电力系统分析习题集(第七章)

1、电力系统分析习题集（第七章）【例7-1】用改进欧拉法求解微分方程其初值为，。【解】：步长取。计算结果见下表。01234500.20.40.60.81.011.186671.348321.498372.627901.7543010.849590.754990.690360.645001.21.356581.499321.631791.756900.20.40.60.81.00.86670.76690.69900.65130.61850.93330.80830.72700.67080.63181.186671.348321.493721.627881.75430这一微分方程的准确解为当时，故误差为

2、【例7-2】用四阶龙格-库塔法求解例7-1中的一阶微分方程。【解】步长取。计算过程及结果如下表所示。00.20.40.60.811 1.18322921.34166681.4832811.6125131.7321410.20.16983420.14907880.13485280.12405460.10.30.50.70.91.11.2677461.4160261.5507071.6745410.18363640.15889300.14201880.12957860.11992400.10.30.50.70.91.09181821.2626761.4126761.5480701.6724750.

3、18172740.15749900.14096000.12874360.11924520.20.40.60.81.01.1817271.3407281.4826271.6120251.7317590.16864780.14880740.13465060.12389700.1153728由以上计算结果可知，当采用龙格-库塔法时，函数值在时为。和准确解相比，其误差为和例7-1相比精确度提高很显著。【例7-3】考虑图7-12所示的9节点电力系统。该系统有3台发电机、3个负荷以及9条支路。支路数据和发电机参数分别列于表7-5和表7-6，正常运行情况下的系统潮流如表7-7所示，系统频率为60Hz。图7-

4、12 9节点系统单线结构图表7-5 支路数据首端母线名末端母线名电阻（标么值）电抗（标么值）容纳之半（标么值）变压器非标准变比450.0100.0850.088460.0170.0920.079570.0320.1610.153690.0390.1700.179780.00850.0720.0745890.01190.10080.1045140.00.05761.0270.00.06251.0390.00.05861.0表7-6 发电机数据发电机母线名1147.280.00.14600.06080.09690.09698.960.02212.800.00.89580.11980.86450.1

5、9696.000.5350.0336.020.01.31250.18131.25780.25005.890.6000.0注：表中所有时间常数的单位为“秒”，阻尼系数及所有电阻、电抗的单位为“标么值”。表6-7 正常运行情况下的系统潮流母线名电压发电机负荷幅值相角（度）有功功率无功功率有功功率无功功率11.0400.00000.71640.270521.02509.28001.63000.066531.02504.66480.8500-0.108641.0258-2.216850.9956-3.98881.25000.500061.0127-3.68740.90000.300071.02583.

6、719781.01590.72751.00000.350091.03241.9667【解】下面针对该系统进行简单模型下的暂态稳定性分析。干扰是在零秒线路57靠近母线7处发生三相接地短路，故障在5个周波（约0.08333秒）由断开线路57而被消除。各发电机用暂态电势保持恒定来模拟，各负荷用恒定阻抗模拟，电力网络用导纳矩阵描述，微分方程用改进欧拉法求解，网络方程用直接法求解。根据图6-8中暂态稳定性分析的基本过程和上节所介绍的方法及计算公式，暂态稳定性分析可归纳如下：1） 初值计算根据式计算各负荷的等值并联导纳，结果如下：负荷（节点5）：负荷（节点6）：负荷（节点8）：根据式计算各发电机的暂态电势

7、、初始功角及其输入机械功率，结果如表7-8所示。另外各发电机角速度的初值为。在以下计算中，不计发电机凸极效应相当于在模型中令，即就是发电机的经典模型。表7-8 发电机的、及发电机不计凸极效应计及凸极效应11.056642.271651.056363.585720.7164121.0502019.731590.7881761.098441.6300031.0169713.166410.7678654.136620.850002） 故障系统与故障后系统描述故障期间的电力网络相当于在7号母线处并联一条阻抗为零的接地支路，这时只要将正常情况下导纳矩阵中的对角元素改为无穷大（可以用大数模拟，例如在实际计

8、算中取为），即可得到故障期间系统的节点导纳矩阵。故障后的电力网络是切除线路57后的情况。由于线路57对导纳矩阵的贡献为式中：，。因此故障后系统的节点导纳矩阵。3） 微分代数方程的数值积分我们仅计算从短路故障开始系统在2秒内的暂态过程。这样，02秒的暂态过程可划分为两个自治系统，即00.08333秒的故障系统和0.083332秒的故障后系统。数值积分采用0.001秒的步长。表7-9列出了不计凸极效应和计及凸极效应时各台发电机的及最大相对摇摆角，后者也如图7-13所示。表7-9 各台发电机的及最大相对摇摆角不计凸极效应计及凸极效应0.000002.2716519.7315913.166413.58

9、57261.0984454.1366217.4599457.512720.042002.2877922.1576414.638563.6901663.5244955.3161719.8698559.834330.083332.3484829.2823718.862484.0027070.6492258.7408426.9338966.646520.133332.4080341.2154025.927574.4840982.6935964.7961938.8073778.209510.183332.5825153.4839533.683205.0931895.3752072.0580450.90

10、14490.282020.233333.1940165.3037841.941826.11234108.0348780.4555662.10977101.922530.283334.5239776.1225850.429077.80288120.2652389.7821171.59862112.462350.333336.7944785.6259158.8093310.37892131.9217199.7492978.83143121.542800.3833310.1641593.6868266.7289213.99684143.06625110.0706583.52267129.069410

11、.4333314.73304100.2963573.8562818.75788153.88636120.5314385.56332135.128470.4833320.54980105.5016879.9165624.71833164.62086131.0217084.95188139.902540.5333327.61528109.3653984.7313931.90242175.50595141.5349181.75011143.603530.5833335.87927111.9524888.2728140.31440186.74309152.1462776.07321146.428690

12、.6333345.23062113.3480590.7258749.94781198.48469162.9854268.11743148.536880.6833355.48563113.7053692.5299660.79180210.83172174.2113858.21973150.039920.7333366.38413113.3138594.3501972.83490223.83807185.9919846.92972151.003170.7833377.60323112.6623296.9430486.06717237.51766198.4868935.05909151.450490

13、.8333388.79177112.45974100.94170100.48127251.85143211.8329323.66797151.370160.8833399.61769113.57983106.66335116.07313266.79269226.1308713.96214150.719560.93333109.81335116.92253114.05272132.84229282.27022241.433727.10918149.427930.98333119.20630123.22495122.79557150.79173298.18912257.737014.0186514

14、7.397401.03333127.73265132.88223132.52906169.92694314.43044274.972265.14958144.503511.08333135.43743145.83696143.03225190.25360330.85111293.0052210.39953140.597521.13333142.46732161.57457154.30791211.77290347.28707311.6416619.10725135.514171.18333149.05619179.23461166.52345234.47339363.56384330.6445

15、630.17842129.090451.23333155.50082197.81063179.84443258.31838379.52044349.7670642.30981121.202061.28333162.12683216.36810194.25664283.22935395.05209368.8034054.24126111.822741.33333169.25129234.19924209.48217309.06874410.17107387.6529464.94795101.102331.38333177.15096250.87411225.02567335.62983425.0

16、7106406.3803673.7231589.441231.43333186.04298266.20363240.30096362.64368440.16296425.2446280.1606577.519281.48333196.07909280.15828254.76080389.80754456.04799444.6723984.0791966.240451.53333207.34945292.78226267.98613416.82762473.42157465.1729885.4328156.593951.58333219.88920304.12763279.73658443.45

17、984492.94327487.2207484.2384249.483431.63333233.68131314.22084289.98411469.53546515.12417511.1440880.5395345.588711.68333248.65233323.06766298.94296494.97070540.25876537.0506974.4153345.288061.73333264.66308330.69730307.08305519.76777568.39552564.8011166.0342248.627751.78333281.50116337.23926315.089

18、28544.01356599.32930594.0362255.7381055.315741.83333298.88496343.01311323.73003567.87632632.61227624.2649844.1281464.735951.88333316.48551348.59530333.64692591.59440667.60021655.0015532.1097976.005811.93333333.96386354.81755345.15535615.45111703.55468685.9012620.8536988.103571.98333351.01413362.6634

19、6358.17917639.73684739.78895716.8275011.64932100.05211图7-13 相对摇摆角与时间的关系曲线由图7-13可以看出，无论计及凸极效应还是不计凸极效应，系统都是暂态稳定的。在计及凸极效应的情况下，最大相对摇摆角为（）。在不计凸极效应的情况下，最大相对摇摆角为（），而第二摆的角度（）比第一摆的角度小。最后，对不计凸极效应和计及凸极效应情况下的临界切除时间进行了计算。得到前者所对应的临界切除时间在之间，后者所对应的临界切除时间在之间。它们对应的摇摆曲线分别如图7-14和图7-15所示。【特例】隐式解法相对于显式解法来说的优点是可以采取较大的步长。这个问题涉及到微分方程数值解的稳定性问题，读者可参看有关文献。我们在这里只用一个简单的例来直观地说明这个问题。设有一阶微分方程初值为时。对于这个微分方程不难求出它的正确解为：这是一个按指数曲线衰减很快的函数，如图7-7所示。当步长取时，用欧拉法计算结果如下