CN101702521A

CN101702521A - State Estimation Method of Power System Considering the Effect of Multi-balancing Machines

Info

Publication number: CN101702521A
Application number: CN200910234999A
Authority: CN
Inventors: 孙国强; 卫志农; 叶芳
Original assignee: Hohai University HHU
Current assignee: Hohai University HHU
Priority date: 2009-11-20
Filing date: 2009-11-20
Publication date: 2010-05-05
Anticipated expiration: 2029-11-20
Also published as: CN101702521B

Abstract

The invention discloses a state estimation method for an electric power system considering the influences of a multi-balancing machine. The method comprises the following steps of: firstly, according to the needs of the system, setting a group of multi-balancing machines for commonly sharing the imbalanced power of the system; secondly, deriving a computational formula of a measurement vector according to the setting of the multi-balancing machines, solving a Jacobian matrix of the measurement vector with respect to a quantity of state, and estimating the state of the system by using a least squares estimation criteria . Because the influences the multi-balancing machine is considered, the imbalanced power of the system is commonly shared by setting a group of multi-balancing machines, and the result of the state estimation accords with the practical situation better. In addition, the method can realize the state estimation considering the influences of the multi-balancing machine only by slightly amending the measurement function and the Jacobian matrix parts of the prior state estimation software. The invention has definite physical meaning, is convenient to realize in the traditional state estimation software and satisfies the requirements of engineering on the estimation precision.

Description

State Estimation Method of Power System Considering the Effect of Multi-balancing Machines

技术领域technical field

发明涉及一种计及多平衡机影响的电力系统状态估计方法，属于电力系统运行和控制技术领域。The invention relates to a power system state estimation method considering the influence of multiple balancing machines, and belongs to the technical field of power system operation and control.

背景技术Background technique

状态估计又称滤波，它利用实时量测系统的冗余度来提高数据精度，自动排除随机干扰所引起的错误信息，估计或预报系统的运行状态。随着西电东送的实施、电力市场的迅速推行，特高压、远距离、交直流混合输电技术在我国电网中发展迅速，电力系统调度中心的自动化水平也需要逐步提高。作为现代大型电力系统各级调度中心能量管理系统(EMS)的重要组成部分，状态估计为EMS提供可靠而完整的系统运行状态的信息，并用这些数据建立各种高级应用软件所需的数据库，被誉为应用软件的“心脏”，因此状态估计是电力系统运行、控制和安全评估等方面的基础。State estimation, also known as filtering, uses the redundancy of the real-time measurement system to improve data accuracy, automatically eliminates error information caused by random interference, and estimates or forecasts the operating state of the system. With the implementation of west-to-east power transmission and the rapid implementation of the power market, UHV, long-distance, AC-DC hybrid transmission technologies are developing rapidly in my country's power grids, and the automation level of power system dispatching centers also needs to be gradually improved. As an important part of the energy management system (EMS) of dispatching centers at all levels in modern large-scale power systems, state estimation provides EMS with reliable and complete information on the system's operating status, and uses these data to establish databases required by various advanced application software. Known as the "heart" of application software, state estimation is the basis of power system operation, control and safety assessment.

在常规状态估计中，通常只设置一个平衡节点，对应一个平衡机，用以平衡系统中的不平衡功率，但这与电力系统的实际情况并不相符。因为在实际电力系统中，当系统出现较小的不平衡功率时，不平衡功率由所有有容量裕度的发电机和所有负荷根据各自的有功频率特性系数来共同分配，也就是一次调频；当系统出现较大的不平衡功率时，不平衡功率除了通过一次调频来调节外，还可以利用具有二次调频能力的发电机组来进行二次调频。因此，当系统不平衡功率较小时，采用设置一个平衡发电机的常规状态估计方法不会对系统运行状态产生太大的影响；而当系统不平衡功率较大时，采用只设置一个平衡发电机的常规状态估计方法是不适当的，由此得到的系统运行状态有可能会偏离实际状态较大。而计及多平衡机影响的状态估计方法则弥补了以上缺点，它通过设置一组平衡发电机组来共同承担系统不平衡功率，使状态估计结果更符合实际情况。目前，多平衡机这个概念多见于电力系统潮流计算中，多平衡机下的电力系统状态估计研究尚未见文献报道。In conventional state estimation, usually only one balance node is set, corresponding to one balance machine, to balance the unbalanced power in the system, but this is not consistent with the actual situation of the power system. Because in the actual power system, when there is a small unbalanced power in the system, the unbalanced power is shared by all generators with capacity margins and all loads according to their respective active frequency characteristic coefficients, that is, primary frequency regulation; when When the system has a large unbalanced power, in addition to adjusting the unbalanced power through primary frequency regulation, the generator set with secondary frequency regulation capability can also be used for secondary frequency regulation. Therefore, when the unbalanced power of the system is small, the conventional state estimation method of setting a balanced generator will not have a great impact on the operating state of the system; and when the unbalanced power of the system is large, using only one balanced generator The conventional state estimation method is inappropriate, and the resulting system operating state may deviate greatly from the actual state. The state estimation method that takes into account the influence of multiple balancing machines makes up for the above shortcomings. It sets a group of balanced generators to share the unbalanced power of the system, so that the state estimation results are more in line with the actual situation. At present, the concept of multi-balancing machines is mostly used in the power flow calculation of power systems, and the research on power system state estimation under multi-balancing machines has not been reported in the literature.

发明内容Contents of the invention

本发明所要解决的技术问题是针对互联电网单平衡机的设置已经难以满足对不平衡功率的平衡要求这一缺陷提供一种计及多平衡机影响的电力系统状态估计方法。The technical problem to be solved by the present invention is to provide a power system state estimation method that takes into account the influence of multiple balancing machines for the defect that the setting of a single balancing machine in the interconnected grid is difficult to meet the balancing requirements for unbalanced power.

本发明为实现上述目的，采用如下技术方案：In order to achieve the above object, the present invention adopts the following technical solutions:

本发明为计及多平衡机影响的电力系统状态估计方法，其特征在于包括以下步骤：The present invention is the power system state estimation method that takes into account the influence of multi-balancing machine, is characterized in that comprising the following steps:

(1)获取电力系统的网络参数，包括：输电线路的支路号、首端节点和末端节点编号、串联电阻、串联电抗、并联电导、并联电纳、变压器变比和阻抗；(1) Obtain the network parameters of the power system, including: the branch number of the transmission line, the number of the first-end node and the end node, series resistance, series reactance, parallel conductance, parallel susceptance, transformer ratio and impedance;

(2)初始化，包括：对状态量设置初值、节点次序优化、形成节点导纳矩阵、设置门槛值，分配内存；(2) Initialization, including: setting the initial value for the state quantity, optimizing the order of nodes, forming a node admittance matrix, setting a threshold value, and allocating memory;

(3)输入遥测数据z，包括电压幅值、发电机有功功率、发电机无功功率、负荷有功功率、负荷无功功率、线路首端有功功率、线路首端无功功率、线路末端有功功率以及线路末端无功功率；(3) Input telemetry data z, including voltage amplitude, generator active power, generator reactive power, load active power, load reactive power, line head end active power, line head end reactive power, line end active power and reactive power at the end of the line;

(4)电力系统有n个节点，其中节点n，…，n-m所在的发电机组成平衡机组，承担电力系统的不平衡功率；(4) There are n nodes in the power system, and the generators where nodes n, ..., n-m are located form a balanced unit to undertake the unbalanced power of the power system;

(5)恢复迭代计数器的迭代次数k＝1；(5) Recover the number of iterations k=1 of the iteration counter;

(6)由现有的状态量x^(k)，按照下式计算各量测量的计算值h(x^(k))；(6) From the existing state quantity x ^(k) , calculate the calculated value h(x ^(k) ) of each quantity measurement according to the following formula;

节点i电压幅值为V_i＝v_i：The voltage amplitude of node i is V _i =v _i :

节点i注入功率：Node i injected power:

$\{\begin{matrix} {P P}_{i i} = = {v v}_{i i} {Σ Σ}_{j j = = 11}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) \\ {Q Q}_{i i} = = {v v}_{i i} {Σ Σ}_{j j = = 11}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) \end{matrix},,$

线路i-j上始端功率：Start-end power on line i-j:

$\{\begin{matrix} {P P}_{ij ij} = = {v v}_{i i}^{22} g g - - {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} - - {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \\ {Q Q}_{ij ij} = = - - {v v}_{i i}^{22} ((b b + + {y the y}_{c c})) - - {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix},,$

线路i-j上末端功率：Terminal power on line i-j:

$\{\begin{matrix} {P P}_{ji the ji} = = {v v}_{j j}^{22} g g - - {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \\ {Q Q}_{ji the ji} = = - - {v v}_{j j}^{22} ((b b + + {y the y}_{c c})) + + {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix},,$

变压器线路i-j上始端功率：Power at the beginning of the transformer line i-j:

$\{\begin{matrix} {P P}_{ij ij} = = - - \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} sin sin {θ θ}_{ij ij} \\ {Q Q}_{ij ij} = = - - \frac{11}{{K K}^{22}} {v v}_{i i}^{22} {b b}_{T T} + + \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} cos cos {θ θ}_{ij ij} \end{matrix},,$

变压器线路i-j上末端功率：Terminal power on transformer line i-j:

$\{\begin{matrix} {P P}_{ji the ji} = = \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} sin sin {θ θ}_{ij ij} \\ {Q Q}_{ji the ji} = = - - {v v}_{j j}^{22} {b b}_{T T} + + \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} cos cos {θ θ}_{ij ij} \end{matrix},,$

式中v_i、v_j分别为节点i、节点j的电压幅值；θ_ij为节点i和节点j的电压相角差；G_ij、B_ij为导纳矩阵的实部和虚部；g、b、y_c分别为线路的电导、电纳、接地电纳；K为变压器的非标准变比；b_T为变压器标准测的电纳；P、Q分别表示发电机的有功功率和无功功率；i、j＝1，2，…，n-m-1，i、j、m、n都为大于零的自然数，m＜n；where v _i and v _j are the voltage amplitudes of node i and node j respectively; θ _ij is the voltage phase angle difference between node i and node j; G _ij and B _ij are the real and imaginary parts of the admittance matrix; g , b, y _c are the conductance, susceptance and grounding susceptance of the line respectively; K is the non-standard transformation ratio of the transformer; b _T is the susceptance measured by the standard transformer; P and Q respectively represent the active power and reactive power of the generator Power; i, j=1, 2,..., nm-1, i, j, m, n are all natural numbers greater than zero, m<n;

(7)对步骤(6)各式分别关于状态量进行求导，得到量测量的雅克比矩阵H(x^(k))：(7) Deriving the various equations in step (6) with respect to the state quantity, and obtaining the Jacobian matrix H(x ^(k) ) of the quantity measurement:

节点注入功率：Node injection power:

$\{\begin{matrix} \frac{&PartialD; &PartialD; {P P}_{i i}}{{&PartialD; &PartialD; v v}_{i i}} = = {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) + + {Σ Σ}_{\underset{j j &NotEqual; &NotEqual; i i}{j j = = 11}}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) + + 22 {G G}_{ii i} {v v}_{i i} \\ \frac{&PartialD; &PartialD; {P P}_{i i}}{{&PartialD; &PartialD; θ θ}_{i i}} = = {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} ((- - {G G}_{ij ij} sin sin {θ θ}_{ij ij} + + {B B}_{ij ij} cos cos {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{\underset{j j &NotEqual; &NotEqual; i i}{j j = = 11}}^{n no - - m m - - 11} {v v}_{j j} ((- - {G G}_{ij ij} sin sin {θ θ}_{ij ij} + + {B B}_{ij ij} cos cos {θ θ}_{ij ij})) \end{matrix}$

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

$\{\begin{matrix} \frac{&PartialD; &PartialD; {P P}_{i i}}{{&PartialD; &PartialD; v v}_{j j}} = = {v v}_{i i} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) \\ \frac{{&PartialD; &PartialD; P P}_{i i}}{{&PartialD; &PartialD; θ θ}_{j j}} = = {v v}_{i i} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) \end{matrix}$

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

$\{\begin{matrix} \frac{{&PartialD; &PartialD; Q Q}_{i i}}{{&PartialD; &PartialD; v v}_{i i}} = = {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) + + {Σ Σ}_{\underset{j j &NotEqual; &NotEqual; i i}{j j = = 11}}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) - - 22 {B B}_{ii i} {v v}_{i i} \\ \frac{{&PartialD; &PartialD; Q Q}_{i i}}{{&PartialD; &PartialD; θ θ}_{i i}} = = {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{\underset{j j &NotEqual; &NotEqual; i i}{j j = = 11}}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} {sin sin θ θ}_{ij ij})) \end{matrix}$

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

$\{\begin{matrix} \frac{{&PartialD; &PartialD; Q Q}_{i i}}{{&PartialD; &PartialD; v v}_{j j}} = = {v v}_{i i} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) \\ \frac{{&PartialD; &PartialD; Q Q}_{i i}}{{&PartialD; &PartialD; θ θ}_{j j}} = = - - {v v}_{i i} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) \end{matrix}$

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

线路i-j上的始端功率Head end power on line i-j

$\{\begin{matrix} \frac{{&PartialD; &PartialD; P P}_{ij ij}}{{&PartialD; &PartialD; v v}_{i i}} = = 22 {v v}_{i i} g g - - {v v}_{j j} g g cos cos {θ θ}_{ij ij} - - {v v}_{j j} b b sin sin {θ θ}_{ij ij} \\ \frac{{&PartialD; &PartialD; P P}_{ij ij}}{{&PartialD; &PartialD; θ θ}_{i i}} = = {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} - - {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix}$

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

$\{\begin{matrix} \frac{{&PartialD; &PartialD; P P}_{ij ij}}{{&PartialD; &PartialD; v v}_{j j}} = = - - {v v}_{i i} g g cos cos {θ θ}_{ij ij} - - {v v}_{i i} b b sin sin {θ θ}_{ij ij} \\ \frac{{&PartialD; &PartialD; P P}_{ij ij}}{{&PartialD; &PartialD; θ θ}_{j j}} = = - - {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix}$

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

$\{\begin{matrix} \frac{{&PartialD; &PartialD; Q Q}_{ij ij}}{{&PartialD; &PartialD; v v}_{i i}} = = - - 22 {v v}_{i i} ((b b + + {y the y}_{c c})) - - {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{j j} b b cos cos {θ θ}_{ij ij} \\ \frac{{&PartialD; &PartialD; Q Q}_{ij ij}}{{&PartialD; &PartialD; θ θ}_{i i}} = = - - {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} - - {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \end{matrix}$

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

$\{\begin{matrix} \frac{{&PartialD; &PartialD; Q Q}_{ij ij}}{{&PartialD; &PartialD; v v}_{j j}} = = - - {v v}_{i i} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} b b cos cos {θ θ}_{ij ij} \\ \frac{{&PartialD; &PartialD; Q Q}_{ij ij}}{{&PartialD; &PartialD; θ θ}_{j j}} = = {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \end{matrix}$

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

(8)求取状态修正量

选取并修正状态量得到

(8) Find the state correction amount

select And modify the state quantity to get

$Δ Δ {\overset{^^}{x x}}^{((k k))} = = {[[{H h}^{T T} (({\overset{^^}{x x}}^{((k k))})) {R R}^{- - 11} H h (({\overset{^^}{x x}}^{((k k))}))]]}^{- - 11} \times \times {H h}^{T T} (({\overset{^^}{x x}}^{((k k))})) {R R}^{- - 11} [[z z - - h h (({\overset{^^}{x x}}^{((k k))}))]]$

${\overset{^^}{x x}}^{((k k + + 11))} = = {\overset{^^}{x x}}^{((k k))} + + Δ Δ {\overset{^^}{x x}}^{((k k))}$

其中，x为电力系统的2n-1维状态变量；是状态量的估计值；z为量测值矢量即遥测数据；h(x)是x的非线性函数；H(x)为量测函数的雅克比矩阵；R^-1为量测矢量的加权阵；T为转秩符号；(k)表示迭代序号；Among them, x is the 2n-1 dimensional state variable of the power system; is the estimated value of the state quantity; z is the measured value vector, that is, the telemetry data; h(x) is the nonlinear function of x; H(x) is the Jacobian matrix of the measured function; R ^-1 is the weighted value of the measured vector array; T is the transfer rank symbol; (k) represents the iteration sequence number;

(9)当

小于设定的收敛标准，结束状态估计，否则返回步骤(6)进行第k+1次估计。(9) when

If it is less than the set convergence standard, end the state estimation, otherwise return to step (6) for the k+1th estimation.

本发明提出的计及多平衡机影响的电力系统状态估计方法，通过设置一组平衡发电机组来共同承担系统的不平衡功率，使得状态估计的结果更符合实际情况，而且只需对现有状态估计软件就量测函数和雅克比矩阵部分稍作修正，即可实现计及多平衡机影响的状态估计，物理意义明确，便于在已有的状态估计软件上实现，并满足工程对估计精度的要求。The method for estimating the state of a power system that takes into account the influence of multi-balancing machines proposed by the present invention sets up a group of balanced generators to share the unbalanced power of the system, so that the result of the state estimation is more in line with the actual situation, and only the existing state The estimation software can realize the state estimation considering the influence of multi-balancing machines by slightly modifying the measurement function and the Jacobian matrix. Require.

附图说明Description of drawings

图1：本发明方法流程图。Fig. 1: Flow chart of the method of the present invention.

图2：本发明采用的元件等值电路图，其中：图(a)是线路∏形等值电路图，图(b)是变压器∏形等值电路图。Fig. 2: the component equivalent circuit diagram that the present invention adopts, wherein: figure (a) is circuit ∏ shape equivalent circuit diagram, and figure (b) is transformer ∏ shape equivalent circuit diagram.

图3：本发明提出的计及多平衡机影响的状态估计方法所应用的两个小算例系统，其中：图(a)是IEEE-14节点系统，图(b)是IEEE-30节点系统。Fig. 3: Two small example systems applied by the state estimation method considering the influence of multi-balancing machines proposed by the present invention, wherein: Fig. (a) is an IEEE-14 node system, and Fig. (b) is an IEEE-30 node system .

具体实施方式Detailed ways

下面结合附图对发明的技术方案进行详细说明：Below in conjunction with accompanying drawing, the technical scheme of invention is described in detail:

1969年美国麻省理工学院的许怀丕(F.C.Schweppe)等人提出了电力系统状态估计的最基本算法——基本加权二乘状态估计算法(WLS)，其基本思想是以量测量和量测估计值之差的平方和最小为目标准则的估计方法。该方法以其模型简单，收敛性能好，估计质量高的特点得到了广泛的应用。本文正是在WLS算法的基础上提出了计及多平衡机影响的电力系统状态估计方法。In 1969, F.C.Schweppe of the Massachusetts Institute of Technology and others proposed the most basic algorithm for power system state estimation - the basic weighted square state estimation algorithm (WLS). The basic idea is to measure and measure the estimated value The minimum sum of squares of the differences is the estimation method of the objective criterion. This method has been widely used for its simple model, good convergence performance and high estimation quality. This paper proposes a power system state estimation method that takes into account the influence of multiple balancing machines on the basis of the WLS algorithm.

如图1所示，在给定网络接线、支路参数和量测系统的条件下，非线性量测方程可表示为：As shown in Figure 1, under the condition of given network connection, branch parameters and measurement system, the nonlinear measurement equation can be expressed as:

z＝h(x)+vz=h(x)+v

式中，z为量测值矢量即遥测数据，绝大多数是通过遥测得到的，也有一小部分是人工设定的；h(x)为由基尔霍夫等基本电路定律所建立的量测函数；x为系统状态变量；v为量测随机误差，假设其服从均值为零、方差为σ²的正态分布。设系统有n个节点，以节点电压幅值和电压相角作为状态变量，节点n，…，n-m所在的发电机符合平衡机的选取要求(即作为平衡机的发电机应有较大的调节余量)，组成平衡机组，则

…，

In the formula, z is the measured value vector, that is, telemetry data, most of which are obtained through telemetry, and a small part is manually set; h(x) is the quantity established by basic circuit laws such as Kirchhoff measurement function; x is the system state variable; v is the measurement random error, assuming that it obeys the normal distribution with mean value zero and variance ^σ2 . Assuming that the system has n nodes, the node voltage amplitude and voltage phase angle are used as state variables, and the generators located at nodes n, ..., nm meet the selection requirements of the balancing machine (that is, the generator as a balancing machine should have a large adjustment margin), to form a balanced unit, then

...,

在电力系统状态估计中，量测量配置的类型要比常规潮流多，不仅包括了各节点的注入功率量测P_i、Q_i，还可以包括支路的功率量测P_ij、Q_ij、P_ji、Q_ji以及节点的电压幅值量测V_i，量测方程如下式所示：In power system state estimation, there are more types of measurement configuration than conventional power flow, including not only the injected power measurements P _i , Q _i of each node, but also the branch power measurements P _ij , Q _ij , P _ji , Q _ji and the node voltage amplitude measurement V _i , the measurement equation is as follows:

节点注入功率：Node injection power:

$\{\begin{matrix} {P P}_{i i} = = {v v}_{i i} {Σ Σ}_{j j = = 11}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} cos cos {θ θ}_{ij ij} + + {B B}_{ij ij} sin sin {θ θ}_{ij ij})) \\ {Q Q}_{i i} = = {v v}_{i i} {Σ Σ}_{j j = = 11}^{n no - - m m - - 11} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) + + {v v}_{i i} {Σ Σ}_{j j = = n no - - m m}^{n no} {v v}_{j j} (({G G}_{ij ij} sin sin {θ θ}_{ij ij} - - {B B}_{ij ij} cos cos {θ θ}_{ij ij})) \end{matrix}$

线路i-j上始端功率：Start-end power on line i-j:

$\{\begin{matrix} {P P}_{ij ij} = = {v v}_{i i}^{22} g g - - {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} - - {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \\ {Q Q}_{ij ij} = = - - {v v}_{i i}^{22} ((b b + + {y the y}_{c c})) - - {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix}$

线路i-j上末端功率：Terminal power on line i-j:

$\{\begin{matrix} {P P}_{ji the ji} = = {v v}_{j j}^{22} g g - - {v v}_{i i} {v v}_{j j} g g cos cos {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b sin sin {θ θ}_{ij ij} \\ {Q Q}_{ji the ji} = = - - {v v}_{j j}^{22} ((b b + + {y the y}_{c c})) + + {v v}_{i i} {v v}_{j j} g g sin sin {θ θ}_{ij ij} + + {v v}_{i i} {v v}_{j j} b b cos cos {θ θ}_{ij ij} \end{matrix}$

$\{\begin{matrix} {P P}_{ij ij} = = - - \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} sin sin {θ θ}_{ij ij} \\ {Q Q}_{ij ij} = = - - \frac{11}{{K K}^{22}} {v v}_{i i}^{22} {b b}_{T T} + + \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} cos cos {θ θ}_{ij ij} \end{matrix}$

变压器线路i-j上末端功率：Terminal power on transformer line i-j:

$\{\begin{matrix} {P P}_{ji the ji} = = \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} sin sin {θ θ}_{ij ij} \\ {Q Q}_{ji the ji} = = - - {v v}_{j j}^{22} {b b}_{T T} + + \frac{11}{K K} {v v}_{i i} {v v}_{j j} {b b}_{T T} cos cos {θ θ}_{ij ij} \end{matrix}$

节点i电压幅值：Node i voltage amplitude:

V_i＝v_i，i、j＝1，2，…，n-m-1V _i =v _i , i, j=1, 2, ..., nm-1

上述量测方程中，v_i、v_j分别为节点i、节点j的电压幅值；θ_ij为节点i和节点j的电压相角差；G_ij、B_ij为导纳矩阵的实部和虚部；g、b、y_c为线路的电导、电纳、接地电纳；K为变压器的非标准变比；b_T为变压器标准测的电纳。In the above measurement equations, v _i and v _j are the voltage amplitudes of node i and node j respectively; θ _ij is the voltage phase angle difference between node i and node j; G _ij and B _ij are the real part of the admittance matrix and Imaginary part; g, b, y _c are the conductance, susceptance, and grounding susceptance of the line; K is the non-standard transformation ratio of the transformer; b _T is the susceptance measured by the transformer standard.

给定量测矢量z以后，状态估计问题就是求使目标函数After the measurement vector z is given, the state estimation problem is to find the objective function

J(x)＝[z-h(x)]^TR^-1[z-h(x)]J(x)＝[zh(x)] ^T R ^-1 [zh(x)]

达到最小时的x的值。其中，R是以σ_i ²为对角元素的量测误差方差阵，在状态估计中取其逆矩阵为量测矢量的加权阵。The value of x at which the minimum is reached. Among them, R is the measurement error variance matrix with σ _i ² as the diagonal element, and its inverse matrix is taken as the weighted matrix of the measurement vector in the state estimation.

由于h(x)是x的非线性函数，所以无法直接计算得出

为了求取

首先要将h(x)线性化。令x₀是x的某一近似值，在x₀附近对h(x)进行泰勒展开，并忽略二次以上的高阶项后，得到：Since h(x) is a nonlinear function of x, it cannot be directly calculated

In order to obtain

The first step is to linearize h(x). Let x ₀ be an approximate value of x, and after performing Taylor expansion on h(x) near x ₀ , and ignoring the higher-order terms above two times, we get:

h(x)≈h(x₀)+H(x₀)Δxh(x)≈h(x ₀ )+H(x ₀ )Δx

式中，Δx＝x-x₀， $H (x_{0}) = \frac{&PartialD; h (x)}{&PartialD; x} |_{x = x_{0}} .$ In the formula, Δx=xx ₀ , $h (x_{0}) = \frac{&PartialD; h (x)}{&PartialD; x} |_{x = x_{0}} .$

以节点注入功率和线路i-j上的始端功率为例，量测矢量的雅克比矩阵H(x)对应的元素为：Taking the node injection power and the head end power on the line i-j as an example, the elements corresponding to the Jacobian matrix H(x) of the measurement vector are:

节点注入功率Node injected power

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

线路i-j上的始端功率Head end power on line i-j

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

(i＝1，2，…，n-m-1)(i=1, 2, . . . , n-m-1)

(j＝1，2，…，n-m-1)(j=1, 2, ..., n-m-1)

由此可见，系统中有m+1个平衡节点，雅克比矩阵将减少2(m+1)阶。It can be seen that there are m+1 balance nodes in the system, and the Jacobian matrix will be reduced by 2(m+1) order.

将h(x)的线性化表达式代入目标函数，通过数学归纳后，该目标函数可利用下式迭代求解：Substituting the linearized expression of h(x) into the objective function, after mathematical induction, the objective function can be solved iteratively using the following formula:

式中(k)表示迭代序号。where (k) represents the iteration number.

如图2所示，图(a)是线路∏形等值电路图，节点i和节点j之间串接导纳g+j′b，节点i、j的输出端分别串接一个接地电纳j′y_c后接地。As shown in Figure 2, figure (a) is a line Π-shaped equivalent circuit diagram, the admittance g+j′b is connected in series between node i and node j, and the output terminals of nodes i and j are respectively connected in series with a ground susceptance j 'y _c followed by grounding.

图(b)是变压器∏形等值电路图，节点i和节点j之间串接

节点i串接一个后接地，j的输出端串接一个后接地。j′表示虚部。Figure (b) is a transformer Π-shaped equivalent circuit diagram, and the node i and node j are connected in series

Node i is connected in series with a After grounding, the output terminal of j is connected in series with a back to ground. j' represents the imaginary part.

下面介绍本发明的两个实施例：Introduce two embodiments of the present invention below:

实施例一：Embodiment one:

本发明采用图3(a)所示的IEEE-14节点的标准算例，分别就单平衡机和多平衡机对算例进行了仿真，系统参数如表1所示，仿真结果如表2所示：The present invention adopts the standard calculation example of the IEEE-14 node shown in Figure 3 (a), and simulates the calculation examples with respect to the single balancing machine and the multi-balancing machine respectively. The system parameters are shown in Table 1, and the simulation results are shown in Table 2. Show:

表1系统参数表Table 1 System parameter list

类型 type 平衡节点balance node 冗余度Redundancy 1 1 1 1 2.152.15 2 2 1，21, 2 2.332.33 33 1，2，3，81, 2, 3, 8 2.802.80

表2仿真结果表Table 2 Simulation result table

类型1Type 1 类型2Type 2 类型3Type 3 JJ 0.0020620.002062 0.0021550.002155 0.0022320.002232 EMIEMI 0.0100310.010031 0.0100310.010031 0.0100310.010031 EEIEEI 0.0070700.007070 0.0069310.006931 0.0067480.006748 EMOEMO 0.0085360.008536 0.0085360.008536 0.0085360.008536 EEOEEO 0.0060310.006031 0.0059020.005902 0.0057360.005736

实施例二：Embodiment two:

本发明采用图3(b)所示的IEEE-30节点的标准算例，分别就单平衡机和多平衡机对算例进行了仿真，系统参数如表3所示，仿真结果如表4所示：The present invention adopts the standard calculation example of the IEEE-30 node shown in Figure 3 (b), and simulates the calculation examples with respect to the single balancing machine and the multi-balancing machine respectively. The system parameters are shown in Table 3, and the simulation results are shown in Table 4. Show:

表3系统参数表Table 3 System parameter table

类型 type 平衡节点balance node 冗余度Redundancy 1 1 1 1 2.072.07 2 2 1，2，51, 2, 5 2.222.22 33 1，2，5，8，111, 2, 5, 8, 11 2.402.40

表4仿真结果表Table 4 Simulation result table

类型1Type 1 类型2Type 2 类型3Type 3 JJ 0.0034820.003482 0.0036650.003665 0.0036810.003681 EMIEMI 0.0088910.008891 0.0088910.008891 0.0088910.008891 EEIEEI 0.0063510.006351 0.0062120.006212 0.0061640.006164 EMOEMO 0.0075130.007513 0.0075130.007513 0.0075130.007513 EEOEEO 0.0052510.005251 0.0051300.005130 0.0050870.005087

其中， $J = Σ_{i = 1}^{m} {[\frac{z_{i} - h_{i} (\hat{x})}{σ_{i}}]}^{2}$ in, $J = Σ_{i = 1}^{m} {[\frac{z_{i} - h_{i} (\hat{x})}{σ_{i}}]}^{2}$

$EMI EMI = = {[[\frac{11}{m m} {Σ Σ}_{i i = = 11}^{m m} {(({EM EM}_{i i}))}^{22}]]}^{\frac{11}{22}},,$ $EEI EEI = = {[[\frac{11}{m m} {Σ Σ}_{i i = = 11}^{m m} {(({EE EE}_{i i}))}^{22}]]}^{\frac{11}{22}}$

$EMO EMO = = {[[\frac{11}{m m} {Σ Σ}_{i i = = 11}^{m m} {(({EM EM}_{i i} / / {σ σ}_{i i}))}^{22}]]}^{\frac{11}{22}},,$ $EEO EEO = = {[[\frac{11}{m m} {Σ Σ}_{i i = = 11}^{m m} {(({EE EE}_{i i} / / {σ σ}_{i i}))}^{22}]]}^{\frac{11}{22}}$

式中，EM_i＝MEA_i-MTR_i为测量误差；EE_i＝EST_i-MTR_i为估计误差；MEA_i为第i个量测量的量测值；EST_i为第i个量测量的估计值；MTR_i第i个量测量的真值。In the formula, EM _i ＝MEA _i -MTR _i is the measurement error; EE _i ＝EST _i -MTR _i is the estimation error; MEA _i is the measurement value of the i-th measurement; EST _i is the estimate of the i-th measurement value; the true value of the i-th quantity measurement of MTR _i .

本发明方法具有如下优势，一方面，从仿真结果可以得知，相比常规的单平衡机状态估计方法，多平衡机状态估计方法的计算精度要高，估计结果更加符合电力系统的实际状态；另一方面，总体方法的构造具有明确的物理意义，保护了现有成熟的状态估计程序，只在量测矢量及其雅克比矩阵部分做了修正，对现有状态估计软件修改很小，容易实现。The method of the present invention has the following advantages. On the one hand, it can be known from the simulation results that, compared with the conventional single balancing machine state estimation method, the calculation accuracy of the multi-balancing machine state estimation method is higher, and the estimation result is more in line with the actual state of the power system; On the other hand, the construction of the overall method has a clear physical meaning, which protects the existing mature state estimation program, and only modifies the measurement vector and its Jacobian matrix. The modification of the existing state estimation software is small and easy accomplish.

Claims

1. A power system state estimation method considering multi-balancing machine influence is characterized by comprising the following steps:

(1) acquiring network parameters of an electric power system, comprising: the branch number, the head end node and the tail end node number of the power transmission line, the series resistance, the series reactance, the parallel conductance, the parallel susceptance, the transformer transformation ratio and the impedance;

(2) initializing, including: setting an initial value for the state quantity, optimizing the node sequence, forming a node admittance matrix, setting a threshold value and distributing a memory;

(3) inputting telemetering data z, including voltage amplitude, generator active power, generator reactive power, load active power, load reactive power, line head end active power, line head end reactive power, line tail end active power and line tail end reactive power;

(4) the power system is provided with n nodes, wherein generators where the nodes n, … and n-m are located form a balance unit and bear unbalanced power of the power system;

(5) recovering the iteration number k of the iteration counter to be 1;

(6) from the existing state quantity x^(k)The calculated value h (x) of each measurement is calculated according to the following formula^(k))；

The voltage amplitude of the node i is V_i＝v_i：

Node i injected power:

starting power on lines i-j:

end power on lines i-j:

the power of the upper starting end of the transformer line i-j is as follows:

power at the upper end of the transformer line i-j:

in the formula v_i、v_jVoltage amplitudes of the node i and the node j are respectively; theta_ijIs the voltage phase angle difference of node i and node j; g_ij、B_ijThe real part and the imaginary part of the admittance matrix; g. b, y_cRespectively the conductance, susceptance and grounding susceptance of the line; k is the nonstandard transformation ratio of the transformer; b_TThe standard susceptance of the transformer is measured; p, Q represent the active and reactive power of the generator, respectively; i. j is 1, 2, …, n-m-1, i, j, m and n are natural numbers which are larger than zero, and m is less than n;

(7) respectively carrying out derivation on the state quantities in the step (6) to obtain a Jacobian matrix H (x) for measuring the quantity^(k))：

And (3) injecting power into the node:

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(i＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mfenced></math>

(j＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><munderover><mi>Σ</mi><mrow><mi>j</mi><mo>=</mo><mi>n</mi><mo>-</mo><mi>m</mi></mrow><mi>n</mi></munderover><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow><mo>+</mo><munderover><mi>Σ</mi><munder><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>&NotEqual;</mo><mi>i</mi></mrow></munder><mrow><mi>n</mi><mo>-</mo><mi>m</mi><mo>-</mo><mn>1</mn></mrow></munderover><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow><mo>-</mo><msub><mrow><mn>2</mn><mi>B</mi></mrow><mi>ii</mi></msub><msub><mi>v</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><munderover><mi>Σ</mi><mrow><mi>j</mi><mo>=</mo><mi>n</mi><mo>-</mo><mi>m</mi></mrow><mi>n</mi></munderover><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow><mo>+</mo><msub><mi>v</mi><mi>i</mi></msub><munderover><mi>Σ</mi><munder><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>&NotEqual;</mo><mi>i</mi></mrow></munder><mrow><mi>n</mi><mo>-</mo><mi>m</mi><mo>-</mo><mn>1</mn></mrow></munderover><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mfenced></math>

(i＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>i</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mi>v</mi></mrow><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mi>ij</mi></msub><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>B</mi><mi>ij</mi></msub><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mfenced></math>

(j＝1，2，…，n-m-1)

starting power on lines i-j

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><msub><mrow><mn>2</mn><mi>v</mi></mrow><mi>i</mi></msub><mi>g</mi><mo>-</mo><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr></mtable></mfenced></math>

(i＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mi>v</mi></mrow><mi>i</mi></msub><mi>g</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>v</mi><mi>i</mi></msub><mi>b</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>P</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mi>v</mi></mrow><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr></mtable></mfenced></math>

(j＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mn>2</mn><mi>v</mi></mrow><mi>i</mi></msub><mrow><mo>(</mo><mi>b</mi><mo>+</mo><msub><mi>y</mi><mi>c</mi></msub><mo>)</mo></mrow><mo>-</mo><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>i</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mi>v</mi></mrow><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>-</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr></mtable></mfenced></math>

(i＝1，2，…，n-m-1)

<math><mfenced open='{' close=''><mtable><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>v</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mrow><mo>-</mo><mi>v</mi></mrow><mi>i</mi></msub><mi>g</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>v</mi><mi>i</mi></msub><mi>b</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr><mtr><mtd><mfrac><msub><mrow><mo>&PartialD;</mo><mi>Q</mi></mrow><mi>ij</mi></msub><msub><mrow><mo>&PartialD;</mo><mi>θ</mi></mrow><mi>j</mi></msub></mfrac><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>g</mi><mi>cos</mi><msub><mi>θ</mi><mi>ij</mi></msub><mo>+</mo><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mi>b</mi><mi>sin</mi><msub><mi>θ</mi><mi>ij</mi></msub></mtd></mtr></mtable></mfenced></math>

(j＝1，2，…，n-m-1)

(8) Determining a state correction amount

Selecting

And correcting the state quantity to obtain

Wherein x is a 2n-1 dimensional state variable of the power system;

is an estimate of the state quantity; z is a vector of measurement values, namely telemetry data; h (x) is a non-linear function of x; h (x) a Jacobian matrix which is a measurement function; r^-1Is measured byMeasuring a weighting array of the vectors; t is a rank conversion symbol; (k) representing an iteration number;

(9) when in useAnd (4) ending the state estimation if the convergence standard is smaller than the set convergence standard, otherwise returning to the step (6) to carry out the (k +1) th estimation.