﻿ 基于不同核支持向量机变权综合的短期负荷预测 Short-Term Load Forecasting Based on Variable Weighted Synthesis of Different Kernel SVM

Statistics and Application
Vol. 09  No. 01 ( 2020 ), Article ID: 34113 , 8 pages
10.12677/SA.2020.91009

Short-Term Load Forecasting Based on Variable Weighted Synthesis of Different Kernel SVM

Dongfen Ma

School of Statistics and Data Science, Xinjiang University of Finance and Economics, Urumqi Xinjiang

Received: Jan. 17th, 2020; accepted: Jan. 31st, 2020; published: Feb. 6th, 2020

ABSTRACT

To improve the accuracy and stability of short-term load forecasting, a method of short-term load forecasting based on different kernel support vector machine (SVM) variable weight synthesis is proposed. In this method, firstly, the load history data is expanded, the feature is selected by correlation analysis, and the historical data is mapped to the input-output relationship view to build the forecasting space. Then, support vector machines of Gaussian kernel, Laplace kernel and Polynomial kernel function are used to study in the forecasting space respectively, and the performance of the model is tested by the 10 fold cross validation. Finally, the variable weight is constructed by using the accuracy and standard deviation of performance test, and the power load forecasting is realized by variable weight synthesis of multiple model. Example analysis shows that compared with methods such as Gaussian kernel support vector machine, partial least squares, decision tree and Bagging, the new method improves accuracy by 0.382%, 3.079%, 3.188% and 2.6%, and stability by 0.383%, 2.452%, 1.781% and 1.43%, respectively.

Keywords:Short-Term Load Forecasting, SVM, Kernel Function, Variable Weight Synthesis of Multiple Models

1. 引言

2. 支持向量机算法原理

$\begin{array}{l}\mathrm{min}\frac{1}{2}{‖\omega ‖}^{2}+C\underset{i=1}{\overset{n}{\sum }}\left({\xi }_{i},{\xi }_{i}{}^{\ast }\right)\\ \left\{\begin{array}{l}{y}_{i}-〈\omega ,\phi \left({x}_{i}\right)〉-b\le \epsilon +{\xi }_{i},\text{}i=1,\cdots ,n;\\ 〈\omega ,\phi \left({x}_{i}\right)〉+b-{y}_{i}\le \epsilon +{\xi }_{i}{}^{\ast },\text{}i=1,\cdots ,n.\\ {\xi }_{i},{\xi }_{i}{}^{\ast }\ge 0\end{array}\end{array}$ (1)

$\begin{array}{l}L=\frac{1}{2}{‖\omega ‖}^{2}+C\underset{i=1}{\overset{n}{\sum }}\left({\xi }_{i},{\xi }_{i}{}^{\ast }\right)-\underset{i=1}{\overset{n}{\sum }}\left({\eta }_{i}{\xi }_{i},{\eta }_{i}{}^{\ast }{\xi }_{i}{}^{\ast }\right)\\ \text{}-\underset{i=1}{\overset{n}{\sum }}{\alpha }_{i}\left(\epsilon +{\xi }_{i}-{y}_{i}+〈\omega ,\phi \left({x}_{i}\right)〉+b\right)\text{}\\ \text{}-\underset{i=1}{\overset{n}{\sum }}{\alpha }_{i}^{\ast }\left(\epsilon +{\xi }_{i}{}^{\ast }-{y}_{i}+〈\omega ,\phi \left({x}_{i}\right)〉+b\right)\end{array}$ (2)

$f\left(x\right)=\underset{i=1}{\overset{n}{\sum }}\left({\stackrel{^}{\alpha }}_{i}-{\stackrel{^}{\alpha }}_{i}^{\ast }\right)k\left(x,{x}_{i}\right)+\stackrel{^}{b}$

3. 基于不同核支持向量机变权综合的短期负荷预测方法

3.1. 主要思路

3.2. 基于相关分析的负荷特征选择

${r}_{i}=r\left({f}_{0},{f}_{i}\right)=\frac{|\underset{k=1}{\overset{m}{\sum }}\left({s}_{t,k}-\frac{1}{m}\underset{k=1}{\overset{m}{\sum }}{s}_{t,k}\right)\left({s}_{ti,k}-\frac{1}{m}\underset{k=1}{\overset{m}{\sum }}{s}_{ti,k}\right)|}{\sqrt{\underset{k=1}{\overset{m}{\sum }}{\left({s}_{t,k}-\frac{1}{m}\underset{k=1}{\overset{m}{\sum }}{s}_{t,k}\right)}^{2}}\sqrt{\underset{k=1}{\overset{m}{\sum }}{\left({s}_{ti,k}-\frac{1}{m}\underset{k=1}{\overset{m}{\sum }}{s}_{ti,k}\right)}^{2}}}$ (3)

${r}_{i}$ 的取值为 ${r}_{i}$ 越接近于1， ${f}_{0}$${f}_{i}$ 之间的相关度越强。设定阈值 $\alpha$，将满足条件的特征所组成的特征集合表示为 ${F}_{t}^{\alpha }:\left\{{f}_{i}|{r}_{i}\ge \alpha \right\}$

3.3. 预测空间构造

${f}_{0}$ 为输出特征，以特征集 ${F}_{t}^{a}$ 内的特征为输入特征，建立预测空间 $\Omega =\underset{i=1}{\overset{n}{\prod }}X\left({f}_{i}\right)×X\left({f}_{0}\right)$

$D=\left\{\left({X}_{t},{s}_{t}\right),t=1,\cdots ,N\right\}$，其中 ${X}_{t}=\left({s}_{t-1},{s}_{t-2},\cdots ,{s}_{t-n}\right)$，N为关系化数据记录条数。

3.4. 核函数选择及模型建立

${k}_{G\text{-}K}\left(u,v\right)=\mathrm{exp}\left(-\gamma {‖u-{v}^{\text{T}}‖}^{2}\right)$ (4)

${k}_{P\text{-}K}\left(u,v\right)={\left(\gamma 〈u,v〉+c\right)}^{p}$ (5)

${k}_{La\text{-}K}\left(u,v\right)=\mathrm{exp}\left(-\gamma ‖u-{v}^{\text{T}}‖\right)$(6)

3.5. 模型性能测试及变权构造

${A}_{i}=1-{E}_{i}=1-\frac{1}{H}\underset{h=1}{\overset{H}{\sum }}{\epsilon }_{ih}=1-\frac{1}{H}\underset{h=1}{\overset{H}{\sum }}\left(\frac{|{y}_{ih}-{\stackrel{^}{y}}_{ih}|}{{y}_{ih}}\right)$ (7)

${\sigma }_{i}=\sqrt{\frac{1}{H}\underset{h=1}{\overset{H}{\sum }}{\left[\left(1-{\epsilon }_{hi}\right)-{A}_{i}\right]}^{2}}$ (8)

${w}_{k}=\frac{\underset{i=1}{\overset{P}{\sum }}{A}_{ki}\left(1-{\sigma }_{ki}\right)}{\underset{k=1}{\overset{3}{\sum }}\underset{i=1}{\overset{P}{\sum }}{A}_{ki}\left(1-{\sigma }_{ki}\right)}$ (9)

${\stackrel{^}{y}}^{*}=\underset{k=1}{\overset{3}{\sum }}{w}_{k}{\stackrel{^}{y}}_{k}$

3.6. 基于不同核函数SVM变权综合的负荷预测

Step 1：确定给定的历史数据集的特征展开参数m，将给定时刻t前m个负荷数据组成电力负荷特征向量 ${X}_{t}$$t=1,2,\cdots ,N-m$

Step 2：设待预测时刻 $t={t}_{z}$$z=1,\cdots ,Z$，确定负荷预测备选特征集，采用相关度分析进行特征选取，组建特征集 ${F}_{t}^{\alpha }$，进而对历史数据集进行关系化展开，构建待预测时刻的预测空间 $\Omega$

Step 3：选用高斯径向基核函数、拉普拉斯核函数及多项式核函数的支持向量机在预测空间上建立预测模型，借助P折交叉验证进行模型测试。

Step 4：依据性能测试结果的准确度 ${A}_{i}$ 及其标准差 ${\sigma }_{i}$ 构造变权，通过变权综合不同核函数的SVM实现 ${t}_{z}$ 的负荷预测。令 $z=z+1$，当 $z>Z$，则预测结束；否则，转Step 2。

4. 算例分析

Figure 1. Correlation between load feature to be forecast and historical load features

$MAPE=\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}\frac{|{y}_{i}-{\stackrel{^}{y}}_{i}|}{{y}_{i}}×100%$ (10)

$SRE=\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left(\frac{|{y}_{i}-{\stackrel{^}{y}}_{i}|}{{y}_{i}}×100%-MAPE\right)}^{2}}$ (11)

Figure 2. Comparison of power load forecasting values

Table 1. MAPE of forecast models

Table 2. SRE of forecast models

5. 总结

Short-Term Load Forecasting Based on Variable Weighted Synthesis of Different Kernel SVM[J]. 统计学与应用, 2020, 09(01): 73-80. https://doi.org/10.12677/SA.2020.91009

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