WO2019184132A1 - 一种基于数据驱动的电网潮流方程线性化求解方法 - Google Patents

一种基于数据驱动的电网潮流方程线性化求解方法 Download PDF

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WO2019184132A1
WO2019184132A1 PCT/CN2018/094553 CN2018094553W WO2019184132A1 WO 2019184132 A1 WO2019184132 A1 WO 2019184132A1 CN 2018094553 W CN2018094553 W CN 2018094553W WO 2019184132 A1 WO2019184132 A1 WO 2019184132A1
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node
equation
vector
matrix
grid
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刘羽霄
张宁
王毅
杨经纬
康重庆
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清华大学
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/04—Circuit arrangements for ac mains or ac distribution networks for connecting networks of the same frequency but supplied from different sources
    • H02J3/06—Controlling transfer of power between connected networks; Controlling sharing of load between connected networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
    • Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
    • Y04S40/00—Systems for electrical power generation, transmission, distribution or end-user application management characterised by the use of communication or information technologies, or communication or information technology specific aspects supporting them
    • Y04S40/20—Information technology specific aspects, e.g. CAD, simulation, modelling, system security

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  • the invention belongs to the field of power flow calculation and the field of data driving technology, and particularly relates to a linearized solution method for power flow equation based on data driving.
  • Power grid power calculation is the basis for power system optimization and analysis.
  • the power flow equation is strongly nonlinear, which increases the complexity of the optimization and control algorithm of the power grid, and the convergence becomes worse.
  • the linearization of the power flow equation can significantly simplify the computational complexity and ensure convergence, and is therefore widely used in power system control, scheduling, and power market clearing algorithms.
  • the current trend equation linearization method is based on a physical model. From the original AC flow equation, based on the characteristics of the grid operating state, such as the connected node voltage phase angle difference is often less than thirty degrees, the voltage amplitude is often close to the standard value, thus making some mathematical assumptions, simplifying communication
  • the power flow equation finally leads to a linearized power flow equation.
  • the regression method is used to discover the relationship between continuous data and is widely used in data-driven power network analysis and optimization.
  • the classical least squares regression method is difficult to adapt to the collinear characteristics of the grid data, and the generalized error of the equation after regression is large.
  • a general linear regression problem can be expressed as:
  • X represents the independent variable data matrix
  • Y represents the dependent variable data matrix
  • A represents the regression parameter matrix
  • N is the number of data samples
  • M is the number of regression parameters
  • [ ⁇ ] T is the matrix transpose
  • Bayesian Linear Regression (BLR) method is based on the Bayesian analysis framework.
  • the basic idea can be summarized as the maximum a posteriori analysis, that is, the parameter that makes the posterior probability the largest, as the parameter obtained by the linear regression algorithm. .
  • the calculation of the posterior probability follows the Bayesian formula, so it is called the Bayesian linear regression method.
  • Bayesian linear regression returns the parameters corresponding to each row of dependent variables in turn:
  • a i [a i1 ...a ij ...a iL ]
  • p(a i ) represents the prior probability distribution
  • a i ) represents the likelihood probability distribution.
  • the prior probability distribution is used to suppress over-fitting, so the problem of matrix morbidity due to data collinearity can also be avoided.
  • the prior probability distribution is set to an elliptical Gaussian distribution:
  • ⁇ j the reciprocal of the standard deviation of the a ij distribution.
  • Bayesian linear regression uses the maximum a posteriori optimization algorithm, and the algorithm uses an iterative method to solve the optimal parameters. For details, see METipping, "Sparse Bayesian learning and the relevance vector machine,” Journal of machine learning research , vol. 1, pp. 211-244, 2001.
  • the object of the present invention is to overcome the deficiencies of the prior art and propose a linearized solution method for power flow equations based on data driving.
  • the invention can improve the accuracy of the calculation of the power flow equation of the power grid and help to reduce the operating cost of the power grid.
  • the invention reduces the calculation amount, improves the calculation flexibility, and is more in line with the actual situation of the power system in the engineering application. .
  • the invention provides a data-driven linearization method for grid power flow equations, characterized in that the method comprises the following steps:
  • P represents the active injection vector of each node of the grid.
  • P represents the active injection vector of each node of the grid.
  • Represents the transpose of the PQ node active injection vector Represents the transpose of the PV node active injection vector, Represents the transposition of the active injection vector of the V ⁇ node;
  • Q represents the reactive injection vector of each node of the grid.
  • Representing the transposition of the reactive power injection vector of the PQ node Represents the transpose of the PV node reactive injection vector, Represents the transposition of the reactive energy injection vector of the V ⁇ node
  • V represents the voltage amplitude vector of each node of the power grid
  • Represents the transpose of the PQ node voltage magnitude vector Represents the transpose of the PV node voltage magnitude vector, Representing the transposition of the V ⁇ node voltage magnitude vector
  • ⁇ represents the voltage phase angle vector of each node of the grid, Represents the transpose of the PQ node voltage phase angle vector, Representing the transpose of the PV node voltage phase angle vector, Representing the transposition of the voltage phase angle vector of the V ⁇ node;
  • step 1-1 Using the results of step 1-1), construct a mapping equation from active injection, reactive injection to voltage phase angle, voltage amplitude, and the expression is as follows:
  • C 1 - C 6 represent a constant matrix in the regression
  • a ij represents a sub-matrix in the regression parameter matrix
  • step 2) according to the mapping equation established in step 1), and the known quantity unknown relationship of each node type, the calculation expression of the solution of the power flow equation is listed;
  • equation (2) When calculating the power flow, for the left dependent variable mapped in equation (2), ⁇ L , ⁇ S , P R and V L are unknowns, V S and V R are known quantities, and for the mapping in equation (2) The right independent variables, P L , P S and Q L are known quantities, Q S and Q R are unknowns; therefore, the formal expression of equation (2) written into the block matrix is as follows:
  • X represents an independent variable data matrix
  • Y represents a dependent variable data matrix
  • A represents a regression parameter matrix
  • x n represents the transposition of the nth row of the matrix X
  • n 1..N
  • y m represents the transposition of the mth row of the matrix Y
  • a m represents the transposition of the mth row of the matrix A
  • m 1...M
  • N represents the number of rows of the matrix X
  • M represents the number of rows of the matrix Y;
  • Bayesian linear regression returns the parameters corresponding to each row of dependent variables in turn:
  • the iterative method is used to solve the parameters corresponding to the dependent variable by the maximum posterior principle.
  • the maximum posterior distribution is proportional to:
  • step 3-2 calculate the solution of the linearized flow equation according to equation (9):
  • the linearized equation has higher calculation accuracy. Since the training data of the present invention is historical data of measurement, reflecting the real operating state of a particular power system, it has higher calculation accuracy. For example, the data-driven method of the present invention can take into account the effects of changes in line parameters due to air humidity and the like. Increasing the accuracy of calculations helps to reduce the cost of grid operation.
  • the invention obtains a linearized power flow calculation equation, which can effectively reduce the calculation amount in the direct calculation and the calculation as the constraint of the optimization problem, thereby enabling the scheduling in operation to be more real-time or capable of allowing simulation, The factors considered in the model such as scheduling are more complete.
  • the linearized power flow model obtained by the method of the invention can be calculated by considering different types of node characteristics, and is in accordance with the actual situation of the power system in the engineering application.
  • FIG. 1 is a schematic diagram of a set of test results in a test result of 300 sets of NREL-118 system according to an embodiment of the present invention.
  • 2 is a histogram of test results of 300 sets of NREL-118 systems in an embodiment of the present invention.
  • the invention provides a data-driven linearization method for grid power flow equations, the method comprising the following steps:
  • mapping equation from active injection, reactive injection to voltage phase angle and voltage amplitude is established, so that the equations according to the mapping method can consider the collinearity before the data and can easily calculate the power flow. . Specific steps are as follows:
  • P represents the active injection vector of each node of the grid.
  • P represents the active injection vector of each node of the grid.
  • Represents the transpose of the PQ node active injection vector Represents the transpose of the PV node active injection vector, Represents the transposition of the active injection vector of the V ⁇ node;
  • Q represents the reactive injection vector of each node of the grid.
  • Representing the transposition of the reactive power injection vector of the PQ node Represents the transpose of the PV node reactive injection vector, Represents the transposition of the reactive energy injection vector of the V ⁇ node
  • V represents the voltage amplitude vector of each node of the power grid
  • Represents the transpose of the PQ node voltage magnitude vector Represents the transpose of the PV node voltage magnitude vector, Representing the transposition of the V ⁇ node voltage magnitude vector
  • ⁇ represents the voltage phase angle vector of each node of the grid, Represents the transpose of the PQ node voltage phase angle vector, Representing the transpose of the PV node voltage phase angle vector, Represents the transpose of the V ⁇ node voltage phase angle vector.
  • the above data is obtained through PMU and SCADA systems.
  • step 1-1 Using the results of step 1-1), construct a mapping equation from active injection, reactive injection to voltage phase angle, voltage amplitude, and the expression is as follows:
  • C 1 - C 6 represent the constant matrix in the regression
  • a ij represents the sub-matrix in the regression parameter matrix.
  • [ ⁇ L ⁇ S P R V L V S V R ] T and [P L P S Q L Q S Q R ] T are known amounts
  • a ij and C 1 to C 6 are The parameters to be returned.
  • the mapping established by the invention is from active injection, reactive injection to voltage phase angle, voltage amplitude mapping, which considers that there are some PQ nodes (such as substation nodes) in the power system, the active and reactive injection is zero. . This situation will make the corresponding term in the regression parameter zero. If the mapping in the opposite direction to the present invention is used, that is, the mapping from voltage phase angle, amplitude to active injection and reactive power injection, it will appear in the process of solving the power flow equation.
  • the matrix is irreversible, and the mapping constructed by the present invention does not cause the matrix to be irreversible.
  • the active injection of the reference node is removed from the independent variable. This is because for most power systems, active power loss is negligible compared to active power injection, that is, the sum of active power of each node is approximately zero. Therefore, the injection of each node has a collinear relationship. In order to ensure that all active injections have strong independence during regression, the present invention removes the active injection of the reference node from the independent variable.
  • step 2) According to the mapping equation established in step 1), and the known quantity unknown relationship of each node type, the calculation expression of the solution of the power flow equation is listed.
  • equation (2) can write the form of the block matrix according to the known unknown partition:
  • the superscript 1...t....T represents the time point of the historical measurement data, and the data of each time point constitutes a set of data, indicating that one group of active and reactive power injections of all nodes in the system at a certain moment. , voltage phase angle, voltage amplitude data.
  • the number of historical measurement data sets should be not less than 2.4 times the number of system nodes.
  • x n represents the transposition of the nth row of the matrix X
  • n 1..N
  • y m represents the transposition of the mth row of the matrix Y
  • a m represents the transposition of the mth row of the matrix A
  • m 1...M
  • N represents the number of rows of the matrix X
  • M represents the number of rows of the matrix Y
  • the values of N and M are determined by the equation (5).
  • Bayesian linear regression returns the parameters corresponding to each row of dependent variables in turn:
  • the solution process uses an iterative method.
  • the specific principles of Bayesian linear regression are detailed in the background section.
  • step 3-2 calculate the solution of the linearized flow equation according to equation (9):
  • the present invention is based on the NREL-118 test system of I. Pena, C. Brancucci and BMHodge, "An Extended IEEE 118-bus Test System with High Renewable Penetration," IEEE Trans. Power Syst., p. 1-1, 2017.
  • the load data and the grid data provided are taken as an example to perform visual verification on the method proposed by the present invention.
  • the active load data is generated based on the weather and load data simulations from 1980 to 2012.
  • the reactive load data is obtained by multiplying the active load value by a randomly generated multiplier, and the multiplier value is between [0.15-0.25]. In this example, 300 sets of data were taken for training, and 300 sets of data were used for testing.
  • the linearized grid power flow equation obtained by the method proposed by the present invention calculates the power flow and compares it with the accurate AC power flow calculation result.
  • the results are shown in Fig. 1 and Fig. 2.
  • Figure 1 focuses on the details and shows one of the 300 test results.
  • the horizontal axis represents the node number, the right side of the vertical axis represents the error, and the left side represents the specific calculated value.
  • Fig. 1(a) shows the test results of the voltage phase angle
  • Fig. 1(b) shows the test results of the voltage amplitude.
  • Figure 2 focuses on the whole and shows a histogram of 300 sets of results.
  • Fig. 1 shows the error
  • Fig. 2 represents the frequency
  • BLR represents the Bayesian linear regression algorithm
  • Figure 2 (a) shows the test of the voltage phase angle
  • Figure 2 (b) shows the test of the voltage amplitude.
  • DLPF is the representative of the traditional model-based linearization method of power flow equations.
  • J.Yang, N.Zhang, C.Kang, and Q.Xia "A State-Independent Linear Power Flow Model with Accurate Estimation of Voltage Magnitude," IEEE Trans. Power Syst., vol. 22, pp. 3607-3617, 2017, and LS is a least squares method, representing a total of unconsidered data. Regression method when linear. The error is measured in absolute error.
  • the BLR method proposed by the present invention is more accurate than the DLPF method based on the model-based power flow linearization method and the LS method which does not consider data collinearity.
  • the error distribution of the BLR method proposed by the present invention is smaller than that of the DLPF and LS methods.
  • the error distribution of LS is very wide, indicating that this method does not consider data collinearity, and it is unstable in actual situations.
  • the present invention suggests that the number of training data sets should be no less than 2.4 times the number of system nodes.

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Abstract

本发明提出一种基于数据驱动的电网潮流方程线性化求解方法，属于电网潮流计算领域以及数据驱动技术领域。该方法首先根据电力系统各个节点的节点类型建立从有功注入、无功注入到电压相角、电压幅值的映射方程，然后根据各个节点类型的已知量未知量关系，列出潮流方程解的计算表达式；利用电网历史量测数据中的有功注入、无功注入、电压幅值、电压相角等数据，提出了考虑到电网数据共线性特点的贝叶斯线性回归电网潮流方程线性化的回归方法，得到线性化的潮流方程的解。本发明可提高电网潮流方程计算的精度，有助于降低电网运行成本，同时本发明减少计算量，提升计算的灵活性，更加符合工程应用中电力系统的实际情况。

Description

一种基于数据驱动的电网潮流方程线性化求解方法
相关申请的交叉引用
本申请要求清华大学于2018年03月29日提交的、发明名称为“一种基于数据驱动的电网潮流方程线性化求解方法”的、中国专利申请号“201810269439.2”的优先权。
技术领域
本发明属于电网潮流计算领域以及数据驱动技术领域，特别涉及一种基于数据驱动的电网潮流方程线性化求解方法。
背景技术
电网潮流计算是电力系统优化与分析的基础。潮流方程是强非线性的，这给电网的优化与控制算法复杂度增加，收敛性变差。对于潮流方程的线性化可以显著地简化计算的复杂度并且能够确保收敛性，因此也被广泛应用于电力系统控制、调度以及电力市场出清的算法中。在现有潮流方程线性化方法是基于物理模型的方法。即从原始的交流潮流方程出发，基于电网运行状态的特征，比如相连接的节点电压相角差往往小于三十度，电压幅值往往接近于标幺值，从而做出一些数学假设，简化交流潮流方程，最终得到线性化的潮流方程。
随着同步向量量测装置(Phasor Measurement Unit,PMU)和数据采集与监视控制(Supervisory Control and Data Acquisition,SCADA)系统在电网中的普及，越来越多的量测数据可以用来建立对电网的辨识与分析。这种利用量测数据对电网进行辨识与分析的数据驱动的方法可以提升电网的辨识与分析的实时性与准确性。
现有的利用数据驱动的方法处理电网潮流的研究很少，文献J.Yu,Y.Weng and R.Rajagopal,"Mapping Rule Estimation for Power Flow Analysis in Distribution Grids,"arXiv preprint arXiv:1702.07948,2017利用非线性的支持向量机回归来发掘潮流分析中变量之间的关系。但是这一研究不是线性的，得到的仍然是非线性潮流模型，不能解决潮流计算复杂度高、收敛性差的问题，另外，这一研究中的电压相角和电压幅值不是解耦的，因此无法处理潮流分析中的PV节点的计算。
电网数据中存在明显的共线性，即不同数据之间的相关性较高。这是由于电网中不同节点的负荷往往同涨同落，不同电网节点之间的电压幅值和相角，由于其物理连接关系， 也会同涨同落。这样的数据共线性特征，会给基于数据驱动的电网潮流方程线性化方法带来一定困难。较强的共线性会造成回归时矩阵病态，从而造成回归结果具有低偏倚高方差的特性，导致回归结果的泛化误差增加。
回归方法用于发现连续数据之间的关系，广泛应用于数据驱动的电力网络分析与优化中。但是经典的最小二乘回归方法难以适应电网数据中的共线性特点，回归后的方程泛化误差较大。一般的线性回归问题可以表示为：
Y＝AX
其中，X表示自变量数据矩阵，Y表示因变量数据矩阵，A表示回归参数矩阵。A,X,和Y都是以行的形式表示的：
X＝[x 1 x 2 ... x N] T Y＝[y 1 y 2 ... y M] T A＝[a 1 a 2 ... a M] T
其中N表示数据样本数，M表示回归参数个数，[·] T表示矩阵转置。
贝叶斯线性回归(Bayesian Linear Regression,BLR)方法的思路依据贝叶斯分析框架，其基本思路可以概括为最大后验分析，即求解使得后验概率最大的参数，作为线性回归算法得到的参数。其中，后验概率的计算遵从贝叶斯公式，因此称为贝叶斯线性回归方法。贝叶斯线性回归依次回归每一行因变量对应的参数：
y i＝a iX+e i,i＝1,2,...,M
其中e i代表因变量y i的噪声，每一个a i都表示一个向量：
a i＝[a i1...a ij...a iL]
其中L表示a i中的元素个数。根据贝叶斯公式，a i的后验概率分布满足：
p(a i|y i,X)∝p(a i)p(y i,X|a i)
其中p(a i)表示先验概率分布，p(y i,X|a i)表示似然概率分布。先验概率分布用来抑制过拟合，因此也可以避免由于数据共线性造成矩阵病态的问题。在本发明中，先验概率分布设置为椭圆高斯分布：
Figure PCTCN2018094553-appb-000001
其中β j表示a ij分布标准差的倒数，基于噪声是高斯分布的假设，似然概率分布可以写成如下形式：
Figure PCTCN2018094553-appb-000002
为了计算参数a i，贝叶斯线性回归使用最大后验优化算法，算法采用迭代的方法求解 最优参数，具体流程详见M.E.Tipping,"Sparse Bayesian learning and the relevance vector machine,"Journal of machine learning research,vol.1,pp.211-244,2001。
发明内容
本发明的目的是为克服已有技术的不足之处，提出一种基于数据驱动的电网潮流方程线性化求解方法。本发明可提高电网潮流方程计算的精度，有助于降低电网运行成本，同时本发明减少了计算量，提升计算的灵活性，更加符合工程应用中电力系统的实际情况。。
本发明提出一种基于数据驱动的电网潮流方程线性化求解方法，其特征在于，该方法包括以下步骤：
1)根据电力系统各个节点的节点类型建立从有功注入、无功注入到电压相角、电压幅值的映射方程；具体步骤如下：
1-1)将电力系统的所有节点划分为PQ,PV,Vθ节点，根据不同节点类型，将电网运行的有功注入、无功注入、电压幅值和电压相角数据分别按照PQ,PV,Vθ节点的顺序排列：
Figure PCTCN2018094553-appb-000003
其中，P表示电网各个节点有功注入向量，
Figure PCTCN2018094553-appb-000004
表示PQ节点有功注入向量的转置，
Figure PCTCN2018094553-appb-000005
表示PV节点有功注入向量的转置，
Figure PCTCN2018094553-appb-000006
表示Vθ节点有功注入向量的转置；Q表示电网各个节点无功注入向量，
Figure PCTCN2018094553-appb-000007
表示PQ节点无功注入向量的转置，
Figure PCTCN2018094553-appb-000008
表示PV节点无功注入向量的转置，
Figure PCTCN2018094553-appb-000009
表示Vθ节点无功注入向量的转置；V表示电网各个节点电压幅值向量，
Figure PCTCN2018094553-appb-000010
表示PQ节点电压幅值向量的转置，
Figure PCTCN2018094553-appb-000011
表示PV节点电压幅值向量的转置，
Figure PCTCN2018094553-appb-000012
表示Vθ节点电压幅值向量的转置；θ表示电网各个节点电压相角向量，
Figure PCTCN2018094553-appb-000013
表示PQ节点电压相角向量的转置，
Figure PCTCN2018094553-appb-000014
表示PV节点电压相角向量的转置，
Figure PCTCN2018094553-appb-000015
表示Vθ节点电压相角向量的转置；
1-2)利用步骤1-1)的结果，构建从有功注入、无功注入到电压相角、电压幅值的映射方程，表达式如下：
Figure PCTCN2018094553-appb-000016
其中，C 1～C 6表示回归中的常量矩阵，A ij表示回归参数矩阵中的子矩阵；
2)根据步骤1)建立的映射方程，以及各个节点类型的已知量未知量关系，列出潮流方程解的计算表达式；
在计算潮流时，对于式(2)中映射的左侧因变量，θ L，θ S，P R和V L是未知量，V S和V R是已知量，对于式(2)中映射的右侧自变量，P L，P S和Q L是已知量，Q S和Q R是未知量；因此，将式(2)写成分块矩阵的形式表达式如下：
Figure PCTCN2018094553-appb-000017
其中，x 1＝[P L,P S,Q L] T和y 2＝[V S,V R] T是已知量，x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T是未知量，
Figure PCTCN2018094553-appb-000018
和
Figure PCTCN2018094553-appb-000019
分别表示式(2)中A ij矩阵的左上、右上、左下、右下部分：
Figure PCTCN2018094553-appb-000020
Figure PCTCN2018094553-appb-000021
Figure PCTCN2018094553-appb-000022
3)获取电力系统历史量测数据，使用贝叶斯线性回归得到电力系统历史量测数据如式(2)所示的映射关系，得到线性化的潮流方程的解；具体步骤如下：
3-1)构建回归模型，表达式如(4)式所示：
Y＝AX  (4)
其中，X表示自变量数据矩阵，Y表示因变量数据矩阵，A表示回归参数矩阵；
对应于式(2)中的映射关系，X、Y和A的表达式分别如下：
Figure PCTCN2018094553-appb-000023
其中，上标1...t....T表示历史量测数据的时间点；
分别将X、Y和A改写为如下形式：
X＝[x 1 x 2 ... x N] T Y＝[y 1 y 2 ... y M] T A＝[a 1 a 2 ... a M] T  (6)
其中，x n代表矩阵X的第n行的转置，n＝1..N，y m代表矩阵Y的第m行的转置，a m代表矩阵A的第m行的转置，m＝1...M；N代表矩阵X的行数，M代表矩阵Y的行数；
3-2)使用贝叶斯线性回归方法得到参数矩阵：
贝叶斯线性回归依次回归每一行因变量对应的参数：
y i＝a iX+e i,i＝1,2,...,M  (7)
采用迭代方法对因变量对应的参数通过最大后验原理求解，最大后验分布正比于：
Figure PCTCN2018094553-appb-000024
3-3)利用步骤3-2)的结果，根据式(9)计算未知量得到线性化的潮流方程的解：
Figure PCTCN2018094553-appb-000025
其中x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T即为线性化的潮流方程的解。
本发明的特点及有益效果在于：
1)不需要系统拓扑和参数信息。在一些地区的配电网中，由于高比例分布式可再生能源的渗透和主动式配电网的普及，配电网真实的系统拓扑、元件参数和控制逻辑往往很难精准建模。而本发明所提出的数据驱动的方法仅仅需要历史量测数据，因此在这种配电网中的应用具有一定优势。
2)线性化后的方程具有更高计算精度。由于本发明的训练数据是量测的历史数据，反映了特定电力系统的真实运行状态，因此具有更高的计算精度。例如，本发明基于数据驱动的方法可以考虑由于空气湿度等原因造成的线路参数变化造成的影响。提高计算精度后有助于降低电网运行中的成本。
3)减少了计算量。本发明得到的是线性化的电网潮流计算方程，在直接计算以及作为优化问题的约束进行计算的问题中，都能够有效减少计算量，从而能够使得运行中的调度更加实时、或者能够允许仿真、调度等模型中考虑的因素更加完整。
4)提升了计算的灵活性。本发明方法得到的线性化电网潮流模型可以考虑不同类型的节点特点进行计算，符合工程应用中电力系统的实际情况。
附图说明
图1为本发明实施例中NREL-118系统300组测试结果中的1组测试结果示意图。
图2为本发明实施例中NREL-118系统300组测试结果的直方图。
具体实施方式
本发明提出的一种基于数据驱动的电网潮流方程线性化求解方法，下面结合附图及具体实施例进一步详细说明如下。
本发明提出的一种基于数据驱动的电网潮流方程线性化求解方法，该方法包括以下步骤：
1)根据电力系统各个节点的节点类型建立从有功注入、无功注入到电压相角、电压幅值的映射方程，使得按照映射方式回归的方程能够考虑数据之前的共线性，并且能够便于计算潮流。具体步骤如下：
1-1)将电力系统的所有节点划分为PQ,PV,Vθ节点，根据不同节点类型，将电网运行的有功注入、无功注入、电压幅值和电压相角数据分别按照PQ,PV,Vθ节点的顺序排列：
Figure PCTCN2018094553-appb-000026
其中，P表示电网各个节点有功注入向量，
Figure PCTCN2018094553-appb-000027
表示PQ节点有功注入向量的转置，
Figure PCTCN2018094553-appb-000028
表示PV节点有功注入向量的转置，
Figure PCTCN2018094553-appb-000029
表示Vθ节点有功注入向量的转置；Q表示电网各个节点无功注入向量，
Figure PCTCN2018094553-appb-000030
表示PQ节点无功注入向量的转置，
Figure PCTCN2018094553-appb-000031
表示PV节点无功注入向量的转置，
Figure PCTCN2018094553-appb-000032
表示Vθ节点无功注入向量的转置；V表示电网各个节点电压幅值向量，
Figure PCTCN2018094553-appb-000033
表示PQ节点电压幅值向量的转置，
Figure PCTCN2018094553-appb-000034
表示PV节点电压幅值向量的转置，
Figure PCTCN2018094553-appb-000035
表示Vθ节点电压幅值向量的转置；θ表示电网各个节点电压相角向量，
Figure PCTCN2018094553-appb-000036
表示PQ节点电压相角向量的转置，
Figure PCTCN2018094553-appb-000037
表示PV节点电压相角向量的转置，
Figure PCTCN2018094553-appb-000038
表示Vθ节点电压相角向量的转置。在实际应用中，上述数据通过PMU、SCADA系统中获取。
1-2)利用步骤1-1)的结果，构建从有功注入、无功注入到电压相角、电压幅值的映射方程，表达式如下：
Figure PCTCN2018094553-appb-000039
其中，C 1～C 6表示回归中的常量矩阵，A ij表示回归参数矩阵中的子矩阵。在式(2)中，[θ L θ S P R V L V S V R] T和[P L P S Q L Q S Q R] T是已知量，A ij和C 1～C 6是待回归的参数。
本发明建立的映射是从有功注入、无功注入到电压相角、电压幅值的映射，这考虑了在电力系统中存在一部分PQ节点(如变电站节点)的有功、无功注入为零的情况。这种情况会使得回归参数中对应项为零，若使用与本发明相反方向的映射，即从电压相角、幅值到有功注入、无功注入的映射，在求解潮流方程的过程中会出现矩阵不可逆的情况，而本发明所构建的映射则不会出现矩阵不可逆的情况。
本发明所建立的映射中，自变量中去除了参考节点的有功注入。这是因为对于绝大多数电力系统，有功网损相比有功注入而言可以忽略不计，即各个节点的有功功率加和近似为零。因此各个节点的注入之间具有共线性关系，为保证在回归时所有的有功注入具有较强的独立性，本发明将参考节点的有功注入从自变量中去除。
2)根据步骤1)建立的映射方程，以及各个节点类型的已知量未知量关系，列出潮流方程解的计算表达式。
在计算潮流时，对于式(2)中映射的左侧因变量，θ L，θ S，P R和V L是未知量，V S和V R是已知量。类似地，对于式(2)中映射的右侧自变量，P L，P S和Q L是已知量，Q S和Q R是未知量。因此，式(2)可以根据已知未知的划分写成分块矩阵的形式：
Figure PCTCN2018094553-appb-000040
其中，x 1＝[P L,P S,Q L] T和y 2＝[V S,V R] T是已知量，x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T是未知量，
Figure PCTCN2018094553-appb-000041
和
Figure PCTCN2018094553-appb-000042
分别表示式(2)中A ij矩阵的左上、右上、左下、右下部分：
Figure PCTCN2018094553-appb-000043
Figure PCTCN2018094553-appb-000044
Figure PCTCN2018094553-appb-000045
3)获取电力系统历史量测数据，使用贝叶斯线性回归得到电力系统历史量测数据如式(2)所示的映射关系，得到线性化的潮流方程的解；具体步骤如下：
3-1)构建统一的回归模型；
为了方便表达，统一的回归模型如(4)式所示：
Y＝AX  (4)
其中，X表示自变量数据矩阵，Y表示因变量数据矩阵，A表示回归参数矩阵。这样的表达方式对应于式(2)中的关系如下：
Figure PCTCN2018094553-appb-000046
其中，上标1...t....T表示历史量测数据的时间点，每一个时间点的数据组成一组数据，表示系统中所有节点某一时刻的1组有功、无功注入，电压相角、电压幅值的数据。对于绝大多数情况，历史量测数据越多，效果越好，没有一个固定的历史量测数据大小要求。本发明根据实际操作经验，建议历史量测数据组数应当不少于系统节点数的2.4倍。
分别将X、Y和A改写为如下形式：
X＝[x 1 x 2 ... x N] T Y＝[y 1 y 2 ... y M] T A＝[a 1 a 2 ... a M] T  (6)
其中，x n代表矩阵X的第n行的转置，n＝1..N，y m代表矩阵Y的第m行的转置，a m代表矩阵A的第m行的转置，m＝1...M；N代表矩阵X的行数，M代表矩阵Y的行数，N和M的取值由式(5)决定。
3-2)使用贝叶斯线性回归方法得到参数矩阵：
贝叶斯线性回归依次回归每一行因变量对应的参数：
y i＝a iX+e i,i＝1,2,...,M  (7)
因变量对应的参数通过最大后验原理求解，最大后验分布正比于：
Figure PCTCN2018094553-appb-000047
求解过程采用迭代方法，贝叶斯线性回归的具体原理详见背景技术部分。
3-3)利用步骤3-2)的结果，根据式(9)计算未知量得到线性化的潮流方程的解：
Figure PCTCN2018094553-appb-000048
其中x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T即为线性化的潮流方程的解。
实施例：
本发明以I.Pena,C.Brancucci and B.M.Hodge,"An Extended IEEE 118-bus Test System with High Renewable Penetration,"IEEE Trans.Power Syst.,p.1-1,2017中的NREL-118测试系统所提供的负荷数据以及网架数据为例对本发明所提出的方法进行视力验证。其中有功负荷数据是根据1980-2012年的天气和负荷数据模拟产生。无功负荷数据根据有功负荷值与随机生成的乘子相乘而得，乘子数值介于[0.15-0.25]之间。本实例共取了300组数据用于训练，300组数据用于测试。
根据本发明提出的方法得到的线性化电网潮流方程计算潮流，并与精确的交流潮流计算结果对比，其结果如图1和图2所示。其中图1侧重于表示细节，展示了300组测试结果中的1组测试结果。横轴表示节点号，纵轴右侧表示误差，左侧表示具体计算数值。而图1(a)表示电压相角的测试结果，图1(b)表示电压幅值的测试结果。图2侧重于表示整体，展示了300组结果的直方图。其中横轴表示误差，纵轴表示频数，BLR代表贝叶斯线性回归算法，图2(a)表示电压相角的测试情况，图2(b)表示电压幅值的测试情况。图1和图2的结果都引入了一些对比方法，其中DLPF是传统的基于模型的潮流方程线性化方法的代表，详细原理见J.Yang,N.Zhang,C.Kang,and Q.Xia,"A State-Independent Linear Power Flow Model with Accurate Estimation of Voltage Magnitude,"IEEE Trans.Power Syst.,vol.22,pp.3607-3617,2017，而LS是最小二乘方法，代表了未考虑数据共线性时候的回归方法。误差用绝对值误差来衡量。
从图1中可以看出，本发明所提出的BLR方法相比于基于模型的潮流线性化方法DLPF以及不考虑数据共线性的LS方法，几乎在每个节点上计算结果都更加精确。从图2可以 看出，在300组测试结果中，本发明所提出的BLR方法的误差分布相比于DLPF和LS方法的误差分布更小。其中LS的误差分布很广，说明这种不考虑数据共线性的方法，在实际情况中表现不稳定。
对于绝大多数情况，训练数据越多，效果越好，没有一个固定的训练数据大小要求。本发明根据实际操作经验，建议训练数据组数应当不少于系统节点数的2.4倍。

Claims (1)

  1. 一种基于数据驱动的电网潮流方程线性化求解方法，其特征在于，该方法包括以下步骤：
    1)根据电力系统各个节点的节点类型建立从有功注入、无功注入到电压相角、电压幅值的映射方程；具体步骤如下：
    1-1)将电力系统的所有节点划分为PQ,PV,Vθ节点，根据不同节点类型，将电网运行的有功注入、无功注入、电压幅值和电压相角数据分别按照PQ,PV,Vθ节点的顺序排列：
    Figure PCTCN2018094553-appb-100001
    其中，P表示电网各个节点有功注入向量，
    Figure PCTCN2018094553-appb-100002
    表示PQ节点有功注入向量的转置，
    Figure PCTCN2018094553-appb-100003
    表示PV节点有功注入向量的转置，
    Figure PCTCN2018094553-appb-100004
    表示Vθ节点有功注入向量的转置；Q表示电网各个节点无功注入向量，
    Figure PCTCN2018094553-appb-100005
    表示PQ节点无功注入向量的转置，
    Figure PCTCN2018094553-appb-100006
    表示PV节点无功注入向量的转置，
    Figure PCTCN2018094553-appb-100007
    表示Vθ节点无功注入向量的转置；V表示电网各个节点电压幅值向量，
    Figure PCTCN2018094553-appb-100008
    表示PQ节点电压幅值向量的转置，
    Figure PCTCN2018094553-appb-100009
    表示PV节点电压幅值向量的转置，
    Figure PCTCN2018094553-appb-100010
    表示Vθ节点电压幅值向量的转置；θ表示电网各个节点电压相角向量，
    Figure PCTCN2018094553-appb-100011
    表示PQ节点电压相角向量的转置，
    Figure PCTCN2018094553-appb-100012
    表示PV节点电压相角向量的转置，
    Figure PCTCN2018094553-appb-100013
    表示Vθ节点电压相角向量的转置；
    1-2)利用步骤1-1)的结果，构建从有功注入、无功注入到电压相角、电压幅值的映射方程，表达式如下：
    Figure PCTCN2018094553-appb-100014
    其中，C 1～C 6表示回归中的常量矩阵，A ij表示回归参数矩阵中的子矩阵；
    3)根据步骤1)建立的映射方程，以及各个节点类型的已知量未知量关系，列出潮流方程解的计算表达式；
    在计算潮流时，对于式(2)中映射的左侧因变量，θ L，θ S，P R和V L是未知量，V S和V R是已知量，对于式(2)中映射的右侧自变量，P L，P S和Q L是已知量，Q S和Q R是未 知量；因此，将式(2)写成分块矩阵的形式表达式如下：
    Figure PCTCN2018094553-appb-100015
    其中，x 1＝[P L,P S,Q L] T和y 2＝[V S,V R] T是已知量，x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T是未知量，
    Figure PCTCN2018094553-appb-100016
    和
    Figure PCTCN2018094553-appb-100017
    分别表示式(2)中A ij矩阵的左上、右上、左下、右下部分：
    Figure PCTCN2018094553-appb-100018
    Figure PCTCN2018094553-appb-100019
    3)获取电力系统历史量测数据，使用贝叶斯线性回归得到电力系统历史量测数据如式(2)所示的映射关系，得到线性化的潮流方程的解；具体步骤如下：
    3-1)构建回归模型，表达式如(4)式所示：
    Y＝AX      (4)
    其中，X表示自变量数据矩阵，Y表示因变量数据矩阵，A表示回归参数矩阵；
    对应于式(2)中的映射关系，X、Y和A的表达式分别如下：
    Figure PCTCN2018094553-appb-100020
    其中，上标1...t....T表示历史量测数据的时间点；
    分别将X、Y和A改写为如下形式：
    X＝[x 1 x 2 ... x N] T Y＝[y 1 y 2 ... y M] T A＝[a 1 a 2 ... a M] T  (6)
    其中，x n代表矩阵X的第n行的转置，n＝1..N，y m代表矩阵Y的第m行的转置，a m代表矩阵A的第m行的转置，m＝1...M；N代表矩阵X的行数，M代表矩阵Y的行数；
    3-2)使用贝叶斯线性回归方法得到参数矩阵：
    贝叶斯线性回归依次回归每一行因变量对应的参数：
    y i＝a iX+e i,i＝1,2,...,M       (7)
    采用迭代方法对因变量对应的参数通过最大后验原理求解，最大后验分布正比于：
    Figure PCTCN2018094553-appb-100021
    3-3)利用步骤3-2)的结果，根据式(9)计算未知量得到线性化的潮流方程的解：
    Figure PCTCN2018094553-appb-100022
    其中x 2＝[Q S,Q R] T和y 1＝[θ L,θ S,P R,V L] T即为线性化的潮流方程的解。
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