電力系統(tǒng)分析習(xí)題集(第七章)

1、精選優(yōu)質(zhì)文檔-傾情為你奉上精選優(yōu)質(zhì)文檔-傾情為你奉上專心-專注-專業(yè)專心-專注-專業(yè)精選優(yōu)質(zhì)文檔-傾情為你奉上專心-專注-專業(yè)電力系統(tǒng)分析習(xí)題集（第七章）【例7-1】用改進(jìn)歐拉法求解微分方程其初值為，?！窘狻浚翰介L取。計算結(jié)果見下表。01234500.20.40.60.81.011.186671.348321.498372.627901.7543010.849590.754990.690360.645001.21.356581.499321.631791.756900.20.40.60.81.00.86670.76690.69900.65130.61850.93330.80830.72700.

2、67080.63181.186671.348321.493721.627881.75430這一微分方程的準(zhǔn)確解為當(dāng)時，故誤差為【例7-2】用四階龍格-庫塔法求解例7-1中的一階微分方程?！窘狻坎介L取。計算過程及結(jié)果如下表所示。00.20.40.60.811 1.1.1.1.1.0.20.0.0.0.0.10.30.50.70.91.11.1.1.1.0.0.0.0.0.0.10.30.50.70.91.1.1.1.1.0.0.0.0.0.0.20.40.60.81.01.1.1.1.1.0.0.0.0.0.由以上計算結(jié)果可知，當(dāng)采用龍格-庫塔法時，函數(shù)值在時為。和準(zhǔn)確解相比，其誤差為和例7-1

3、相比精確度提高很顯著?！纠?-3】考慮圖7-12所示的9節(jié)點電力系統(tǒng)。該系統(tǒng)有3臺發(fā)電機(jī)、3個負(fù)荷以及9條支路。支路數(shù)據(jù)和發(fā)電機(jī)參數(shù)分別列于表7-5和表7-6，正常運行情況下的系統(tǒng)潮流如表7-7所示，系統(tǒng)頻率為60Hz。圖7-12 9節(jié)點系統(tǒng)單線結(jié)構(gòu)圖表7-5 支路數(shù)據(jù)首端母線名末端母線名電阻（標(biāo)么值）電抗（標(biāo)么值）容納之半（標(biāo)么值）變壓器非標(biāo)準(zhǔn)變比450.0100.0850.088460.0170.0920.079570.0320.1610.153690.0390.1700.179780.00850.0720.0745890.01190.10080.1045140.00.05761.0270

4、.00.06251.0390.00.05861.0表7-6 發(fā)電機(jī)數(shù)據(jù)發(fā)電機(jī)母線名1147.280.00.14600.06080.09690.09698.960.02212.800.00.89580.11980.86450.19696.000.5350.0336.020.01.31250.18131.25780.25005.890.6000.0注：表中所有時間常數(shù)的單位為“秒”，阻尼系數(shù)及所有電阻、電抗的單位為“標(biāo)么值”。表6-7 正常運行情況下的系統(tǒng)潮流母線名電壓發(fā)電機(jī)負(fù)荷幅值相角（度）有功功率無功功率有功功率無功功率11.0400.00000.71640.270521.02509.2800

5、1.63000.066531.02504.66480.8500-0.108641.0258-2.216850.9956-3.98881.25000.500061.0127-3.68740.90000.300071.02583.719781.01590.72751.00000.350091.03241.9667【解】下面針對該系統(tǒng)進(jìn)行簡單模型下的暫態(tài)穩(wěn)定性分析。干擾是在零秒線路57靠近母線7處發(fā)生三相接地短路，故障在5個周波（約0.08333秒）由斷開線路57而被消除。各發(fā)電機(jī)用暫態(tài)電勢保持恒定來模擬，各負(fù)荷用恒定阻抗模擬，電力網(wǎng)絡(luò)用導(dǎo)納矩陣描述，微分方程用改進(jìn)歐拉法求解，網(wǎng)絡(luò)方程用直接法求解。

6、根據(jù)圖6-8中暫態(tài)穩(wěn)定性分析的基本過程和上節(jié)所介紹的方法及計算公式，暫態(tài)穩(wěn)定性分析可歸納如下：初值計算根據(jù)式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-81)計算各負(fù)荷的等值并聯(lián)導(dǎo)納，結(jié)果如下：負(fù)荷（節(jié)點5）：負(fù)荷（節(jié)點6）：負(fù)荷（節(jié)點8）：根據(jù)式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-74) GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-

7、80)計算各發(fā)電機(jī)的暫態(tài)電勢、初始功角及其輸入機(jī)械功率，結(jié)果如表7-8所示。另外各發(fā)電機(jī)角速度的初值為。在以下計算中，不計發(fā)電機(jī)凸極效應(yīng)相當(dāng)于在模型中令，即就是發(fā)電機(jī)的經(jīng)典模型。表7-8 發(fā)電機(jī)的、及發(fā)電機(jī)不計凸極效應(yīng)計及凸極效應(yīng)11.056642.271651.056363.585720.7164121.0502019.731590.7881761.098441.6300031.0169713.166410.7678654.136620.85000故障系統(tǒng)與故障后系統(tǒng)描述故障期間的電力網(wǎng)絡(luò)相當(dāng)于在7號母線處并聯(lián)一條阻抗為零的接地支路，這時只要將正常情況下導(dǎo)納矩陣中的對角元素改為無窮大（可以用

8、大數(shù)模擬，例如在實際計算中取為），即可得到故障期間系統(tǒng)的節(jié)點導(dǎo)納矩陣。故障后的電力網(wǎng)絡(luò)是切除線路57后的情況。由于線路57對導(dǎo)納矩陣的貢獻(xiàn)為式中：，。因此故障后系統(tǒng)的節(jié)點導(dǎo)納矩陣。微分代數(shù)方程的數(shù)值積分我們僅計算從短路故障開始系統(tǒng)在2秒內(nèi)的暫態(tài)過程。這樣，02秒的暫態(tài)過程可劃分為兩個自治系統(tǒng)，即00.08333秒的故障系統(tǒng)和0.083332秒的故障后系統(tǒng)。數(shù)值積分采用0.001秒的步長。表7-9列出了不計凸極效應(yīng)和計及凸極效應(yīng)時各臺發(fā)電機(jī)的及最大相對搖擺角，后者也如圖7-13所示。表7-9 各臺發(fā)電機(jī)的及最大相對搖擺角不計凸極效應(yīng)計及凸極效應(yīng)0.000002.2716519.7315913.1

9、66413.5857261.0984454.1366217.4599457.512720.042002.2877922.1576414.638563.6901663.5244955.3161719.8698559.834330.083332.3484829.2823718.862484.0027070.6492258.7408426.9338966.646520.133332.4080341.2154025.927574.4840982.6935964.7961938.8073778.209510.183332.5825153.4839533.683205.0931895.3752072.05

10、80450.9014490.282020.233333.1940165.3037841.941826.11234108.0348780.4555662.10977101.922530.283334.5239776.1225850.429077.80288120.2652389.7821171.59862112.462350.333336.7944785.6259158.8093310.37892131.9217199.7492978.83143121.542800.3833310.1641593.6868266.7289213.99684143.06625110.0706583.5226712

11、9.069410.4333314.73304100.2963573.8562818.75788153.88636120.5314385.56332135.128470.4833320.54980105.5016879.9165624.71833164.62086131.0217084.95188139.902540.5333327.61528109.3653984.7313931.90242175.50595141.5349181.75011143.603530.5833335.87927111.9524888.2728140.31440186.74309152.1462776.0732114

12、6.428690.6333345.23062113.3480590.7258749.94781198.48469162.9854268.11743148.536880.6833355.48563113.7053692.5299660.79180210.83172174.2113858.21973150.039920.7333366.38413113.3138594.3501972.83490223.83807185.9919846.92972151.003170.7833377.60323112.6623296.9430486.06717237.51766198.4868935.0590915

13、1.450490.8333388.79177112.45974100.94170100.48127251.85143211.8329323.66797151.370160.8833399.61769113.57983106.66335116.07313266.79269226.1308713.96214150.719560.93333109.81335116.92253114.05272132.84229282.27022241.433727.10918149.427930.98333119.20630123.22495122.79557150.79173298.18912257.737014

14、.01865147.397401.03333127.73265132.88223132.52906169.92694314.43044274.972265.14958144.503511.08333135.43743145.83696143.03225190.25360330.85111293.0052210.39953140.597521.13333142.46732161.57457154.30791211.77290347.28707311.6416619.10725135.514171.18333149.05619179.23461166.52345234.47339363.56384

15、330.6445630.17842129.090451.23333155.50082197.81063179.84443258.31838379.52044349.7670642.30981121.202061.28333162.12683216.36810194.25664283.22935395.05209368.8034054.24126111.822741.33333169.25129234.19924209.48217309.06874410.17107387.6529464.94795101.102331.38333177.15096250.87411225.02567335.62

16、983425.07106406.3803673.7231589.441231.43333186.04298266.20363240.30096362.64368440.16296425.2446280.1606577.519281.48333196.07909280.15828254.76080389.80754456.04799444.6723984.0791966.240451.53333207.34945292.78226267.98613416.82762473.42157465.1729885.4328156.593951.58333219.88920304.12763279.736

17、58443.45984492.94327487.2207484.2384249.483431.63333233.68131314.22084289.98411469.53546515.12417511.1440880.5395345.588711.68333248.65233323.06766298.94296494.97070540.25876537.0506974.4153345.288061.73333264.66308330.69730307.08305519.76777568.39552564.8011166.0342248.627751.78333281.50116337.2392

18、6315.08928544.01356599.32930594.0362255.7381055.315741.83333298.88496343.01311323.73003567.87632632.61227624.2649844.1281464.735951.88333316.48551348.59530333.64692591.59440667.60021655.0015532.1097976.005811.93333333.96386354.81755345.15535615.45111703.55468685.9012620.8536988.103571.98333351.01413

19、362.66346358.17917639.73684739.78895716.8275011.64932100.05211圖7-13 相對搖擺角與時間的關(guān)系曲線由圖7-13可以看出，無論計及凸極效應(yīng)還是不計凸極效應(yīng)，系統(tǒng)都是暫態(tài)穩(wěn)定的。在計及凸極效應(yīng)的情況下，最大相對搖擺角為（）。在不計凸極效應(yīng)的情況下，最大相對搖擺角為（），而第二擺的角度（）比第一擺的角度小。最后，對不計凸極效應(yīng)和計及凸極效應(yīng)情況下的臨界切除時間進(jìn)行了計算。得到前者所對應(yīng)的臨界切除時間在之間，后者所對應(yīng)的臨界切除時間在之間。它們對應(yīng)的搖擺曲線分別如圖7-14和圖7-15所示?！咎乩侩[式解法相對于顯式解法來說的優(yōu)點是可以采

20、取較大的步長。這個問題涉及到微分方程數(shù)值解的穩(wěn)定性問題，讀者可參看有關(guān)文獻(xiàn)。我們在這里只用一個簡單的例來直觀地說明這個問題。設(shè)有一階微分方程 MACROBUTTON MTPlaceRef * MERGEFORMAT SEQ MTEqn h * MERGEFORMAT (7- SEQ MTEqn c * Arabic * MERGEFORMAT 28)初值為時。對于這個微分方程不難求出它的正確解為：這是一個按指數(shù)曲線衰減很快的函數(shù)，如圖7-7所示。當(dāng)步長取時，用歐拉法計算結(jié)果如下表所示。時段順序00.0001-100-2.5l0.025-1.51503.7520.0502.25-225-5.62

21、530.075-3.375可以看出，上表所列函數(shù)值隨時間在作振蕩的變化，而且振蕩的幅值愈來愈大，如圖7-7中虛線所示。從數(shù)學(xué)上來說，這種情況表示歐拉法數(shù)值解本身已經(jīng)不穩(wěn)定。當(dāng)采用隱式歐拉法計算時，就不會出現(xiàn)這種情況。首先將式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-28)化為差分方程因此可以得到當(dāng)時上式變?yōu)橐虼丝梢缘玫较卤碛嬎愕慕Y(jié)果：時段順序00.000110.02520.05030.075上表所列函數(shù)值隨時間單調(diào)衰減，如圖7-7中點劃線所示。圖7-7 微分方程不同解法的圖例為了進(jìn)一步說明以上現(xiàn)象與步長的關(guān)

22、系，我們把微分方程式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-28)寫成更一般的形式 MACROBUTTON MTPlaceRef * MERGEFORMAT SEQ MTEqn h * MERGEFORMAT (7- SEQ MTEqn c * Arabic * MERGEFORMAT 29)式中：常數(shù)具有時間的量綱，在工程上叫做時間常數(shù)。當(dāng)采用歐拉法時，將式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-29)代入遞推公式 GOTOBUTTON ZEqnNum * MERGEFORMAT REF ZEqnNum ! * MERGEFORMAT (7-9)，得因此 MACROBUTTON MTPlaceRef * MERGEFORMAT SEQ MTEqn h * MERGEFORMAT (7-