A High-Order Derivation Method for Distribution Network Transient Data

JIANG Wentao; LI Shimian; FENG Zongcong; RAN Xiaokang; CHEN Yong

doi:10.13718/j.cnki.xdzk.2021.12.022

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Nowadays, all kinds of algorithms in distribution network need high-order derivation of data. Due to the limitation of their own theories, the traditional high-order derivation methods are difficult to calculate accurately in both low-order and high-order cases. In order to improve the calculation accuracy, a new high-order derivative method is proposed in this paper. Based on the polynomial fitting method, by analyzing the transient characteristics of voltage and current signals in the distribution network, the base function is selected for fitting reasonably, and the coefficient of the base function is calculated by the least square method to fit the objective function. The new fitting function is used to replace the original discrete data for derivative operation, which not only ensures the anti-interference ability, but also overcomes the limitations of the traditional polynomial fitting method in high-order cases. Simulation results show that this method has high accuracy in high-order and low-order cases, and has good accuracy and anti-interference ability.

Corresponding author: CHEN Yong

Received Date: 08/07/2020

Available Online: 20/12/2021

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Abstract: Nowadays, all kinds of algorithms in distribution network need high-order derivation of data. Due to the limitation of their own theories, the traditional high-order derivation methods are difficult to calculate accurately in both low-order and high-order cases. In order to improve the calculation accuracy, a new high-order derivative method is proposed in this paper. Based on the polynomial fitting method, by analyzing the transient characteristics of voltage and current signals in the distribution network, the base function is selected for fitting reasonably, and the coefficient of the base function is calculated by the least square method to fit the objective function. The new fitting function is used to replace the original discrete data for derivative operation, which not only ensures the anti-interference ability, but also overcomes the limitations of the traditional polynomial fitting method in high-order cases. Simulation results show that this method has high accuracy in high-order and low-order cases, and has good accuracy and anti-interference ability.

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电力系统安全可靠的运行离不开其底层数据的支持，在电力系统相关数据的处理和计算中，高阶求导的计算随处可见，如时域下分布参数线路模型的推导和计算^[1-2]、电力系统等值参数模型的建立^[3]、电力系统负荷模型辨识研究^[4]、故障测距^[5-6]等. 高阶求导^[7-8]已经渗透到电力系统的方方面面，它不仅和电网相关模型紧密相关，还与电力系统状态评估、故障处理、保护策略有着密切联系^[9-11]. 由此可见，高阶导数的准确计算在电力系统中是十分重要的基础工作.

电力系统中我们常处理的数据大都是电压电流信号的采样值，即离散数据，此时连续函数的求导公式不再适用，目前常用的方法有差分求导法和多项式拟合求导法. 差分求导法是利用导数的定义和微分的思想，近似求取某点处的导数，采样频率越大，其精度就越高，实现简单方便^[12]. 但受边界效应的影响，采样区间首端或尾端的导数值不确定，即自由度减1，N点采样值只能确定N-1个一阶导数值，阶次每高一阶，自由度便少1，其根本原因是在对连续函数进行截断和采样处理时，必然会造成首尾信息缺失，而且不可避免，更关键的是差分法抗干扰能力极差，微弱的干扰便能造成很大的误差. 多项式拟合法^[13-15]用一串线性无关的多项式(x，x²，…，xⁿ)对离散数据进行拟合，将离散数据转化成连续函数，以连续函数逼近的方式代替求解，此方法受拟合精度的影响，若拟合精度低，则误差大. 此外，求导阶次受拟合多项式最高次数制约，n次多项式的n阶导数为常数，n-1阶导数为一元函数，以此类推，其求导阶次在接近多项式多高阶次后误差明显.

针对上述弊端，本文提出一种正弦多项式拟合的方法求取电网离散数据的高阶导数值，本方法在多项式拟合的基础上，针对电力系统电压电流暂态分量和稳态分量的特征，合理选取正弦多项式函数为基底，具有良好的抗干扰性和可靠性.

2. 算法实现

将式(2)展开可得：

其中：

设共采样m+1个点，将式(5)中表示时间的变量t改写为离散采样时间t_i(i=0，1，…，m)后，代入式(4)，则求误差平方和‖δ‖₂²最小值问题可转换成求多元函数

的极值点O(a₀^*，a₁^*，…，a_2n+3^*). 令

由此得：

则式(7)可表示为

为了方便用MATLAB实现，可将式(8)用矩阵表示^[19].

创建矩阵A，B，

其中φ_i(t_j)表示函数φ_i(t)在t_j时刻的值.

所以式(8)可表示为

式(9)是关于正弦多项式系数a_j的线性方程组，由于多项式φ_j(j=0，1，…，2n+3)线性无关，故方程组系数矩阵A>*A′满秩，有唯一解，记为点O(a₀^*，a₁^*，…，a_2n+3^*).

4. 结论

传统高阶求导计算方法因其自身局限性无法保证计算准确性；差分法的边界效应对首尾区间的影响在高阶次时无法忽视和避免，也没有抗干扰能力，在微弱干扰环境下便不再适用；多项式拟合法在实际应用中多项式必然是有限项，无法避免求导阶次受限于多项式最高次数的问题，无法在高阶次情况下保证精度，给计算结果带来巨大的偏差，对要求计算快速可靠将造成十分严重的影响. 本文提出的正弦多项式拟合法在含直流衰减或高频衰减的情形下都具有优于传统方法的特性，一方面在大幅提高高阶次的精度的同时，在低阶次也可将误差控制得极小；另一方面由于算法中核心方程组具有唯一解，可使拟合函数的参数具有较强的稳定性，有利于表征相关物理特性，因而具有非常高的准确性和抗干扰能力.

Figure (3) Reference (19)

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A High-Order Derivation Method for Distribution Network Transient Data

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