電力負(fù)荷預(yù)測(cè)中英文對(duì)照外文翻譯文獻(xiàn)

#中英文資料翻譯基于改進(jìn)的灰色預(yù)測(cè)模型的電力負(fù)荷預(yù)測(cè)［摘要］盡管灰色預(yù)測(cè)模型已經(jīng)被成功地運(yùn)用在很多領(lǐng)域，但是文獻(xiàn)顯示其性能仍能被提

高。為此，本文為短期負(fù)荷預(yù)測(cè)提出了一個(gè)GM（1,1）—關(guān)于改進(jìn)的遺傳算法（GM（1,

1）-IGA）。由于傳統(tǒng)的GM（1，1）預(yù)測(cè)模型是不準(zhǔn)確的而且參數(shù)a的值是恒定的，為了

解決這個(gè)問(wèn)題并提高短期負(fù)荷預(yù)測(cè)的準(zhǔn)確性，改進(jìn)的十進(jìn)制編碼遺傳算法（GA）適用于探

求灰色模型GM（1，1）的最佳a(bǔ)值。并且，本文還提出了單點(diǎn)線性算術(shù)交叉法，它能極

大地改善交叉和變異的速度。最后，用一個(gè)日負(fù)荷預(yù)測(cè)的例子來(lái)比較GM（1，1）-IGA模型和傳統(tǒng)的GM（1，1）模型，結(jié)果顯示GM（1，1）-IGA擁有更好地準(zhǔn)確性和實(shí)用性。關(guān)鍵詞：短期的負(fù)荷預(yù)測(cè)，灰色系統(tǒng)，遺傳算法，單點(diǎn)線性算術(shù)交叉法第一章緒論日峰值負(fù)荷預(yù)測(cè)對(duì)電力系統(tǒng)的經(jīng)濟(jì)，可靠和安全戰(zhàn)略都起著非常重要的作用。特別是用于每日用電量的短期負(fù)荷預(yù)測(cè)（STLF）決定著發(fā)動(dòng)機(jī)運(yùn)行，維修，功率互換和發(fā)電和配電任務(wù)的調(diào)度。短期負(fù)荷預(yù)測(cè)（STLF）旨在預(yù)測(cè)數(shù)分鐘，數(shù)小時(shí)，數(shù)天或者數(shù)周時(shí)期內(nèi)的電力負(fù)荷。從一個(gè)小時(shí)到數(shù)天以上不等時(shí)間范圍的短期負(fù)荷預(yù)測(cè)的準(zhǔn)確性對(duì)每一個(gè)電力單位的運(yùn)行效率有著重要的影響，因?yàn)樵S多運(yùn)行決策，比如：合理的發(fā)電量計(jì)劃，發(fā)動(dòng)機(jī)運(yùn)行，燃料采購(gòu)計(jì)劃表，還有系統(tǒng)安全評(píng)估，都是依據(jù)這些預(yù)測(cè)M。傳統(tǒng)的負(fù)荷預(yù)測(cè)模型被分為時(shí)間序列模型和回歸模型［2,3,4］。通常，這些模型對(duì)于日常的短期負(fù)荷預(yù)測(cè)是有效的，但是對(duì)于那些特別的日子就會(huì)產(chǎn)生不準(zhǔn)確的結(jié)果際,7］。此外，由于它們的復(fù)雜性，為了獲得比較滿意的結(jié)果需要大量的計(jì)算工作?；疑到y(tǒng)理論最早是由鄧聚龍?zhí)岢鰜?lái)的［8,9,10］，主要是模型的不確定性和信息不完整的分析，對(duì)系統(tǒng)研究條件的分析，預(yù)測(cè)以及決策?；疑到y(tǒng)讓每一個(gè)隨機(jī)變量作為一個(gè)在某一特定范圍內(nèi)變化的灰色量。它不依賴(lài)于統(tǒng)計(jì)學(xué)方法來(lái)處理灰色量。它直接處理原始數(shù)據(jù)，來(lái)尋找數(shù)據(jù)內(nèi)在的規(guī)律［11］。灰色預(yù)測(cè)模型運(yùn)用灰色系統(tǒng)理論的基本部分。此外，灰色預(yù)測(cè)可以說(shuō)是利用介于白色系統(tǒng)和黑色系統(tǒng)之間的灰色系統(tǒng)來(lái)進(jìn)行估計(jì)。信息完全已知的系統(tǒng)稱(chēng)為白色系統(tǒng)；相反地，信息完全未知的系統(tǒng)稱(chēng)為黑色系統(tǒng)?；疑Ｐ虶M（1，1）（即一階單變量灰色模型）是灰色理論預(yù)測(cè)中主要的模型，由少量數(shù)據(jù)（4個(gè)或更多）建立，仍然可以得到很好地預(yù)測(cè)結(jié)果［12］?；疑A(yù)測(cè)模型組成部分是灰色微分方程組一一特性參數(shù)變化的非常態(tài)微分方程組，或者灰色差分方程組一一結(jié)構(gòu)變化的非常態(tài)差分方程組，而不是一階微分方程組或者常規(guī)情況下的差分方程組回。灰色模型GM（1，1）有一個(gè)參數(shù)a，它在很多文章里經(jīng)常被設(shè)為0.5,這個(gè)常數(shù)a可能不是最理想的，因?yàn)椴煌膯?wèn)題可能需要不同的a值，否則可能產(chǎn)生錯(cuò)誤的結(jié)果。為了修正前面提到的錯(cuò)誤，本文嘗試用遺傳算法來(lái)估算a值。JohnHolland首先描述了遺傳算法（GA），以一個(gè)抽象的生物進(jìn)化來(lái)提出它們，并且給出了一個(gè)理論的數(shù)學(xué)框架作為歸化［14］。一個(gè)遺傳算法相對(duì)于其他函數(shù)優(yōu)化方法的顯著特征是尋找一個(gè)最佳的解決方案來(lái)著手，此方案不是以一個(gè)單一逐次改變的結(jié)構(gòu)，而是給出一組使用遺傳算子來(lái)建立新結(jié)構(gòu)的解決措施［15］。通常，二進(jìn)制表示法應(yīng)用于許多優(yōu)化問(wèn)題，但是本文的遺傳算法（GA）米用改進(jìn)的十進(jìn)制編碼表示方案。本文打算用改進(jìn)的遺傳算法（GM（1,1）-IGA）來(lái)解決電力系統(tǒng)中短期負(fù)荷預(yù)測(cè)（STLF）中遇到的問(wèn)題。傳統(tǒng)的GM（1,1）預(yù)測(cè)模型經(jīng)常設(shè)定參數(shù)a為0.5，因此背景值z(mì)（i）（k）可能不準(zhǔn)確。為了克服以上弊端，用改進(jìn)的十進(jìn)制編碼的遺傳算法來(lái)獲得理想的參數(shù)a值，從而得到較準(zhǔn)確的背景值z(mì)（i）（k）。而且，提出了單點(diǎn)線性算術(shù)交叉法。它能極大地改善交叉和變異的速度，使提出的GM（1,1）-IGA能更準(zhǔn)確地預(yù)測(cè)短期日負(fù)荷。本文結(jié)構(gòu)如下：第二章介紹灰色預(yù)測(cè)模型GM（1，1）；第三章用改進(jìn)的遺傳算法來(lái)估算a；第四章提出了GM（1，1）-IGA來(lái)實(shí)現(xiàn)短期日負(fù)荷預(yù)測(cè)；最后，第五章得出結(jié)論。第二章灰色預(yù)測(cè)模型GM（1,1）本章重點(diǎn)介紹灰色預(yù)測(cè)的機(jī)理?；疑Ｐ虶M（1,1）是時(shí)間序列預(yù)測(cè)模型，它有3個(gè)基本步驟：（1）累加生成，（2）累減生成，（3）灰色建模。灰色預(yù)測(cè)模型利用累加的原理來(lái)創(chuàng)建微分方程。本質(zhì)上講，它的特點(diǎn)是需要很少的數(shù)據(jù)?；疑Ｐ虶M（1，1），例如：?jiǎn)巫兞恳浑A灰色模型，總結(jié)如下:第'步:記原始數(shù)列:x（第'步:記原始數(shù)列:x（o）=Cx（o）（1）,x（o）（2）,x（o）（3）,...,x（o）x（o）是n階離散序列。x（0）（m）是m次時(shí)間序列，但m必須大于等于4。在原始序列x（o）的基礎(chǔ)上，通過(guò)累加的過(guò)程形成了一個(gè)新的序列x（1）。而累加的目的是提供構(gòu)建模型的中間數(shù)據(jù)和減弱變化趨勢(shì)。x（1）定義如下：有x(1)(1)=x(o)(1).x(1)(k)=fx(0)(m),k=2,3...nr次累加序列。x(r)=cr(1),x》(2,3()3,(x()>nr次累加序列。第二步：設(shè)定a值來(lái)預(yù)測(cè)Z（1）（k）通過(guò)GM（1，1），我們可以建立下面的一階灰色微分方程:dx(1)+ax(1)=bdt它的差分方程是x(o)(k)+az(1)(k)=b。a稱(chēng)為發(fā)展系數(shù)，b稱(chēng)為控制變量。以微分的形式表示導(dǎo)數(shù)項(xiàng)，我們可以得到:dx(1)dt=皿+1)-皿)=x(1)(k+1)-x(1dx(1)dt在一個(gè)灰色GM（1，1）模型建立前，一個(gè)適當(dāng)?shù)腶值需要給出以得到一個(gè)好的背景值z(mì)（1）（k）。背景值序列定義如下：其中,z(1)(k)=a*x(1)(k)+(l-a)*x(1)(k一1),k=2,3...n,0<a<1為方便起見(jiàn)，a值一般被設(shè)為0.5，z(1)(k)推導(dǎo)如下：/、x(1)(k)+x(1)(k-1)z(1)(k)=LJ2然而，這個(gè)常量a可能不是最理想的，因?yàn)椴煌膱?chǎng)合可能需要不同的a值。而且,不管是發(fā)展系數(shù)a還是控制變量b都由z（1）（k）值確定。由于系數(shù)a是常量，原始灰色信息的白化過(guò)程可能被抑制。因此，GM（1,1）模型中預(yù)測(cè)x（o）（k）值的準(zhǔn)確性將會(huì)嚴(yán)重的降低。為了修正以上不足，系數(shù)a必須是基于問(wèn)題特征的變量，因此我們用遺傳算法來(lái)估算a值。第三步：構(gòu)建累加矩陣B和系數(shù)向量x。應(yīng)用普通最小二乘法（OLS）來(lái)獲得發(fā)展系數(shù)a，nb。如下：于是有z(1)(2)z(1)(3)二x（o）（2）,x（o）（3）,,x（o）*Bt*Xn第四步：獲得一階灰色微分方程的離散形式，如下:解得x（1）為(b、x(1)(k+1)=x(o)(1)一丁Ia丿*e-ak+纟ax(o)為x(o)(k+1)=x(1)(k+1)-x(1)(k)=(e-a一1)*x(0)(1)—2*e-ak第三章運(yùn)用改進(jìn)GA估算-值為了預(yù)測(cè)出準(zhǔn)確的灰色模型GM(1,1),殘差校驗(yàn)是必不可少的。因此，本文中所提出的目標(biāo)函數(shù)的方法可以確保預(yù)測(cè)值誤差是最小。目標(biāo)函數(shù)定義為最小平均絕對(duì)百分比誤差，如下：minMAPE=才||e(k)|k=1且,e(k)且,e(k)=x(0)(k)-x(o)(k)x(o)(k)X100%x(0)(k)為原始數(shù)據(jù)，x(0)(k)為預(yù)測(cè)值，n是該數(shù)列的維數(shù)。從上面描述構(gòu)建的GM(1,1)，我們可以得到：在GM(1,1)中參數(shù)a的值能夠決定z⑴的值；不管是發(fā)展系數(shù)a還是控制變量b都由z(i)(k)值確定。更重要的是，x(0)的結(jié)果由a，b決定，因此整個(gè)模型選擇過(guò)程最重要的部分就是a的值。在a和殘差之間有著某些復(fù)雜的非線性關(guān)系，這些非線性是很難通過(guò)解析來(lái)解決的，因此選擇最理想的a值是GM(1,1)的難點(diǎn)。遺傳算法是一個(gè)隨機(jī)搜索算法，模擬自然選擇與演化。它能廣泛應(yīng)用正是基于后面兩個(gè)基本方面：計(jì)算代碼非常簡(jiǎn)單并且還提供了一個(gè)強(qiáng)大的搜索機(jī)制。它們函數(shù)相對(duì)獨(dú)立，意味著它們不會(huì)被函數(shù)的屬性所限制，例如:連續(xù)性，導(dǎo)數(shù)的存在，等等。盡管二進(jìn)制法經(jīng)常應(yīng)用于許多優(yōu)化問(wèn)題，但是在本文我們采用改進(jìn)十進(jìn)制編碼法方案來(lái)解決。在數(shù)值函數(shù)優(yōu)化方面，改進(jìn)的十進(jìn)制編碼法相對(duì)于二進(jìn)制編碼法擁有很大的優(yōu)勢(shì)。這些優(yōu)勢(shì)簡(jiǎn)要的敘述如下：第一步：GA的效率提高了，因此，沒(méi)有必要將染色體轉(zhuǎn)換為二進(jìn)制類(lèi)型。第二步：由于有效的內(nèi)部電腦浮點(diǎn)表示，需要較少的內(nèi)存。第三步：甄別二進(jìn)制或其它值不會(huì)使精度降低，并且有更大的自由來(lái)使用不同的遺傳算子。我們利用改進(jìn)的十進(jìn)制碼代表性方法來(lái)尋找在灰色GM(1,1)模型中最佳系數(shù)的a值。本文中，我們提出單點(diǎn)線性算術(shù)交叉法，并且利用它來(lái)獲得a值；它能極大地提高交叉和變異的速度。改進(jìn)的十進(jìn)制碼代表性方法的步驟如下：編碼：假設(shè)aw【0,1］是二進(jìn)制字符串的C位，然后由右至左每隔n位轉(zhuǎn)換為十進(jìn)制。(nvC,n和C的值要確保精度)

隨機(jī)化種群：選擇一個(gè)整數(shù)M作為種族的大小，然后隨機(jī)地從集合［o,l］選擇M點(diǎn)，如a(i,0)(i=1,2,…,M)，這些點(diǎn)組成個(gè)體的原始種群，該序列被定義為：P(0)=仁(1,0),a(2,0),…,a(M,0)}評(píng)估適應(yīng)度：在選擇的過(guò)程中，個(gè)體a(i,k)被選擇參與新個(gè)體的繁殖。擁有高度地適應(yīng)度F(a(i,k))的個(gè)體a(i,k)逐代衍化和發(fā)展。適應(yīng)度函數(shù)是F(a(i,k))=<Cmax-fUO'f"<Cmax=￡(攵(0)(a(i,k))—X?))10，其它，=1'if(0)C(i,k))是從個(gè)體a(i,k)獲得的預(yù)測(cè)值。c是迭代最小二乘總和的最大值。imax第四步：選擇：在本文中，我們根據(jù)它們的適應(yīng)度函數(shù)F(a(i,k))分別地計(jì)算出個(gè)體選定的概率()F(a(I,K))/，然后我們通過(guò)輪盤(pán)選擇法，使繁殖的各自概Pik=八F(a(i,k))i=1率是P(k)，最后我們拿原始的個(gè)體來(lái)生成下一代的P(k+1)。第五步：交叉和變異：編碼和交叉是相關(guān)的；我們利用了十進(jìn)制碼表示法，因此我們提出了一種新的交叉算子“單點(diǎn)線性算術(shù)交叉”。1)選擇合適的兩個(gè)有交叉概率p的個(gè)體。2)c2)為這兩個(gè)選擇的個(gè)體，我們?nèi)匀徊捎秒S機(jī)抽樣方法以得到交叉算子。例如:■z、…ziki（k+1）il,z???zzz、j1j2jkj（k+1）3)交叉互相交換它們的正確的字符串。位在左側(cè)的交叉可以通過(guò)以下計(jì)算算法:a：基因分析：z=卩*z+（1-卩）*zikikikz=卩*z+（1—卩）*zjkjkjkb：交換后基因：z=卩*z+（1-卩）*zikikjkz=卩*z+（1-卩）*zjkjkik卩e[0,l]稱(chēng)為交叉系數(shù)，每次根據(jù)隨機(jī)的交叉系統(tǒng)來(lái)選擇。4）變異：下面是一個(gè)新的變異方案：當(dāng)變異算子被選擇，新的基因值是一個(gè)在域權(quán)重的隨機(jī)數(shù)，它是用原始基因值得到的加權(quán)總和。如果變異算子的值是z，變異值是：iz=a*r+（1-Q）*ziia是變異系數(shù)，ae[0,l]or是一個(gè)隨機(jī)數(shù)，rgTz,z]。每當(dāng)進(jìn)行變異操作時(shí)，r-imax-imin會(huì)被隨機(jī)的挑選。因此，新的后代可以通過(guò)交叉和變異操作來(lái)創(chuàng)建。第六步：推出原則：選擇當(dāng)前的一代個(gè)體來(lái)繁殖下一代個(gè)體，然后求出適應(yīng)度值并判斷算法是否符合退出條件。如果符合條件，這個(gè)a值就是最佳的，否則回到第四步，直到種群內(nèi)所有個(gè)體達(dá)到統(tǒng)一標(biāo)準(zhǔn)或幾代個(gè)體的數(shù)量超過(guò)最大值100。第四章?負(fù)荷預(yù)測(cè)案例在本章，我們?cè)囍鴮?duì)GM(1,1)-關(guān)于改進(jìn)的遺傳算法進(jìn)行性能評(píng)估。第一步：m天的日負(fù)荷數(shù)據(jù)序列定義為(x(k)|k二1,2,…,n},我們測(cè)量了每個(gè)小時(shí)的電力負(fù)荷，于是負(fù)荷序列向量就是一個(gè)24維數(shù)據(jù)。1點(diǎn)：X=((i)|i=1,2,...,m}010124點(diǎn)：X=fx(i)li=1,2,...,m}2424式中m是所建模型的天數(shù)，X是日負(fù)荷數(shù)據(jù)序列的第j點(diǎn)。j10009509008508007507006506005505000510152025圖1.原始數(shù)據(jù)和預(yù)測(cè)值Hour(h)第二步：我們利用改進(jìn)的遺傳算法為各自X的負(fù)荷數(shù)據(jù)序列來(lái)選擇?值。接著，我們可以算出a和b,然后我們利用GM(1,1)-IGA來(lái)預(yù)測(cè)第m+1天中的第j點(diǎn)的負(fù)荷，于

是我們可以得到X.（m+1），最后第m+1天地24個(gè)預(yù)測(cè)值構(gòu)成了這個(gè)負(fù)荷數(shù)據(jù)序列l(wèi)x（m+l）|j=1,2,...,24｝。j這有一個(gè)GM（1，1）-關(guān)于改進(jìn)的遺傳算法（GM（1，1）-IGA）的例子，兩種預(yù)測(cè)日負(fù)荷數(shù)據(jù)曲線（7月26號(hào)）和原始的日負(fù)荷曲線同時(shí)在圖1中畫(huà)出。第三步：我們可以利用GM（1,1）-遺傳算法的四個(gè)指標(biāo)來(lái)檢驗(yàn)精度，包括相對(duì)誤差，均方差率，小誤差概率和關(guān)聯(lián)度誤差。如果相對(duì)誤差和均方差率較低，或者小誤差概率和關(guān)聯(lián)度誤差較大，GM（1,1）-GA的準(zhǔn)確性檢驗(yàn)是較好的【16。設(shè)置模擬殘差x（0）（k）為s(k)=x(o)(k)-x(o)(k)，k=1，2，?，n設(shè)置模擬的相對(duì)剩余為A(A(k)=s(k)/x(o)(k)|,k=1，2，?，n設(shè)置x(o)平均值為x=1工x(o)(k)nk=1設(shè)置x(o)的方差為S2=1工((0)01nk=1設(shè)置殘差平均值為s=1工s（k）nk=1設(shè)置殘差方差為S2=-工（s（k）-S>2nk=1因此，GM（1，1）-IGA的校驗(yàn)值如下:1）.平均相對(duì)誤差為a=1HA（k）nk=12).均方差率為c=S:S123）.小誤差概率為3）.小誤差概率為p=p(s(k)—S<0.6745S4).關(guān)聯(lián)度為s=(1+|S+|s|)/(1+|s|+|s|+|s—s|)其中,

S二藝Cx(°)(k)-x(°)(1))+Cx(°)(n)-x(°)(1)]k=22k=2根據(jù)上述公式，GM（1,1）-IGA的指標(biāo)的校驗(yàn)值見(jiàn)表1。二藝(X(°)(k)—x(°)(1))+1G(°)(n)2k=2根據(jù)上述公式，GM（1,1）-IGA的指標(biāo)的校驗(yàn)值見(jiàn)表1。表1GM-IGA和GM的四個(gè)指標(biāo)GM-GAGM平均相對(duì)誤差0.0000900.0001均方差率0.00390.0073小誤差概率10.92關(guān)聯(lián)度0.980.90通過(guò)表1可以看出，GM-GA所以指標(biāo)的精確度都是一級(jí)的，因此這個(gè)GM（1，1）-IGA可以被用來(lái)預(yù)測(cè)短期負(fù)荷。第四步：在圖1中，我們可以得到GM（1，1）-IGA的預(yù)測(cè)負(fù)荷數(shù)據(jù)曲線比GM（1，1）的曲線更接近于原始的日負(fù)荷數(shù)據(jù)曲線。進(jìn)一步分析，本文選擇相對(duì)誤差作為標(biāo)準(zhǔn)來(lái)評(píng)價(jià)兩種模式。兩種模型的偏差值如下，GM（1，1）的平均誤差為2.285%，然而，GM（1，1）-IGA的平均誤差為0.914%。訓(xùn)訓(xùn)第五章?結(jié)論本文提出了GM（1,1）-關(guān)于改進(jìn)的遺傳算法（GM（1,1）-IGA）來(lái)進(jìn)行短期負(fù)荷預(yù)測(cè)。采用十進(jìn)制編碼代表性方案，改進(jìn)的遺傳算法用于獲得GM（1,1）模型中的最優(yōu)值。本文也提出了單點(diǎn)線性算術(shù)交叉法，它能極大地提高交叉和變異的速度，因此GM（1,1）-IGA可以準(zhǔn)確地預(yù)測(cè)短期日負(fù)荷。GM（1，1）-IGA的特點(diǎn)是簡(jiǎn)單、易于開(kāi)發(fā)，因此，它在電力系統(tǒng)中作為一個(gè)輔助工具來(lái)解決預(yù)測(cè)問(wèn)題是適宜的。-4-60510152025Hour(h)2(％)Hour(h)圖2.GM(1，1)的偏差值2(％)Hour(h)圖3.GM(1,1)-IGA的偏差值致謝這項(xiàng)工作是由國(guó)家自然科學(xué)基金部分支持。（70671039）參考文獻(xiàn)P.GuptaandK.Yamada,“AdaptiveShort-TermLoadForecastingofHourlyLoadsUsingWeatherInformation,”IEEETr.OnPowerApparatusandSystems.VolPas-91,pp2085-2094,1972.D.W.Bunn,E.D.Farmer,“ComparativeModelsforElectricalLoadForecasting”.JohnWiley&Son,1985,NewYork.AbdolhosienS.Dehdashti,JamesRTudor,MichaelC.Smith,“ForecastingOfHourlyLoadByPatternRecognition-ADeterministicApproach,”IEEETr.OnPowerApparatusandSystems,Vol.AS-101,No.9Sept1982.S.RahrnanandRBhamagar,“AnexpertSystemBasedAlgorithmforShort-TermLoadForecast,”IEEETr.OnPowerSystems,Vol.AS-101,No.9Sept.1982M.T.Hagan,andS.M.Behr,“TimeSeriesApproachtoShort-TermLoadForecasting,”IEEETrans.onPowerSystem,Vol.2,No.3,pp.785-791,1987.XieNaiming,LiuSifeng.“ResearchonDiscreteGreyModelandItsMechanism”.IEEETr.System,ManandCybernetics,Vol1,2005,pp:606-610J.L.Deng,“Controlproblemsofgreysystems,”SystemsandControlLetters,vol.1,no.5,pp.288-294,1982.J.L.Deng,Introductiontogreysystemtheory,J.GreySyst.1(1)(1989)1-24J.L.Deng,PropertiesofmultivariablegreymodelGM(1N),J.GreySyst.1(1)(1989)125-141.J.L.Deng,Controlproblemsofgreysystems,Syst.ControlLett.1(1)(1989)288-294.Y.P.Huang,C.C.Huang,C.H.Hung,Determinationofthepreferredfuzzyvariablesandapplicationstothepredictioncontrolbythegreymodelling,TheSecondNationalConferenceonFuzzyTheoryandApplication,Taiwan,1994,pp.406-409.S0aeroandMRIrving,“AGeneticAlgorithmForGeneratorSchedulingInPowerSystems,”IEEETr.ElectricalPower&EnergySystems,Vol18.Nol,ppl9-261996.Edmund,T.H.HengDiptiSrinivasanA.C.Liew.“ShortTermLoadForecastingUsingGeneticAlgorithmAndNeuralNetworks”.IEEECatalogueNo:98EX137pp576-581Chew,J.M.,Lin,Y.H.,andChen,J.Y.,"TheGreyPredictorControlinInvertedPendulumSystem",JournalofChinaInstituteofTechnologyandCommerce,Vol.ll,pp.17-26,1995[15]J.GreySyst.,“Introductiontogreysystemtheory,”vol.1,no.1,pp.1-24,1989ApplicationofImprovedGreyPredictionModel

forPowerLoadForecasting[Abstract]Althoughthegreyforecastingmodelhasbeensuccessfullyutilizedinmanyfields,literaturesshowitsperformancestillcouldbeimproved.Forthispurpose,thispaperputforwardaGM(1,"-connectionimprovedgeneticalgorithm(GM(1,1)-IGA)forshort-termloadforecasting(STLF).WhileTraditionalGM(1,1)forecastingmodelisnotaccurateandthevalueofparameteraisconstant,inordertosolvethisproblemandenhancetheaccuracyofshort-termloadforecasting(STLF),theimproveddecimal-codegeneticalgorithm(GA)isappliedtosearchtheoptimalavalueofgreymodelGM(1,1).What'smore,thispaperalsoproposestheone-pointlinearityarithmeticalcrossover,whichcangreatlyimprovethespeedofcrossoverandmutation.Finally,adailyloadforecastingexampleisusedtotesttheGM(1,1)-IGAmodelandtraditionalGM(1,1)model,resultsshowthattheGM(1,1)-IGAhadbetteraccuracyandpracticality.Keywords:Short-termLoadForecasting,GreySystem,GeneticAlgorithm,One-pointLinearityArithmeticalCrossover.IntroductionDailypeakloadforecastingplaysanimportantroleinallaspectsofeconomic,reliable,andsecurestrategiesforpowersystem.Specifically,theshort-termloadforecasting(STLF)ofdailyelectricityusageiscrucialinunitcommitment,maintenance,powerinterchangeandtaskschedulingofbothpowergenerationanddistributionfacilities.Short-termloadforecasting(STLF)aimsatpredictingelectricloadsforaperiodofminutes,hours,daysorweeks.Thequalityoftheshort-termloadforecastswithleadtimesrangingfromonehourtoseveraldaysaheadhasasignificantimpactontheefficiencyofoperationofanypowerutility,becausemanyoperationaldecisions,suchaseconomicdispatchschedulingofthegeneratingcapacity,unitcommitment,schedulingoffuelpurchaseaswellassystemsecurityassessmentarebasedonsuchforecasts[1].Traditionalshort-termloadforecastingmodelscanbeclassifiedastimeseriesmodelsorregressionmodels[2,3,4].Usually,thesetechniquesareeffectivefortheforecastingofshort-termloadonnormaldaysbutfailtoyieldgoodresultsonthosedayswithspecialevents[5,6,7].Furthermore,becauseoftheircomplexities,enormouscomputationaleffortsarerequiredtoproduceacceptableresults.Thegreysystemtheory,originallypresentedbyDeng[8,9,10],focusesonmodeluncertaintyandinformationinsufficiencyinanalyzingandunderstandingsystemsviaresearchonconditionalanalysis,forecastinganddecisionmaking.Thegreysystemputseachstochasticvariableasagreyquantitythatchangeswithinagivenrange.Itdoesnotrelyonstatisticalmethodtodealwiththegreyquantity.Itdealsdirectlywiththeoriginaldata,andsearchestheintrinsicregularityofdata[11].Thegreyforecastingmodelutilisestheessentialpartofthegreysystemtheory.Therewith,greyforecastingcanbesaidtodefinetheestimationdonebytheuseofagreysystem,whichisinbetweenawhitesystemandablack-boxsystem.Asystemisdefinedasawhiteone訐theinformationinitisknown;otherwise,asystemwillbeablackbox訐nothinginitisclear.ThegreymodelGM(1,1)isthemainmodelofgreytheoryofprediction,i.e.asinglevariablefirstordergreymodel,whichiscreatedwithfewdata(fourormore)andstillwecangetfineforecastingresult[12].Thegreyforecastingmodelsaregivenbygreydifferentialequations,whicharegroupsofabnormaldifferentialequationswithvariationsinbehaviorparameters,orgreydifferenceequationswhicharegroupsofabnormaldifferenceequationswithvariationsinstructure,ratherthanthefirst-orderdifferentialequationsorthedifferenceequationsinconventionalcases[13].ThegreymodelGM(1,l)hasparameterawhichwasoftensetto0.5inmanyarticles,andthisconstantamightnotbeoptimal,becausedifferentquestionsmightneeddifferentavalue,whichproduceswrongresults.Inordertocorrecttheabove-mentioneddefect,thispaperattemptstoestimateabygeneticalgorithms.Geneticalgorithms(GA)werefirstlydescribedbyJohnHolland,whopresentedthemasanabstractionofbiologicalevolutionandgaveatheoreticalmathematicalframeworkforadaptation[14].ThedistinguishingfeatureofaGAwithrespecttootherfunctionoptimizationtechniquesisthatthesearchtowardsanoptimumsolutionproceedsnotbyincrementalchangestoasinglestructurebutbymaintainingapopulationofsolutionsfromwhichnewstructuresarecreatedusinggeneticoperators[15].Usually,thebinaryrepresentationwasappliedtomanyoptimizationproblems,butinthispapergeneticalgorithms(GA)adoptedimproveddecimal-coderepresentationscheme.ThispaperproposedGM(1,1)-improvedgeneticalgorithm(GM(1,1)-IGA)tosolveshort-termloadforecasting(STLF)problemsinpowersystem.ThetraditionalGM(1,1)forecastingmodeloftensetsthecoefficientato0.5,whichisthereasonwhythebackgroundvaluez(1)(k)maybeunsuitable.Inordertoovercometheabove-mentioneddrawbacks,theimproveddecimal-codegeneticalgorithmwasusedtoobtaintheoptimalcoefficientavaluetosetproperbackgroundvaluez(1)(k).Whatismore,theone-pointlinearityarithmeticalcrossoverwasputforward,whichcangreatlyimprovethespeedofcrossoverandmutationsothattheproposedGM(1,1)-IGAcanforecasttheshort-termdailyloadsuccessfully.Thepaperisorganizedasfollows:section2proposesthegreyforecastingmodelGM(1,1):section3presentsEstimateawithimprovedgeneticalgorithm:section4putsforwardashort-termdailyloadforecastingrealizedbyGM(1,1)-IGAandfinally,aconclusionisdrawninsection5.GreypredictionmodelGM(1,1)

Thissectionreviewstheoperationofgreyforecastingindetails.ThegreymodelGM(1,1)isatimeseriesforecastingmodel.Ithasthreebasicoperations:(1)accumulatedgeneration,(2)inverseaccumulatedgeneration,and(3)greymodeling.Thegreyforecastingmodelusestheoperationsofaccumulatedtoconstructdifferentialequations.Intrinsicallyspeaking,ithasthecharacteristicsofrequiringlessdata.ThegreymodelGM(1,1),i.e.,asinglevariablefirst-ordergreymodel,issummarizedasfollows:Stepl:Denotetheinitialtimesequencebyx(0)=Co)(1),x(o)(2),x(o)(3),...,x(o)(n))x(0)isthegivendiscreten-th-dimensionalsequence.x(0)(m)isthetimeseriesdataattimem,nmustbeequaltoorlargerthan4.Onthebasisoftheinitialsequencex(0),anewsequencex(1)issetupthroughtheaccumulatedgeneratingoperationinordertoprovidethemiddlemessageofbuildingamodelandtoweakenthevariationtendency,sox(1)isdefinedas:x(1)=(x(1)(1),x(1)(2),x(1)(3),...x(1)(n))andxG)=，…x（r）therWherex(】)(1)=x(o)(1),andx(1)(k)=￡x(0)(m),k=andxG)=，…x（r）therStep2:Tosettheavaluetofinez(1)(k)AccordingtoGM(1,1),wecanformthefollowingfirst-ordergreydifferentialequation:dx(1)+ax(1)=bdtAnditsdifferenceequationisx(o)(k)+az(1)(k)=b.WhereawascalledthedevelopingcoefficientofGM,andbwascalledthecontrolvariable.Denotingthedifferentialcoefficientsubentryintheformofdifference,wecanget:dt處=x(1)(k+1)-x(1)(k)=x(1)(k+1)-x(1)(k)dtBeforeagreyGM(1,1)modelwassetup,aproperavalueneededtobeassignedforabetterbackgroundvaluez(1)(k).Thesequenceofbackgroundvalueswasdefinedas:z(1)=t(1)(1),z(1)(2),...,z(1)(n)}Amongthemz(1)(k)=a*x(1)(k)+(1-a)*x(1)(k-1),k=2,3…n,0<a<1Forconvenience,theavaluewasoftensetto0.5,thez(1)(k)wasderivedas:22However,thisconstantamightnotbeoptimalbecausethedifferentquestionsmightneeddifferentavalue.And,bothdevelopingcoefficientaandcontrolvariablebweredeterminedbythez(1)(k).Theprocessoftheoriginalgreyinformationforwhiteningmaybesuppressedresultedfromthecoefficientawasconstant.Hence,theaccuracyofpredictionvaluex'(0)(k)inGM(1,1)modelwouldseriouslybedecreased.Inordertocorrectthedefect,thecoefficientamustbeavariablebasedonthefeatureofproblems,soweestimateabygeneticalgorithms.Step3:ToconstructaccumulatedmatrixBandcoefficientvectorXn.ApplyingtheOrdinaryLeastSquare(OLS)methodobtainsthedevelopingcoefficienta,bwasasfollows:■-zd)(2)1「B_-z(1)(3)1LJ—-z(】)(n)1andx_X0)(2),X0()(x)(n_L/\一Soa,Soa,b_\Bt*BJ-1*Bt*XStep4:Toobtainthediscreteformoffirst-ordergreydifferentialequation,asfollows:Thesolutionofx(1)isx(1x(1)(k+1)_([、

x(0)(1)—天a,.b*e-ak+aAndthesolutionofx(0)isaAndthesolutionofx(0)isa—1)*[x(0)(1)-b*e-akEstimateawithimprovedGAInordertoestimatetheaccuracyofgreymodeGM(1,1),theresidualerrortestwasessential.Therefore,theobjectivefunctionoftheproposedmethodinthispaperwastoensurethattheforecastingvalueerrorswereminimum.Theobjectivefunctionwasdefinedasmeanabsolutepercentageerror(MAPE)minimizationasfollows:minMAPE=￡||e(k)||k=1Where,e(k)=Where,e(k)=x(0)(k)-x(0)(k)

x(0)(k)x100%x(0)(k)isoriginaldata,x(0)(k)isforecastingvalue,nisthenumberofsequencedata.However,fromtheabovedescriptionoftheestablishmentofGM(1,1),wecanget:InGM(1,,thevalueofparameter以candeterminez(1),and,bothdevelopingcoefficientaandcontrolvariablebweredeterminedbythez(1)(k).Whatismore,thesolutionofx(0)wasdeterminedbyaandb,sothekeypartofthewholemodelselectingprocesswasthevalueofa.Thereiskindofcomplicatednonlinearrelationshipbetweenaandresidualerrors,andthisnonlinearitywashardtosolvebyresolution,sotheoptimalselectionofawasthedifficultpointofGM(1,1).Geneticalgorithmisarandomsearchalgorithmthatsimulatesnaturalselectionandevolution.Itisfindingwidespreadapplicationasaconsequenceoftwofundamentalaspects:thecomputationalcodeisverysimpleandyetprovidesapowerfulsearchmechanism.Theyarefunctionindependentwhichmeanstheyarenotlimitedbythepropertiesofthefunctionsuchascontinuity,existenceofderivatives,etc.Althoughthebinaryrepresentationwasusuallyappliedtomanyoptimizationproblems,inthispaper,weusedtheimproveddecimal-coderepresentationschemeforsolution.Theimproveddecimal-coderepresentationintheGAoffersanumberofadvantagesinnumericalfunctionoptimizationoverbinaryencoding.Theadvantagescanbebrieflydescribedasfollows:Stepl:EfficiencyofGAisincreasedasthereisnoneedtoconvertchromosomestothebinarytype,Step2:Lessmemoryisrequiredasefficientfloating-pointinternalcomputerrepresentationscanbeuseddirectly,Step3:Thereisnolossinprecisionbydiscriminationtobinaryorothervalues,andthereisgreaterfreedomtousedifferentgeneticoperators.Weutilizedtheimproveddecimal-coderepresentationschemeforsearchingoptimalcoefficientavalueingreyGM(1,1)model.Inthispaper,weproposedone-pointlinearityarithmeticalcrossoverandutilizedittoselectthevalueofa;itcangreatlyimprovethespeed

ofcrossoverandmutation.Thestepsoftheimproveddecimal-coderepresentationschemeareasfollows:Coding：Supposeae[0,1]isabinarystringofCbits,thenleteverynbitstransformadecimalfromrighttoleft.(nvC,thevaluesofnandCareensuredbyprecision)Randomizepopulation:SelectoneintegerMasthesizeofthepopulation,andthenselectMpointsstochasticallyfromtheset[0,1],asa(i,0)(i=1,2,M),thesepointscomposetheindividualsoftheoriginalpopulation,thesequenceisdefinedas:P(0)={a(1,0),a(2,0),a(M,0)}inthereproductionofnewindividuals.Theindividuala(a(i,k))hasthepriorityF(ainthereproductionofnewindividuals.Theindividuala(a(i,k))hasthepriorityF(a(i,k))二andX(o)(a(i,k))isthevalueofforecastingwiisthemaximumofthesumofiterativesquares.yandadvancestothenextgeneration.—f(a(i,k)),f(a(i,k))<yandadvancestothenextgeneration.—f(a(i,k)),f(a(i,k))<c宅Imax=乙一0,cmaxisgainedbytheindividuala(i,k).cmaxStep4:Selection:Inthispaper,wecalculateindividualselectedprobability()F(a(l,K))/=八F(a(i,k))=1respectivelyaccordingtotheirfitnessfunctionsF(a(i,k)),thenweadopttheroulettewheelselectionscheme,sothatthepropagatedprobabilityofrespectiveindividualisp(k),afterthatwetaketheinbornindividualtocomposethenextgenerationp(k+1).Step5:CrossoverandMutation:Codingandcrossoverarecorrelative;weutilizedthedecimal-coderepresentation,soweproposeanewcrossoveroperator“one-pointlinearityarithmeticalcrossover”Selectthefittwoindividualswithprobabilityofcrossoverp.cForthetwoselectedindividuals,westilladopttherandomselectionmeanstoensurethecrossoveroperator.Forexample::,z???zzz、i1i2iki(k+1),z???zzz、j1j2jkj(k+1)crossover:Weexchangetheirrightstringseachother.Thebitontheleftofcrossovercanbecalculatedthroughthefollowingalgorithm:a:Geneanalysis:z=卩*z+（1-卩）*zz=卩*z+（1-卩）*zjjjb:Exchangethebackgene:z=卩*z+（1-卩）*zz=卩*z+（1-卩）*zjkr-ijkikThe卩eL0,lJiscalledcrossovercoefficient,itischoseneachtimebyrandomcrossoveroperation.4）Mutation:Thereisanewmutationoperation:whenthemutationoperatorwaschosen,thenewgenevalueisthatarandomnumberwithinthedomainofweight,whichisoperatedintoaweightedsumwithoriginalgenevalue.IfthevalueofmutationoperatorisZi,themutationvalueis:z=a*r+（1-a）*z,zimax」imin.ItAndaisthemutationcoefficient,ae[0,1].risarandomnumber,,zimax」imin.ItStep6:Quitprinciple:Selecttheremainingindividualsinthecurrentgenerationtoreproducetheindividualsinthenextgeneration,thenevaluatethefitnessvalueandjudgewhetherthealgorithmfulfilsthequitcondition.Ifitiscertifiable,inthiscasetheavalueisoptimalsolution,elserepeatfromStep4untilallindividualsinpopulationmeettheconvergencecriteriaorthenumberofgenerationsexceedsthemaximumof100.4.Loadpredictionexample

Inthissection,wetrytoevaluatetheperformanceofGM(1,“connectionimprovedgeneticalgorithm.First:Thedailyloaddatasequencesofmdaysaredefinedasix(k)|k=1,2,...,n},wemeasuredthepowerloadeachhour,andtheloadsequencevectorisatwenty-four-dimensionaldata.=fx=fx(i)|i=1,2,.,m}=fx(i)li=1,2,m}02f(0)?“}x(i)i=1,2,.,m)=fx(i)|i=1,2,...,m}0102thetimeofday:XIjthetimeofday:X:j24thetimeofday:X2424'Wheremisthenumberofmodelingdays,Xjisthedailyloaddatasequenceofthej-thtimeofday.{x{x(m+1)|jj=1,2,...,24Second:WeutilizeimprovedgeneticalgorithmtoselectthevalueofaforrespectiveloaddatasequenceXj.Afterthat,wecancalculateaandb,thenweutilizeGM(1,1)-IGAtopredicttheloadforecastingofthej-thtimeofthe(m+1)-thday,sowecouldgetXj(m+1),andthetwenty-fourforecastingvaluesofthe(m+1)-thdaystructuretheloaddatasequenceTherewasanexampleofGM(l,l)-connectionimprovedgeneticalgorithm(GM(l,l)-IGA),boththetwoforecastingdailyloaddatacurves(July26)andtheoriginaldailyloaddatacurveweredrawnsimultaneouslyonFig1.Thirdly:WecanusefourindexesofthisGM(1,1)-GAtoverifytheprecise,includingoftherelativeerror,theratioofmeansquareerror,themicroerrorprobabilityandtherelevancedegree.TheaccuracyverificationofGM(1,1)-GAisbetteriftherelativeerrorandtheratioofmeansquareerrorislower,orthemicroerrorprobabilityandtherelevancedegreeislarger[16].Setthesimulatedresidualofx(0)(k)iss(k)=x(o)(k)-X(o)(k)k=1,2,-SetthesimulatedrelativeresidualisA(k)=s(k”x(0)(k)|,Setthemeanofx(0)isx=—工x(o)(k)nk=1k=1,2,?Setthevarianceofx(0)isS2=—工(x(o)(k)一x)1nk=1Setthemeansofresidualerroriss=1工s(k)nk=1SetthevarianceofresidualerrorisS22=1工(s(k)-?nk=1SothecheckvalueofthisGM(1,1)-GAisasfollowed:1).themeanrelativeerrorisa=1工A(k)n3.)themicroerrorprobabilityisp=p(s(k)-S<0.6745S)k=12).theratioofmeansquareerrorisc=S:Sf24).therelevancedegreeiss=G+|S+|s|),，，(1+|S+|s|+|s-s|)Thereamong,s=藝(x(o)(k)-x(0(1))+—C()o(n)-x()(1))2k=2s=藝G(o)(k)-x(o)(1))+G(o)(n)—x(o)(1))k=2Onthebasisofaboveformula,theindexesofverificationofGM(1,1)-GAandGMisinTable1.Table1ThefourindexesofGM-GAandGMGM-GAGMThemeanrelativeerror0.0000900.0001ratioofmeansquareerror0.00390.0073microerrorprobability10.92therelevancedegree0.980.90Accordingtotable1,theallprecisionindexesofGM-GAarefirstdegree,sothisGM(1,1)-GAcanbeusedtopredicttheshort-termload.Fourth:AtFig1,wecangetthattheforecastingloaddatacurveofGM(1,1)-GAwasmoreclosedtotheoriginaldailyloaddatacurvethanGM(1,l)'s.Forfurtheranalysis,thispaperselectsrelativeerrorsasacriteriontoevaluatethetwomodels.Theerrorfiguresoftwomodelsareasfollows,andtheaverageerrorofGM(1,1)was2.285%,otherwise,theaverageerrorofGM(1,1)-IGAwas0.914%.5.Conclusion

ThispaperproposesGM(1,1)connectionimprovedge