中英文资料外文翻译文献

基于改进的灰色预测模型的电力负荷预测

[摘要]尽管灰色预测模型已经被成功地运用在很多领域，但是文献显示其性能仍能被提高。为此，本文为短期负荷预测提出了一个GM（1，1）—关于改进的遗传算法（GM（1，1）-IGA）。由于传统的GM（1，1）预测模型是不准确的而且参数的值是恒定的，为了解决这个问题并提高短期负荷预测的准确性，改进的十进制编码遗传算法（GA）适用于探求灰色模型GM（1，1）的最佳值。并且，本文还提出了单点线性算术交叉法，它能极大地改善交叉和变异的速度。最后，用一个日负荷预测的例子来比较GM（1，1）-IGA模型和传统的GM（1，1）模型，结果显示GM(1，1)-IGA拥有更好地准确性和实用性。

关键词：短期的负荷预测，灰色系统，遗传算法，单点线性算术交叉法

第一章 绪论

日峰值负荷预测对电力系统的经济，可靠和安全战略都起着非常重要的作用。特别是用于每日用电量的短期负荷预测（STLF）决定着发动机运行，维修，功率互换和发电和配电任务的调度。短期负荷预测（STLF）旨在预测数分钟，数小时，数天或者数周时期内的电力负荷。从一个小时到数天以上不等时间范围的短期负荷预测的准确性对每一个电力单位的运行效率有着重要的影响，因为许多运行决策，比如：合理的发电量计划，发动机运行，燃料采购计划表，还有系统安全评估，都是依据这些预测。传统的负荷预测模型被分为时间序列模型和回归模型。通常，这些模型对于日常的短期负荷预测是有效的，但是对于那些特别的日子就会产生不准确的结果。此外，由于它们的复杂性，为了获得比较满意的结果需要大量的计算工作。

灰色系统理论最早是由邓聚龙提出来的，主要是模型的不确定性和信息不完整的分析，对系统研究条件的分析，预测以及决策。灰色系统让每一个随机变量作为一个在某一特定范围内变化的灰色量。它不依赖于统计学方法来处理灰色量。它直接处理原始数据，来寻找数据内在的规律。灰色预测模型运用灰色系统理论的基本部分。此外，灰色预测可以说是利用介于白色系统和黑色系统之间的灰色系统来进行估计。

信息完全已知的系统称为白色系统；相反地，信息完全未知的系统称为黑色系统。灰色模型GM（1，1）（即一阶单变量灰色模型）是灰色理论预测中主要的模型，由少量数据（4个或更多）建立，仍然可以得到很好地预测结果。灰色预测模型组成部分是灰色微分方程组——特性参数变化的非常态微分方程组，或者灰色差分方程组——结构变化的非常态差分方程组，而不是一阶微分方程组或者常规情况下的差分方程组。灰色模型GM（1，1）有一个参数，它在很多文章里经常被设为0.5，这个常数可能不是最理想的，因为不同的问题可能需要不同的值，否则可能产生错误的结果。为了修正前面提到的错误，本文尝试用遗传算法来估算值。

John Holland 首先描述了遗传算法（GA），以一个抽象的生物进化来提出它们，并且给出了一个理论的数学框架作为归化。一个遗传算法相对于其他函数优化方法的显著特征是寻找一个最佳的解决方案来着手，此方案不是以一个单一逐次改变的结构，而是给出一组使用遗传算子来建立新结构的解决措施。通常，二进制表示法应用于许多优化问题，但是本文的遗传算法（GA）采用改进的十进制编码表示方案。

本文打算用改进的遗传算法（GM(1,1)-IGA）来解决电力系统中短期负荷预测（STLF）中遇到的问题。传统的GM（1，1）预测模型经常设定参数为0.5，因此背景值可能不准确。为了克服以上弊端，用改进的十进制编码的遗传算法来获得理想的参数值，从而得到较准确的背景值。而且，提出了单点线性算术交叉法。它能极大地改善交叉和变异的速度，使提出的GM（1，1）-IGA能更准确地预测短期日负荷。

本文结构如下：第二章介绍灰色预测模型GM（1，1）；第三章用改进的遗传算法来估算；第四章提出了GM（1，1）-IGA来实现短期日负荷预测；最后，第五章得出结论。

第二章 灰色预测模型GM（1，1）

本章重点介绍灰色预测的机理。灰色模型GM(1,1)是时间序列预测模型，它有3个基本步骤：（1）累加生成，（2）累减生成，（3）灰色建模。灰色预测模型利用累加的原理来创建微分方程。本质上讲，它的特点是需要很少的数据。

灰色模型GM（1，1），例如：单变量一阶灰色模型，总结如下：

第一步：记原始数列：

=

是n阶离散序列。是m次时间序列，但m必须大于等于4。在原始序列的基础上，通过累加的过程形成了一个新的序列。而累加的目的是提供构建模型的中间数据和减弱变化趋势。定义如下：

有 ，

则 是r次累加序列。

第二步：设定值来预测

通过GM（1，1），我们可以建立下面的一阶灰色微分方程：

它的差分方程是。

a称为发展系数，b称为控制变量。

以微分的形式表示导数项，我们可以得到：

在一个灰色GM（1，1）模型建立前，一个适当的值需要给出以得到一个好的背景值。背景值序列定义如下：

其中，

为方便起见，值一般被设为0.5，推导如下：

然而，这个常量可能不是最理想的，因为不同的场合可能需要不同的值。而且，不管是发展系数a还是控制变量b都由值确定。由于系数是常量，原始灰色信息的白化过程可能被抑制。因此，GM（1，1）模型中预测值的准确性将会严重的降低。为了修正以上不足，系数必须是基于问题特征的变量，因此我们用遗传算法来估算值。

第三步：构建累加矩阵B和系数向量。应用普通最小二乘法（OLS）来获得发展系数a，b。如下：

于是有

第四步：获得一阶灰色微分方程的离散形式，如下：

解得为

为

第三章 运用改进GA估算值

为了预测出准确的灰色模型GM（1，1），残差校验是必不可少的。因此，本文中所提出的目标函数的方法可以确保预测值误差是最小。目标函数定义为最小平均绝对百分比误差，如下：

且，

为原始数据，为预测值，n是该数列的维数。从上面描述构建的GM（1，1），我们可以得到：在GM（1，1）中参数的值能够决定的值；不管是发展系数a还是控制变量b都由值确定。更重要的是，的结果由a，b决定，因此整个模型选择过程最重要的部分就是的值。在和残差之间有着某些复杂的非线性关系，这些非线性是很难通过解析来解决的，因此选择最理想的值是GM(1,1)的难点。

遗传算法是一个随机搜索算法，模拟自然选择与演化。它能广泛应用正是基于后面两个基本方面：计算代码非常简单并且还提供了一个强大的搜索机制。它们函数相对独立，意味着它们不会被函数的属性所限制，例如:连续性,导数的存在，等等。尽管二进制法经常应用于许多优化问题，但是在本文我们采用改进十进制编码法方案来解决。在数值函数优化方面，改进的十进制编码法相对于二进制编码法拥有很大的优势。这些优势简要的叙述如下：

第一步：GA的效率提高了，因此，没有必要将染色体转换为二进制类型。

第二步：由于有效的内部电脑浮点表示，需要较少的内存。

第三步：甄别二进制或其它值不会使精度降低，并且有更大的自由来使用不同的遗传算子。

我们利用改进的十进制码代表性方法来寻找在灰色GM（1，1）模型中最佳系数的值。本文中，我们提出单点线性算术交叉法，并且利用它来获得值；它能极大地提高交叉和变异的速度。改进的十进制码代表性方法的步骤如下：

（1）编码：假设是二进制字符串的C位，然后由右至左每隔n位转换为十进制。（n，n和C的值要确保精度）

（2）随机化种群：选择一个整数M作为种族的大小，然后随机地从集合选择M点，如，这些点组成个体的原始种群，该序列被定义为：

（3）评估适应度：在选择的过程中，个体被选择参与新个体的繁殖。拥有高度地适应度F（）的个体逐代衍化和发展。适应度函数是

是从个体获得的预测值。是迭代最小二乘总和的最大值。

第四步：选择：在本文中，我们根据它们的适应度函数分别地计算出个体选定的概率，然后我们通过轮盘选择法，使繁殖的各自概率是，最后我们拿原始的个体来生成下一代的。

第五步：交叉和变异：编码和交叉是相关的；我们利用了十进制码表示法，因此我们提出了一种新的交叉算子“单点线性算术交叉”。

1）选择合适的两个有交叉概率的个体。

2）为这两个选择的个体，我们仍然采用随机抽样方法以得到交叉算子。例如：

3）交叉

①互相交换它们的正确的字符串。

②位在左侧的交叉可以通过以下计算算法：

a：基因分析：

b：交换后基因：

称为交叉系数，每次根据随机的交叉系统来选择。

4）变异：下面是一个新的变异方案：当变异算子被选择，新的基因值是一个在域权重的随机数，它是用原始基因值得到的加权总和。如果变异算子的值是，变异值是：

是变异系数，。r是一个随机数，。每当进行变异操作时，r会被随机的挑选。因此，新的后代可以通过交叉和变异操作来创建。

第六步：推出原则：选择当前的一代个体来繁殖下一代个体，然后求出适应度值并判断算法是否符合退出条件。如果符合条件，这个值就是最佳的，否则回到第四步，直到种群内所有个体达到统一标准或几代个体的数量超过最大值100。

第四章．负荷预测案例

在本章，我们试着对GM（1，1）-关于改进的遗传算法进行性能评估。

第一步：m天的日负荷数据序列定义为，我们测量了每个小时的电力负荷，于是负荷序列向量就是一个24维数据。

1点：

2点：

j点：

24点：

式中m是所建模型的天数，是日负荷数据序列的第j点。

图1. 原始数据和预测值

第二步：我们利用改进的遗传算法为各自的负荷数据序列来选择值。接着，我们可以算出a和b，然后我们利用GM（1，1）-IGA来预测第m+1天中的第j点的负荷，于是我们可以得到，最后第m+1天地24个预测值构成了这个负荷数据序列。

这有一个GM（1，1）-关于改进的遗传算法（GM（1，1）-IGA）的例子，两种预测日负荷数据曲线（ 7月26号 ）和原始的日负荷曲线同时在图1中画出。

第三步：我们可以利用GM （ 1,1 ） -遗传算法的四个指标来检验精度，包括相对误差，均方差率，小误差概率和关联度误差。如果相对误差和均方差率较低，或者小误差概率和关联度误差较大，GM （ 1,1 ）-GA的准确性检验是较好的。

设置模拟残差为

， k=1，2，…，n

设置模拟的相对剩余为

k=1，2，…，n

设置平均值为

设置的方差为

设置残差平均值为

设置残差方差为

因此，GM（1，1）-IGA的校验值如下：

1).平均相对误差为

2).均方差率为

3).小误差概率为

4).关联度为

其中，

根据上述公式，GM（1，1）-IGA的指标的校验值见表1。

表1 GM-IGA和GM的四个指标

通过表1可以看出，GM-GA所以指标的精确度都是一级的，因此这个GM（1，1）-IGA可以被用来预测短期负荷。

第四步：在图1中，我们可以得到GM（1，1）-IGA的预测负荷数据曲线比GM（1，1）的曲线更接近于原始的日负荷数据曲线。进一步分析，本文选择相对误差作为标准来评价两种模式。两种模型的偏差值如下，GM（1，1）的平均误差为2.285%，然而，GM（1，1）-IGA的平均误差为0.914%。

第五章．结论

本文提出了GM（1，1）-关于改进的遗传算法（GM（1，1）-IGA）来进行短期负荷预测。采用十进制编码代表性方案，改进的遗传算法用于获得GM（1，1）模型中的最优值。本文也提出了单点线性算术交叉法，它能极大地提高交叉和变异的速度，因此GM（1，1）-IGA可以准确地预测短期日负荷。GM（1，1）-IGA的特点是简单、易于开发，因此，它在电力系统中作为一个辅助工具来解决预测问题是适宜的。

图2.GM（1，1）的偏差值

图3.GM（1，1）-IGA的偏差值

致谢

这项工作是由国家自然科学基金部分支持。（70671039）

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Application of Improved Grey Prediction Model

for Power Load Forecasting

[Abstract] Although the grey forecasting model has been successfully utilized in many fields, literatures show its performance still could be improved. For this purpose, this paper put forward a GM (1, 1)-connection improved genetic algorithm (GM (1, 1)-IGA) for short-term load forecasting (STLF). While Traditional GM (1,1) forecasting model is not accurate and the value of parameter is constant, in order to solve this problem and enhance the accuracy of short-term load forecasting (STLF), the improved decimal-code genetic algorithm (GA) is applied to search the optimal value of grey model GM (1, 1). What’s more, this paper also proposes the one-point linearity arithmetical crossover,which can greatly improve the speed of crossover and mutation. Finally, a daily load forecasting example is used to test the GM (1, 1)-IGA model and traditional GM (1, 1) model, results show that the GM (1, 1)-IGA had better accuracy and practicality.

Keywords: Short-term Load Forecasting, Grey System,Genetic Algorithm, One-point Linearity Arithmetical Crossover.

1. Introduction

Daily peak load forecasting plays an important role in all aspects of economic, reliable, and secure strategies for power system. Specifically, the short-term load forecasting (STLF) of daily electricity usage is crucial in unit commitment, maintenance, power interchange and task scheduling of both power generation and distribution facilities. Short-term load forecasting (STLF) aims at predicting electric loads for a period of minutes, hours, days or weeks. The quality of the short-term load forecasts with lead times ranging from one hour to several days ahead has a significant impact on the efficiency of operation of any power utility, because many operational decisions, such as economic dispatch scheduling of the generating capacity, unit commitment, scheduling of fuel purchase as well as system security assessment are based on such forecasts [1]. Traditional short-term load forecasting models can be classified as time series models or regression models [2,3,4]. Usually, these techniques are effective for the forecasting of short-term load on normal days but fail to yield good results on those days with special events [5,6,7]. Furthermore, because of their complexities, enormous computational efforts are required to produce acceptable results.

The grey system theory, originally presented by Deng[8,9,10], focuses on model uncertainty and information insufficiency in analyzing and understanding systems via research on conditional analysis, forecasting and decision making. The grey system puts each stochastic variable as a grey quantity that changes within a given range. It does not rely on statistical method to deal with the grey quantity. It deals directly with the original data, and searches the intrinsic regularity of data[11]. The grey forecasting model utilises the essential part of the grey system theory.Therewith, grey forecasting can be said to define the estimation done by the use of a grey system, which is in between a white system and a black-box system.

A system is defined as a white one if the information in it is known; otherwise, a system will be a black box if nothing in it is clear. The grey model GM (1, 1) is the main model of grey theory of prediction, i.e. a single variable first order grey model, which is created with few data (four or more) and still we can get fine forecasting result [12]. The grey forecasting models are given by grey differential equations, which are groups of abnormal differential equations with variations in behavior parameters, or grey difference equations which are groups of abnormal difference equations with variations in structure, rather than the first-order differential equations or the difference equations in conventional cases [13]. The grey model GM (1, l) has parameter which was often set to 0.5 in many articles,and this constant might not be optimal, because different questions might need different value,which produces wrong results. In order to correct the above-mentioned defect, this paper attempts to estimate by genetic algorithms.

Genetic algorithms (GA) were firstly described by John Holland, who presented them as an abstraction of biological evolution and gave a theoretical mathematical framework for adaptation [14].The distinguishing feature of a GA with respect to other function optimization techniques is that the search towards an optimum solution proceeds not by incremental changes to a single structure but by maintaining a population of solutions from which new structures are created using genetic operators[15].Usually, the binary representation was applied to many optimization problems, but in this paper genetic algorithms (GA) adopted improved decimal-code representation scheme.

This paper proposed GM(1,1)-improved genetic algorithm (GM(1,1)-IGA)to solve short-term load forecasting (STLF) problems in power system. The traditional GM (1, 1) forecasting model often sets the coefficient to 0.5, which is the reason why the background value z(1)(k) may be unsuitable. In order to overcome the above-mentioned drawbacks, the improved decimal-code genetic algorithm was used to obtain the optimal coefficient value to set proper background value z(1)(k).What is more, the one-point linearity arithmetical crossover was put forward, which can greatly improve the speed of crossover and mutation so that the proposed GM(1,1)-IGA can forecast the short-term daily load successfully.

The paper is organized as follows: section 2 proposes the grey forecasting model GM(1,1): section 3 presents Estimate with improved genetic algorithm:section 4 puts forward a short-term daily load forecasting realized by GM(1,1)-IGA and finally, a conclusion is drawn in section 5.

2. Grey prediction model GM (1,1)

This section reviews the operation of grey forecasting in details. The grey model GM(1,1) is a time series forecasting model. It has three basic operations: (1) accumulated generation, (2) inverse accumulated generation, and (3) grey modeling. The grey forecasting model uses the operations of accumulated to construct differential equations.Intrinsically speaking, it has the characteristics of requiring less data.

The grey model GM(1,1), i.e., a single variable first-order grey model, is summarized as follows:

Step1: Denote the initial time sequence by

=

x(0) is the given discrete n-th-dimensional sequence.x(0)(m) is the time series data at time m , n must be equal to or larger than 4. On the basis of the initial sequence x(0) , a new sequence x(1) is set up through the accumulated generating operation in order to provide the middle message of building a model and to weaken the variation tendency, so x(1) is defined as:

Where ，

and and is the r times accumulated series.

Step2: To set the value to fine z(1)(k)

According to GM (1, 1), we can form the following first-order grey differential equation:

And its difference equation is.Where a was called the developing coefficient of GM,and b was called the control variable.

Denoting the differential coefficient subentry in the form of difference, we can get:

Before a grey GM (1, 1) model was set up, a proper value needed to be assigned for a better background value z(1)(k). The sequence of background values was defined as:

Among them

For convenience, the value was often set to 0.5,the z(1)(k) was derived as:

However, this constant might not be optimal because the different questions might need different value. And, both developing coefficient a and control variable b were determined by the z(1)(k). The process of the original grey information for whitening may be suppressed resulted from the coefficient was constant. Hence, the accuracy of prediction value xˆ(0)(k) in GM (1, 1) model would seriously be decreased. In order to correct the defect, the coefficient

must be a variable based on the feature of problems,so we estimate by genetic algorithms.

Step3: To construct accumulated matrix B and coefficient vector X n . Applying the Ordinary Least Square (OLS) method obtains the developing coefficient a , b was as follows:

and

So

Step4: To obtain the discrete form of first-order grey differential equation, as follows:

The solution of x(1) is

And the solution of x(0) is

3. Estimate with improved GA

In order to estimate the accuracy of grey mode GM(1, 1), the residual error test was essential. Therefore,the objective function of the proposed method in this paper was to ensure that the forecasting value errors were minimum. The objective function was defined as mean absolute percentage error (MAPE) minimization as follows:

Where，

x(0)(k) is original data, is forecasting value,n is the number of sequence data. However, from the above description of the establishment of GM (1, 1), we can get: In GM (1, 1), the value of parameter can determine z(1) , and, both developing coefficient a and control variable b were determined by the z(1)(k).What is more, the solution of x(0) was determined by a and b,so the key part of the whole model selecting process was the value of .There is kind of complicated nonlinear relationship between and residual errors,and this nonlinearity was hard to solve by resolution, so the optimal selection of was the difficult point of GM（1,1）.

Genetic algorithm is a random search algorithm that simulates natural selection and evolution. It is finding widespread application as a consequence of two fundamental aspects: the computational code is very simple and yet provides a powerful search mechanism.They are function independent which means they are not limited by the properties of the function such as continuity, existence of derivatives, etc. Although the binary representation was usually applied to many optimization problems, in this paper, we used the improved decimal-code representation scheme for solution. The improved decimal-code representation in the GA offers a number of advantages in numerical function optimization over binary encoding. The advantages can be briefly described as follows:

Step1:Efficiency of GA is increased as there is no need to convert chromosomes to the binary type,

Step2:Less memory is required as efficient floating-point internal computer representations can be used directly,

Step3:There is no loss in precision by discrimination to binary or other values, and there is greater freedom to use different genetic operators.

We utilized the improved decimal-code representation scheme for searching optimal coefficient value in grey GM (1, 1) model. In this paper, we proposed one-point linearity arithmetical crossover and utilized it to select the value of ; it can greatly improve the speed of crossover and mutation. The steps of the improved decimal-code representation scheme are as follows:

（1）Coding：Suppose is a binary string of C bits, then let every n bits transform a decimal from right to left.(n

（2）Randomize population: Select one integer M as the size of the population, and then select M points stochastically from the set,as, these points compose the individuals of the original population, the sequence is defined as:

（3）Evaluate the fitness: In the selection step,individuals are chosen to participate in the reproduction of new individuals. The individual with the highest fitness F（） has the priority and advances to the next generation. The fitness function is

and is the value of forecasting which is gained by the individual. is the maximum of the sum of iterative squares.

Step4: Selection: In this paper, we calculate individual selected probability respectively according to their fitness functions, then we adopt the roulette wheel selection scheme, so that the propagated probability of respective individual is p(k) ,after that we take the inborn individual to compose the next generation p(k +1).

Step5: Crossover and Mutation: Coding and crossover are correlative; we utilized the decimal-code representation, so we propose a new crossover operator “one-point linearity arithmetical crossover”

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