爬升玩具美术教学设计

爬升玩具美术教学设计（精选6篇）

爬升玩具美术教学设计 篇1

爬升玩具

【教学内容】 设计·应用 【教材分析】

本课教学以“爬升玩具”为主题，利用身边的硬纸筒、塑料瓶、彩色纸和剪子、胶水进行爬升玩具的创造、设计和制作，并利用挂起的绳子进行游戏，使玩具升起来。【教学目标】

1.了解爬升玩具的制作方法。2.进行爬升玩具的创作和设计。

3.感受爬升玩具的创作、设计、制作的乐趣，提高学生选择材料、色彩进行创作和动手制作的能力。【教学重点】

学习爬升玩具的创作、设计。【教学难点】

作品的新颖，制作的精美、实用。【教具】

课本、爬升玩具、硬纸筒、塑料瓶、彩色纸和剪子、胶水 【学具】

课本、硬纸筒、塑料瓶、彩色纸和剪子、胶水 【课时】 第一课时 【教学过程】

一、组织教学：

检查学生学具的准备情况。稳定学生情绪。

二、导入：

课件展示几种“飞”的玩具。研究飞起来的原理。继续研究课本中爬升玩具是怎样升起来的？

（体验玩具的乐趣，并研究其原理。培养学生的探究科学的精神。）引出课题。

三、讲授新知：

爱心

用心

专心 1

想一想什么会飞？你想做一个什么样的爬升玩具？ 学生进行畅想讨论。课件展示，启发构思：

想一想，可以利用什么材料来制作爬升玩具呢？

小组分析爬升玩具实物：想一想怎样巧妙利用材料来制作一个漂亮有趣的爬升玩具？ 教师出示爬升玩具，小组分析：这些爬升玩具利用了什么形状的材料制作的？你还能想出用什么材料来制作？

说一说，什么可以制作成形象的爬升玩具？你们有更好的创意吗？ 可以利用哪些制作方法来制作一个漂亮的爬升玩具？ 教师进行简单示范制作。

比一比哪个小组研究出来的爬升玩具制作方法又快又好。

四、作品创作：

设计制作一个造型新颖有趣的爬升玩具。学生创作。3.教师巡回辅导。【作业设计】

设计制作一个造型新颖有趣的爬升玩具。

第二课时

【教学内容】 设计·应用 【教材分析】

本课教学以“爬升玩具”为主题，利用身边的硬纸筒、塑料瓶、彩色纸和剪子、胶水进行爬升玩具的创造、设计和制作，并利用挂起的绳子进行游戏，使玩具升起来。【教学目标】

1.了解爬升玩具的制作方法。2.进行爬升玩具的创作和设计。

3.感受爬升玩具的创作、设计、制作的乐趣，提高学生选择材料、色彩进行创作和动手制作的能力。【教学重点】

学习爬升玩具的创作、设计。【教学难点】

作品的新颖，制作的精美、实用。

爱心

用心

专心 2

【教具】

课本、爬升玩具、硬纸筒、塑料瓶、彩色纸和剪子、胶水 【学具】

课本、硬纸筒、塑料瓶、彩色纸和剪子、胶水 【课时】 第二课时 【教学过程】

一、组织教学：

检查学生学具的准备情况。稳定学生情绪。

二、导入：

复习上节课学习的内容，让学生将自己准备的东西准备好。

出示有关玩具，提高学生欣赏能力，让学生观察这些作品的特点，增进学生的创新意识。引出课题

三、作品创作：

教师引导学生将上一课时未完成的作品进一步补充完整。

学生进行创作时可能会出现想象不够丰富，作品容易出现雷同的情况，教师在教学过程中多设置一个训练环节，以竞赛的方式，就某一创新鼓励学生比赛说出设计内容。教师巡回辅导。

四、作品展示：

1．全班进行飞行旅程的游戏，请同学展示作品的同时介绍设计构思。

五、课后延伸：

教师引导学生收拾整理工具、清洁课桌及卫生。引导学生小结本课学习情况。

你在本课学到了什么？学得怎样？还想学些什么？ 【作业设计】

继续补充完善爬升玩具作品。

爱心

用心

专心 3

爬升玩具美术教学设计 篇2

下面就我利用信息技术教学《中国民间玩具》谈谈我的浅显探析:

一、利用信息技术走近民间玩具, 激发创作兴趣

爱因斯坦说:“兴趣是最好的老师。”传统的美术教学一般都是老师不厌其烦地教, 学生枯燥无味地学, 导致学生的学习效率低下, 甚至产生厌学心理。而运用多媒体软件等进行范画制作和作业的完成, 不仅有利于激发学生的学习兴趣, 还可使学生获得最佳的学习效率。美术教学中运用电教手段对诱发学生的情趣、产生学习的兴趣有着很大的优势。还能够创设良好的教学情境, 加深学生的感观刺激, 牢牢地抓住学生的注意力, 激发他们的学习兴趣, 在教育教学活动中起到事半功倍的效果。如在《中国民间玩具》的教学上, 我通过多媒体灵活生动地切换图片影像:拨浪鼓、空竹、陀螺、布老虎、布公鸡、纸鱼、手风车……同学们目不转睛地欣赏着这些造型简单而夸张, 色彩鲜艳而逼真的图片, 发出啧啧的赞叹声。我趁机又转换民间玩具爱好者参与游戏比赛的录像片断, 学生们欣赏着有趣的抖空竹、踢毽子、打陀螺……看着这既激烈又有趣的民间玩具的活动场面, 美术课堂教学一下子进入轻松、愉悦的氛围。多媒体的运用把学生带到了民间玩具的活动情境之中, 为美术课堂创设了愉悦和谐的情境。从同学们专注、好奇的眼神里, 老师感觉到了他们跃跃欲试的心情。他们多么想也能就地取材, 利用泥土、碎布、木、草、面等随手可得的材料自己做一做、玩一玩啊!运用多媒体于美术教学中, 的确能有效地培养学生学习兴趣、充分调动学生的学习积极性。

夸美纽斯说过:“兴趣是创造一个欢乐和光明的教学环境的主要途径之一。”兴趣是引导学生进入知识殿堂的向导, 一旦学生对某事有兴趣, 心理上就会处在一种亢奋状态, 学习起来便感到其乐无穷。多媒体教学手段让学生走近了民间玩具, 由此产生了对民间玩具的喜爱之情。

二、利用多媒体自制民间玩具, 启迪创造性思维

俗话说:“生活是创作的源泉。”现代美术教育已经不再是单纯的技能技巧的教学, 而是把培养学生丰富的想象力、创造性思维作为主要目标, 置于技法之上。作为学习成果之一的美术作业, 也应体现学生的创造性, 鼓励求异, 反对千篇一律。可以说:没有想象力培训的美术课就不是真正意义上的美术课。而恰当使用多媒体, 能较好地达到这一要求。

美术教学中运用电教手段不但能诱发学生的情趣、产生学习的兴趣, 同时对增强想象力和思维能力有很大的优势, 并通过学生的尝试发展了求异思维。民间玩具虽作为博孩子们以欢乐之具, 但深层的意义则是寄托着人们对美好生活的向往、追求和对人生的希望, 在创作玩具的过程中需要有中华文化的渗透。然而现在的孩子离古老的民间玩具越来越远, 取而代之的是商场出售的价格高昂的塑料、电动类玩具, 失去了材料取之于生活自己动手动脑自制的玩具其纯朴、益智等特色。

作为基础教育的小学美术老师, 在当代的小学生中渗透一种回归本土文化、关注本土文化、发扬本土文化的意识是非常有必要的。如何启迪学生自己动脑动手自制民间玩具, 利用多媒体手段启发学生积极思维, 通过各种有效的教学方式使学生思维处于活跃状态, 进而培养学生思维的敏捷性、独立性、创造性?在《中国民间玩具》一课的教学中, 我反复地利用投影仪播放各种不同的民间玩具和其他资料, 使学生有“材”可用, 放开眼界, 提高创造欲。看到同学们把课前我布置搜集的可用于制作民间玩具的布料、绒线、易拉罐、鸡毛、蟹壳类等较为普通的废旧材料摆在桌上, 准备动手制作时, 我根据电脑上收集的图片, 结合他们带来的材料进行分析交流, 启发他们的创造性思维, 学生纷纷提出自己的创意:鸡毛可以制作成传统的鸡毛毽子或是吊饰品, 稻草可以编制小动物、草鞋, 布料可以做沙包袋等。在设计制作的过程中, 启发他们多种材料的结合法, 如稻草和布料可结合起来做成小人。对于稍难制作的玩具, 鼓励他们小组合作。一会儿, 一件件新奇的玩具展现在大家眼前, 有利用贝壳、田螺壳做成的项链、手链, 有利用稻草、布料做成的布娃娃, 有利用硬纸或塑料做成的手风车, 有利用蛋壳、沙子做成的不倒翁……然后对他们的作品进行了赞赏性点评, 并将它们的作品直接投影在银幕上, 由于扩大了可见度, 学生积极参与, 充满了好奇与喜悦。大家共同探究 , 一起完善 , 看看其制作是否合理, 寓意是否深刻。课堂上同学们兴致勃勃 , 感受到了制作民间玩具的乐趣。

三、运用信息技术创新民间玩具, 拓宽美术思路

网络时代能给我们提供大量的信息资源, 丰富我们的知识面, 开阔视野, 节省时间。教师如何由以前的主导者变为组织者、指导者、鼓励者?我鼓励学生与家长一道共同收集民间美术实物、图片、视频等资料, 并联系实际引入民间美术有关的玩具, 为学生发现美、了解美、欣赏美、评价美以及表现自我提供一种充满乐趣和信心的途径。在激发学生创作欲望的同时, 也提供了创作的依据。电教手段延伸了课堂、课后, 同学们上网搜索不同类型的民间玩具, 在爸爸妈妈的指导下, 他们制作了很多可爱的民间玩具, 并有创作发挥, 如:不同颜色的绣球, 用竹篾及纸张制作的各种形态的风筝, 还有用稻田里的泥巴捏成的小泥人……同学们课余自己动手制作的民间玩具, 别有一番情趣。

在各种玩具充斥市场的今天, 孩子们通过信息技术等渠道了解和自己动手制作中国民间玩具, 更激发了他们对民间艺术的兴趣 , 也培养了孩子们对祖国传统艺术的传承与创新精神。

爬升玩具美术教学设计 篇3

教学重点：

感受民间玩具造型、色彩特点与审美情趣。

教学难点：

能用语言描述民间玩具的艺术特色。

教学过程：

1、活动一：教师问：“你发现什么？它们叫什么？”

欣赏课本提供的玩具图例，找出它们的名字。

2、活动二：它们是用什么材料做的`？

A、教师提供“泥、纸、竹、皮、面、木、布”等多种材料，让学生进行猜想与假设。

B、感受民间艺人心灵手巧及民间艺术特色。

3、活动三：它们“想说”些什么？

A、让学生讨论民间玩具的产地、寓意及有关的故事。

4、活动四：民间玩具美在哪里？为什么？

A、提供描述的感觉词语（如：材料美、装饰美、对称美、质朴、饱满、逼真），玩具中哪些地方使你产生这种感觉？

B、民间玩具造型、色彩特点与审美情趣。

5、活动五：学习民间艺人，可利用身边的废旧材料制作小玩具或描绘自己感兴趣的民间玩具。

6、布置作业。

A、尝试收集大家画的民间玩具。

B、布置师生共同收集民间小玩具集市。

教学后记：基本能从儿童的角度去欣赏书中的作品。

爬升玩具美术教学设计 篇4

活动目标：

1.学习制作蛋壳玩具的方法，引导幼儿设计制作出富有特色的蛋壳玩具，提高幼儿的想象能力，培养幼儿的创造性思维。

2.让幼儿利用各种材料进行涂、粘、剪、画、贴，培养幼儿的动手操作能力。

活动准备：

1.蛋壳若干、彩笔、剪刀、碎花布、双面胶、胶水、海绵纸、皱纹纸、即时贴、各色卡纸、毛线等。

2.教师自制蛋壳玩具若干

3.课件

活动过程：

一、创设情境、激趣导入

“今天，小猴请我们到他家做客，并有好消息告诉我们，我们马上出发吧!”在《我是汽车小司机》背景音乐下，师幼做驾车的动作进活动室。

“咦，小朋友们快来看呀，老师捡到了一个蛋壳。”(事先在活动室地上放好一个蛋壳)

“你们知道这个小小的蛋壳可以做成什么吗?”(幼儿畅所欲言，发散幼儿的思维)

“小朋友们想法可真多，下面听老师给你们讲一个关于蛋壳的故事。”

幼儿通过听故事，知道虽然蛋壳对于小动物们有不同的用处，但他们最终都丢弃了蛋壳，只有小猴子感到蛋壳是有用的。小猴“废物利用”，把蛋壳做成了漂亮的玩具。通过这个短小的故事，教育幼儿，只要肯动脑筋，我们认为没有用的东西，同样会变成美好的东西，萌发幼儿的环保与变废为宝的意识，同时，调动幼儿的兴趣，引出主题。

在《我是汽车小司机》音乐下，教师带领幼儿进入小猴的“环保小屋”中参观小猴举行的的蛋壳玩具展览会。

注：以“去小猴家做客----参观小猴的蛋壳玩具展览会----参加展览会”的故事情节贯穿活动的始终。

二、参观蛋壳玩具

1.参观各种各样的蛋壳玩具

“小猴家到了，让我们一起来看一下小猴的蛋壳玩具吧!”为了便于幼儿观看，把玩具放在一个高桌子上，给幼儿充分的视觉空间，围着桌子自由观看。在幼儿观看的过程中，教师可指导幼儿从玩具的外形特征、所用材料等方面观察。

2.幼儿自由讨论

教师提出问题，幼儿带着问题，再次观察蛋壳玩具，并展开讨论，通过自由讨论，加深了对蛋壳玩具的认识。如：

“小猴用蛋壳都做成了哪些蛋壳玩具?”(彩蛋、各种动物和娃娃)

“做这些蛋壳玩具都用到了哪些材料?”引导幼儿可从蛋壳玩具的头发、五官、衣服、身体等方面分别说一下用到了什么材料。

“这些蛋壳玩具是用什么方法做出来的?”

总结：彩绘，直接用色彩在蛋壳表面画上美丽的图案，做成可爱的彩蛋。也可以用剪贴、添加的方法做各种各样的动物和娃娃。

三、探究制作过程

“你们猜猜蛋壳玩具是怎么做出来的?如果给你一个蛋壳，你会做成什么，你该怎样做呢?”

幼儿讨论、回答(引导幼儿可按一定的顺序做。比如先给蛋壳玩具画上五官，再给玩具做上身体，然后分别装饰头部和身体。教师要充分调动幼儿的积极性，发挥幼儿的想像力，不限制幼儿的思维，让幼儿有充分发挥的空间，用不同的方法装饰，力求做出更具特色的蛋壳玩具。)。

四、制作蛋壳玩具

“小猴告诉我们的好消息就是：我们小朋友做出的蛋壳玩具也可以参加他的展览会呢，我们赶快来制作一个蛋壳玩具吧。”激起幼儿的制作欲望，放一段轻音乐，幼儿在轻音乐下完成自己的制作。教师巡回指导，帮助个别幼儿在制作过程中遇到的困难，运用儿童化的语言指导幼儿，给幼儿以相互启发。

五、展览、评价作品

1.让几个幼儿介绍他的蛋壳玩具是怎么制作的? “你做了一个什么样的蛋壳玩具?用到了什么材料?你是怎么做出来的?”

2.你喜欢哪个玩具，为什么?

3.对创造性强的作品给予表扬，让幼儿欣赏，对一般的作品找出其优点，鼓励幼儿的进步

六、拓展活动，进行环保教育

1.观看课件

大屏幕放映蛋壳的.其他用途：蛋壳贴画、蛋壳雕刻、蛋壳艺术等图片，以拓宽幼儿的知识面，增长幼儿见识。

2.环保教育

“今天，我们知道蛋壳可做出这么多漂亮的作品，有这么多的用处，回家后，可告诉爸爸妈妈，吃剩下的蛋壳不要扔掉，不但做出漂亮的玩具，而且可减少很多垃圾，这样为环保还做了一份贡献呢。这是不是一件很美的事情呀?好，从现在开始，让我们人人都做一个环保小卫士吧!”

最后在背景音乐《我是汽车小司机》中，师幼离开活动室，结束活动。

活动反思：

《蛋壳玩具》这节美工活动以故事情境(去小猴家做客----参观小猴的蛋壳玩具展览会----参加展览会)贯穿始终，让幼儿在轻松愉快的氛围下完成自己的制作活动。幼儿用教师所提供的材料进行画、剪、贴、粘，制做出漂亮的蛋壳玩具，发展了幼儿对美的表现力、想像力和创造力，同时体验到了变废为宝的乐趣，萌发了幼儿的环保意识。

结冰对爬升性能的影响研究 篇5

飞机在结冰气象条件下飞行时, 飞机迎风部件表面结冰现象比较常见。结冰改变了飞机气动外形, 从而改变了气动力, 影响到了飞机的稳定性与操纵性, 如果不重视飞机结冰问题, 就有可能会对飞行产生严重的不利影响, 甚至可能产生导致机毁人亡。

国外很早就注意到飞机结冰问题, 对飞机结冰进行了大量的试验研究和数值计算分析, 形成了很多成熟的数值分析软件和完备的冰风洞试验条件, 国内对飞机结冰的研究起步比较晚, 且多数为对机翼等部件结冰机理的研究, 很少能定量地分析结冰对飞行性能的影响, 综上所述, 很有必要进行结冰对飞机飞行性能的影响研究, 为将来设计考虑结冰情况提供参考。

结冰并不改变飞机的运动数学模型。本文通过将飞机作为一个质点进行受力分析, 建立飞机运动的动力学方程。结冰对飞机气动数据的影响量, 通过其他机型试验的结果给出, 在此基础上, 采用数值计算的方法分析了结冰对飞机飞行性能的影响并给出了相应结论。

1 计算公式及方法

1.1 动力学方程

飞行性能计算中, 将飞机视为一个质点, 飞机在垂直平面内无侧滑时受力情况如图1所示, 按照航迹坐标系建立飞行的运动方程, 将力分解到沿速度方向和垂直于速度方向, 利用牛顿第二定律可得出动力学方程如下:

式中:α机身迎角;V飞行速度;T发动机可用推力;D阻力;L升力;φT推力作用线与机身轴线夹角;θ航迹角;G飞机重量;t时间;g重力加速度。

1.2 最大爬升率爬升

根据爬升率定义可求的爬升率:

联立 (2) 式、 (3) 式将θ消去化简可得爬升率计算公式, 见式 (4) :

飞机以最大爬升率方式爬升时, θ和V变化较小, 可以近似认为dθ/dt=0, d V/d H=0。

2 算例

通常情况下, 结冰导致飞机的气动特性变差, 从而影响到飞机的飞行性能。冰风洞和飞行试验的结果表明, 大多数情况下, 结冰后飞机的失速速度约增大5%~10%, 全机的升力系数约减小5%~15%, 阻力系数约增大15%~50%。

在本文给出的算例中, 假定升力系数减少10%, 阻力系数增大32.5%, 可用推力下降5%。计算结果见表1。

3 计算结果分析及讨论

由于结冰导致剩余功率减少, 使得飞机的爬升率、爬升角、爬升梯度都有所减小, 且减小的比例与剩余功率减小的比例基本一致, 由表1可见, 爬升率减少量为30%~60%。

4 结论

结冰改变了飞机气动外形, 使飞机爬升性能变差, 最大爬升率减少量为30%~60%;

因此, 在民机的设计中, 应着力开展结冰对飞机气动特性的影响和防除冰装置的研制的研究设计工作, 或者在飞机运营过程中, 应根据气象雷达合理的避开结冰区, 使飞机设计更合理、飞行更经济、更安全。

摘要：本文结合XX型号的数据, 通过受力分析, 建立飞机运动方程, 计算分析了结冰与非结冰时飞机的最大爬升率。定量给出了结冰对飞机爬升性能的影响, 可以为今后飞机方案设计考虑结冰的情况提供一定参考。

关键词：结冰,爬升性能,飞行力学

参考文献

[1]金长江, 范立钦, 周士林.飞机飞行性能计算[M].北京:国防工业出版社, 1983:58-120.

[2]张锡金, 主编.飞机设计手册:第六册[M].北京:航空工业出版社, 2006.

[3]顾诵芬, 解思适.飞机总体设计[M].北京:北京航空航天大学出版社, 2001:89-200.

爬升玩具美术教学设计 篇6

关键词：安全约束机组组合,电力生产调度,爬升约束,Lagrangian松弛,Benders分解可行性定理

The security-constrained unit commitment (SCUC) now remains to be one of the most important daily functions for independent system operators (ISOs) to clear the electric power market and for generation companies (GENCOs) to analyze generation costs and determine bidding strategies[1—3].The objective of the SCUC problem is to minimize the total bid cost in an electric power market or to minimize the total operating cost in the traditional power systems while satisfying system-wide constraints including system load demand, system spinning reserve and related security constraints, and individual unit constraints such as minimum/maximumgenerationlevel, minimum up/down time, ramping rate constraints and so on.

Since the SCUC problem is an NP-hard mixed integer-programming problem, it is extremely difficult to obtain an accurate optimal solution within acceptable time since the time spent on obtaining the optimal solution increases exponentially.The security constraints such as transmission are not considered in generation scheduling problems in many researches[5—8, 10, 12]due to mathematical difficulties[9].With the enhancement of computational capability and storage capacity of computers, the unit commitment (UC) problem and the security-constrained unit commitment (SCUC) problem can be considered in one solution process[9].Lagrangian Relaxation (LR) is one of the most successful methods for obtaining near optimal solutions to problems of separable structures[4].When Lagrangian Relaxation method is applied to solve power generation scheduling problems[5—16], the system constraints, e.g., load demand balance, spinning reserve requirement and or security constraint such as transmission constraints are all relaxed and added to the Lagrangian function by introducing several dual variables called Lagrange multipliers.In order to improve the convergence of the Lagrangian Relaxation method, some penalty term associated with some system constraints are also added to the Lagrangian function, the augmented Lagrangian function is then formulated[9,10,13].The procedure for solving SCUC problems within Lagrangian or augmented Lagrangian framework is in general divided into two stages[6,9,10,13,14]:one is to solve the dual problem of the primal SCUC problem to obtain a dual security-constrained unit commitment (i.e., SCUC, which maybe infeasible at some scheduling hours) and then some heuristic strategy[5—8, 10—13]is used to adjust them to a feasible one at each scheduling hour;the other is to allocate power generation economically among all generating units at each hour (economic dispatch, i.e., ED) .An integer programming method, which is called the direct method, is applied to obtain a feasible SCUC and near-optimal power generation schedule over 24 hours per day in literature[14]by adjusting some states of some units at some hours such that the increased opportunity cost attains minimization and all system constraints can be satisfied.

It is clear that the core to develop an effective method for solving SCUC problem within (augmented) Lagrangian relaxation framework is how to obtain a feasible SCUC.First of all, conditions for checking the feasibility of a SCUC at each hour must be known before solving economic dispatch problem.An integer programming problem can be formulated based on the proposed feasibility conditions, in which the decision variables are the unit states.A method such as cutting plane method is then applied to solve this integer programming problem.Thus, conditions will be very important for judging the feasibility of a SCUC and for constructing an integer programming problem to obtain a near optimal feasible SCUC by adjustment of some states of some units in a dual SCUC such that the“opportunity cost”[14]attains minimization.Better conditions can reduce the searching region of the algorithm and computation burden.Such conditions is derived and rigorously proved in our previous paper[17,18].The necessary and sufficient condition for checking the feasibility of a UC considering the economic re-dispatch of units with ramping limits is proposed in paper[17].However, the security constraint such as transmission is not taken into account.On the other hand, although the transmission constraints are included in system constraints in the literature[18], the economic re-dispatch of power generation of units with ramping constraints is not consider, i.e., the power generation of the units with ramping constraints remains the value in the dual solution.Serious consequences ensue:a feasible SCUC may be judged wrongly as an infeasible one and a better feasible scheduled generation dispatch can not be obtained.In order to overcome the two serious issues, some conditions for checking the feasibility of a SCUC considering the economic re-dispatch of units with ramping constraints are proposed in this paper.In particular, a very efficient numerical computational method for judging the feasibility of a SCUC is also proposed based on Bender decomposition feasibility theorem in previous paper[21].Thus, a very efficient method for solving SCUC problem with ramping constraints can be developed efficiently based on these conditions.

Numerical tests showthe efficiency and effectiveness of the conditions for judging the feasibility of a SCUC at each hour and for constructing a feasible SCUC.

1 Formulation of SCUC Problems

For the convenience of presentation, some notations are defined as follows.

T:commitment horizon in hours;

I:number of units with the index i denoting the i-th unit;

Pi (t) :power generation by unit i at time t;

ui (t) :binary variable:1 if unit i is turned on or kept on during the time period t, else 0;

xi (t) :the number of time periods that unit i has been up or down;

:the minimum number of time periods for which unit i must be up;

:the minimum number of time periods for which unit i must be down;

Ci[Pi (t) ]:fuel cost of producing power Pi (t) for thermal unit i;

Si[xi (t) , ui (t) ]:startup/shutdown cost for unit i;

D (t) :total demand of the whole power system during time period t;

Pr (t) :the spinning reserve requirement during time period t;

ri (t) :is the spinning reserve contribution to the system during time period t, is the maximum spinning reserve requirement;

:the maximum generation of unit i at generating time t, if unit ihas no ramp rate limit;

:the minimum generation of unit i at generating time t, if unit i has no ramp rate limit;

Δi:the maximum ramp rate of unit i;

:the (real) power flow limit on pransmission line l;

Tl:the matrix relating generator output to power flow on transmission line l.

The objective of the unit commitment problem is to minimize the total operating cost as the following mixed integer-programming problem:

subject to.

1.1 System level constraints

1.1.1 System demand constraint

where Dk (t) is the demand at bus k.

1.1.2 Spinning reserve constraint

where is the maximum spinning reserve requirement.

1.1.3 Transmission security constraints:

1.2 Individual unit constraints

1.2.1 The minimum up/down time constraint

1.2.2 The relation between the unit state and unit up/down decision

1.2.3 Generation constraint

1.2.4 Ramp rate constraints

1.2.5 Minimum generation level limit at the first/last generating hour t

2 Conditions for Checking the Feasibility of a SCUC

The objective of developing necessary or sufficient conditions for a SCUC to be feasible is to construct a feasible SCUC schedule.If a SCUC is infeasible, certain criteria such as the minimization rule of“opportunist cost”[14]can be used to adjust it to a feasible one.So the condition for determining the feasibility of a SCUC is crucial.Similar conditions were presented in papers[18]without considering the economic redispatch of power generation of units with ramping constraints in solving the economic dispatch problem.Some new conditions for checking the feasibility of a SCUC with ramping constraints in this paper are proposed which extend the results in our previous works[17,18].

For presentation clarity, the SCUC problem is simplified as the following mixed integer-programming problem:

where y is an n-dimensional continuous variable and z is an m-dimensional discrete variable, and g (y, z) is a vector function.The discrete variable, z=z0∈Z, is said to be quasi-feasible if the continuous variable can be found such that y=y0∈Y and g (y0, z0) ≥0.

Feasibility Theorem based on Benderdecomposition[19,20]:For a mixed integer-programming problem,

with fixed z=z0∈Z, the necessary and sufficient condition for the point z0to be quasi-feasible with respect to the primal problem equa. (10) is that the following inequalities satisfy for arbitrary convex combination vectorσsuch that

where thecomponents ofσare nonnegative and the sum of them is 1, otherwise z0is said to be quasi-infeasible.

In our previous papers[18], it is assumed that units with ramping constraints are not involved in final economic dispatch of power generation at each hour, i.e., the power generation of such unit remains the value in dual solution.Thus, serious outcome may arise, i.e., a quasi-feasible SCUC may be judged wrongly as an infeasible one and a best scheduling generation dispatch may not be obtained.Therefore, the quasi-feasible conditions of a SCUC associated with the units with ramping constraints must be taken into con-sideration.

For the analysis, the units are classified into three categories at timet:E2tis the set of units each of which has minimum generation level constraint at the first/last generating hour but without ramping constraint, and which is being at the first/last generating state at hour t;E3tis the set of units with ramping constraints, each of which is being at generating state at hour t;E1tis the set of the rest units which is at the generating state at hour t.The set E1tand E3tis further classified into two types according to the sign ofΓlias follows, respectively:

The units in set, E3t, have to be categorized into four types:E03t, E13t, E23tand E33t, where concretely

where first_last (i) =1 denotes that unit i has the minimum generation level limit at the first or last generating hour, otherwise first_last (i) =0.

Some propositions or theorems which describe the necessary and or sufficient conditions for a SCUC at hour t to be quasi-feasible are presented and proved in detail as follows, respectively.

Proposition 1[17](1): (1) A SCUC at hour t is loadreserve-feasible, i.e., system demand constraints (2) and spinning reserve requirements (3) hold, if and only if the following all inequalities hold

Proposition 2:The necessary and sufficient condition for a SCUC at hour t to be single-transmissionline-feasible, i.e., the transmission constraint (4) holds for a single transmission line l∈{1, 2, …, L}, is that the inequalities (22) and (23) are all satisfied:

where El+1t, El-1t, El+3tand El-3tare defined in equas (13) and (14) , l=1, …, L.

Proof.According to Feasibility Theorem (13) , we only show that the following (24) is nonnegative for an arbitrary convex combination vector (λlt, μlt) , λlt≥0, μlt≥0, l=1, …, L andλlt+μlt=1.In fact, Ll (P, μlt) =

We will prove this proposition for the two cases as follows.

Case 1:For 0≤μlt≤0.5, we have

Therefore, the necessary and sufficient condition for the equation (25) is nonnegative for the case 1 is that the inequality equa. (26) holds forμlt=0 and forμlt=0.5, that is

Case 2:For 0.5≤μlt≤1, we have

Therefore, the necessary and sufficient condition for the equation (25) to be nonnegative for the case 2is that the inequality (27) is nonnegative forμlt=0.5 and forμlt=1, that is

This proves that (25) is nonnegative for an arbitrary value of the convex coefficient vector, (λlt, μlt) , λlt≥0, μlt, l=1, …, L, andλlt+μlt=1.Based on Feasibility Theorem (12) , transmission constraint (4) is satisfied.Q.E.D.

Based on the above Proposition 1 and Proposition2, the necessary condition for a SCUC at hour t to be quasi-feasible is given as the following theorem 1.

Theorem 1 The necessary condition for a SCUC at hour t to be quasi-feasible is that the following inequalities are all satisfied,

where El+1t, El-1t, El+3tand El-3tare defined in (13) and (14) ;l=1, …, L.

The individual unit constraints from (5) ~ (9) are naturally satisfied when solving the subproblems in the course of dual iteration.

Proposition 1 is in fact a necessary and sufficient condition for a traditional unit commitment (UC) at hour t to be quasi-feasible without transmission constraints.

Since theorem 1 determining a region in which a quasi-feasible SCUC lies is only a necessary condition, a necessary and sufficient condition must be developed such that a feasible power generation scheduling can be found in this region.Theorems 2 and 3 below will settle this issue.

Theorem 2The necessary and sufficient condition for a SCUC to be quasi-feasible at hour t is that the following linear programming problem is feasible,

where

Proof.Let

If all expressions above are substituted into system constraints (2) ~ (4) , then all the system constraints are changed to (35) ~ (38) , and inequalities (39) ~ (41) hold naturally.

The optimal objective is to maximizing the total spinning reserve, that is

or

which is equivalent to equa. (34) .

Sufficiency.Assume that the linear programming problem (34) subject to constraints (35) ~ (41) is feasible, then it must have an optimal solution (ΔPi1, ΔPi2;ΔPj;ΔPk;ΔPm1, ΔPm2) , where i∈E1t, j∈E13t, k∈E23t, m∈E33t.We will prove that individual unit constraint holds

for each i∈E1t∪E13t∪E23t∪E33t.

and , ΔPi2'<ΔPi2.It is-very clear that the constructed new solution is also a feasible solution of smaller objective function value than the original solution.This is a contradiction.Similar result can be obtained for i∈E33t.

The discussion above can help to prove that individual reserve constraints (52) hold for all i∈E1t∪E13t∪E23t∪E33tat the optimal solution.Therefore, all the system constraints and the individual reserve constraints are satisfied.

Necessity.Assume that a SCUC is quasi-feasible at hour t, i.e., there must have a dispatch of power generation level Pi (t) ;i∈E1t∪E13t∪E23t∪E33t, which can be expressed as (46) ~ (49) , to be a feasible solution of the linear programming problem (34) subject to (35~41) .Q.E.D.

The following Theorem 2 gives a very efficient method[21]for judging the feasibility of a SCUC at hourt.

Theorem 3[21]A SCUC at hour t is quasi-feasible if and only if the optimal value of the following nonlinear programming is nonnegative:

where

and

These three Theorems above are very important for determining the quasi-feasibility of a SCUC and constructing a feasible power generation schedule, of which, Theorem 1 can be used to adjust an quasi-infeasible SCUC to a quasi-feasible one, Theorem 2 is dedicated to check the quasi-feasibility of a SCUC by solving a small-size linear programming and Theorem 3is to judge the feasibility of a SCUC by solving a simple dual problem.Numerical test[21]shows that theorem 3is very efficient since there only needs 2 seconds or so to judge the quasi-feasibility of a SCUC for 24 hours in a power system with 31 buses, 16 units.On the other hand, the Proposition 1 is in fact a necessary and sufficient condition for a UC without consideration of the security constraints[17].However, Theorem 1 is only a necessary condition for a SCUC at hour t to be quasifeasible.In order to construct a feasible SCUC schedule, the dual SCUC must be adjusted such that all the inequalities in Theorem 1 hold.Unfortunately, we cannot determine the quasi-feasibility of a dual SCUC at hour t even though all the inequalities in Theorem 1hold.Though does so, Theorem 1 is still a good necessary condition for adjusting a quasi-infeasible SCUC to a quasi-feasible one.Therefore, Theorem 2 or Theorem 3 has to be used to determine the quasi-feasibility of a SCUC.

Our previous paper[18]proposed similar conditions without considering theeconomic re-dispatch of power of units with ramping constraints.If done in such a way, the probability of obtaining a quasi-feasible SCUC is lowered and a better power generation schedule may not be constructed.

3 Numerical Testing

Example 1 An example is presented in this section to demonstrate the efficiency of the method for solving a SCUC problem by adjusting the infeasible dual solution at each hour such that the new condition, i.e., Proposition 1 and proposition 2 are satisfied and the theorem 2 or 3 hold.This example is from[14]and the system parameters are summarized in tables 1~5.The transmission line parameters are given in table 1.The percentage of system load drawn by each load bus is given in table 2 with Dk (t) =D (t) σk, where k is the index of load bus and the system loads are listed in table 3.The system spinning reserve is defined as10%of the system load at each hour.The basic unit parameters and the initial states of units are shown in table 4 of which units 4~7 and 9 having ramping constraints, units 1 and 4 having minimum generation constraints at the first/last up hour:the corresponding value of ramp rate of units are 200 MW, 180 MW, 180MW, 200 MW, 200 MW, respectively.The power grid is illustrated in fig.1.

The fuel cost curve of uniti and the start cost curve are defined by

and

respectively.The parametersai1, ai2, bi1, bi2are given in table 5.

The dual SCUCs at iterations 17, 21, 25~45, 47~50 are quasi-feasible judged by Theorems 2 or 3.The dual SCUCs at iterations 3 and 7 are judged to be infeasible similarly, the first of which isinfeasible at hours 6, 23~24, and the second at hour 22, which can be adjusted to quasi-feasible ones by using the new method based on the conditions in this paper, while all these dual SCUCs are all wrongly judged to be“quasiinfeasible”if using the old conditions in paper[18].The best dual SCUC, which is quasi-feasible and is listed in table 6 (‘1’stands for‘up’state, ‘0’for‘down’state) , occurs at 26thiteration.The quality of the obtained SCUC schedule in term of duality gap is given in table 7.

Example 2 This example is the same as the example1 except that the ramping constraint of unit 9 is canceled.The dual SCUC (table 8) only at 4thiteration among 50 iterations, which is quasi-infeasible judged by old conditions similar to theorem 2 in paper[18]at hours:1~7, 9, 14~17, 19~24, can be adjusted to a quasi-feasible SCUC (See Table 9) .However, the dual SCUCs at 37 iterations are quasi-feasible or infeasible judged by theorems 2 or 3, 12 of which are quasiinfeasible at one or more hours and can all be adjusted to quasi-feasible ones.The best dual SCUC, which is quasi-feasible and is listed in table 10, occurs at 34th iteration.The comparison of quality of the obtained SCUC power generation schedule in term of duality gap is given in Table 11.

Testing examples show that the new conditions for determining the feasibility of a SCUC are efficient and effective.

4 Conclusion

In this paper, the necessary condition, i.e., proposition 1 (which is in fact a necessary andsufficient condition for judging the quasi-feasibility of a UC without considering security constraints) or the necessary conditions, i.e., Theorem 1 and a necessary and sufficient condition (Theorem 2 or Theorem 3) for checking the quasi-feasibility of a SCUC considering the economic re-dispatch of power generation of units with ramping constraints are proposed.The conditions including proposition 1~2, theorem 1 and 3 are rigorously proved based on the Bender’s decomposition feasibility theorem[19,20], which are crucial for judging the quasi-feasibility of a SCUC and for constructing a feasible SCUC power generation schedule.Since Theorem 1 is only a necessary condition, Theorem 2 or theorem 3 should be used to find a quasi-feasible SCUC.