似计算输出随机变量的部分数字特征。与解析法不同,近似法从理论上无法得到输出随机变量完整的概率分布信息。近些年常用的两种近似法为点估计和无迹变
解析法和近似法的计算精度受新型电力系统不确定性高维度、强相关特点影响较大,主要在于两类方法从原理上只能计及不确定性概率模型的部分低阶数字特
数据驱动的输出不确定性计算方法通过构造代理模型降低计算量,主要包括基于多项式混沌展开的代理模型和神经网络代理模型。
多项式混沌展开是利用正交多项式基函数表示输出随机变量,并采用少量输入和输出样本确定多项式系数,从而构建代理模型的方法。确定多项式系数的方法有侵入式Galerkin投影法和非侵入式回归法,后者也常称为随机响应面或随机配置点法。为了高效求解非正态输入随机变量的情况,采用不同类型正交多项式基函数的广义多项式混沌展开被提出,且在电力系统不确定性分析中得到了广泛应用,并衍生出了众多改进算法,包括稀疏多项式混沌展开、低秩逼近、高斯过程回归等,以缓解维数灾问题,三种方法的特点和比较可参考文献[
对于神经网络代理模型,近年来也取得了快速发展。文献[
数据驱动方法受新型电力系统高维度、多样化不确定性影响较大。其中,多项式混沌法计算量随着不确定性输入维数的增加呈指数增长,高维下面临严重的“维数灾”问
首先,不确定性量化分析方法最为常见的应用场景是方程问题。其中,以潮流方程为核心的概率潮流问题最为典型,包括经典概率潮
对于线性方程问题,本文调研的方法均可采用;而对于一般的非线性方程问题,则解析法无法直接使用,需要进行线性化,例如,直流潮流、线性化交流潮
其次,不确定性量化分析方法还经常应用于优化问题。其中,最为典型的是概率最优潮
对于优化问题,解析法也无法直接使用,可在基准点处导出优化模型的KKT(Karush-Kuhn-Tucher)条件,并对其进行线性化,再采用解析
最后,不确定性量化分析方法还可用于微分方程问题,包括电力系统暂态稳定、小干扰稳定、电压稳定、频率稳定
不确定性量化分析的过程及方法如
图4 不确定性量化分析过程及方法
Fig.4 Uncertainty quantitative analysis process and methods
为适用于未来新型电力系统高维度、强相关、多样化不确定性场景,不确定性静态建模与量化分析还需要在多个方面进一步开展研究,如
1)新型电力系统新特征的不确定性模型与量化分析模
2)不确定因素联合概率/复杂相关性的准确建模和量化分析方法,其重点在于权衡建模分析的准确度和计算复杂
3)面向短期预测和系统运行的后验不确定性高效建模及量化分析方法。与规划层面更加关注精度不同,短期运行层面对不确定性建模与分析的效率要求更高。然而,预测的影响因素(气象波动特征、季节、人类社会行为活动等)具有多元化、时变性特
4)数据问题。新型电力系统的海量多源异构数据对不确定性建模和分析中数据的挖掘能力、处理能力和鲁棒性提出了更高的要求。对于普遍存在的系统运行数据缺失、错误的问
5)不确定性建模分析与决策的有效结合。当前电力系统规划与运行决策针对不确定性的描述仍然集中于参数化、非条件化的假
本文系统阐述了电力系统不确定性静态建模和量化分析的基本内涵和框架,明确了新型电力系统不确定性建模和分析的难点,梳理并归纳了近几年快速发展的不确定性静态建模和量化分析方法,总结了未来的研究方向。主要结论如下:
1)电力系统众多不确定性可按先验不确定性和后验不确定性静态建模,两者以条件概率联系;
2)不确定性量化分析方法可归纳为模型驱动和数据驱动两类,可用于电力系统方程、优化、微分方程问题等输出随机变量概率分布的求取;
3)新型电力系统不确定性呈现高维度、强相关、多样化的特点,给其建模和分析带来了挑战;
4)未来应结合中国新型电力系统特征,着力提高不确定性建模和分析方法的广度、计算效率、计算准确度并推动实际应用。
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