F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
The mode shape, as indicated by the right eigenvector, gives the relative phase of each state in a particular mode. However, it does not give the influence of each state on the mode. We would like to be able to obtain the influence of states on modes because then we will know which states (machines) to control in order to increase damping of a certain problem mode. Let's define a new state variable (" xi ") as follows: x q x q T k k T k k & & = ⇒ = ξ ξ Notice that ξ k is a scalar since the transpose of the left eigenvector is a 1×n and the state vector is an n×1. Thus, ξ k is a combination of all of the states, but the manner in which all of the other states is combined is through the left eigenvector elements of the k th mode. An important attribute of ξ k , and the reason why it is of great interest here, is that it is a state which is associated with the k th mode and no other mode. We can prove this as follows. Start with the system state equations: x A x = & Pre-multiply both sides by T k q. x A q x q T k T k = & (P-1) Recall that the left eigenvector is defined as T k k T k q A q λ = (P-2) Notice that the left-hand-side of eq. (P-2), is on the right-hand-side of eq. (P-1). Substituting the right-hand-side of eq. (P-2) into the right-hand-side of eq. (P-1), we obtain:
模态振型，如右特征向量所示，给出了特定模态中每个状态的相对相位。然而，它没有给出每种状态对模态的影响。我们希望能够获得状态对模态的影响，因为这样我们就知道要控制哪些状态(机器)以增加某个问题模态的阻尼。让我们定义一个新的状态变量(“xi”)如下:x q x q T k k T k k & & =⇒= ξ ξ注意，ξ k是一个标量，因为左特征向量的转置是1×n，状态向量是n×1。因此，ξ k是所有状态的组合，但是所有其他状态的组合方式是通过第k个模态的左特征向量元素。ξ k的一个重要属性，也是我们对它非常感兴趣的原因，是它是一个与第k阶模态无关的状态。我们可以这样证明。从系统状态方程开始:x A x = &两边先乘以tkq x A q x q T k T k k = & (P-1)回想一下左特征向量被定义为tkk T k q A q λ = (P-2)注意到方程(P-2)的左边，在方程(P-1)的右边。将方程(P-2)的右侧代入方程(P-1)的右侧，得到:
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{"title":"Power System Outlines","authors":"Georgios Tzounas","doi":"10.1201/9780429325311-1","DOIUrl":"https://doi.org/10.1201/9780429325311-1","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127590553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
{"title":"Differential Equations with Singular Pencil","authors":"F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas","doi":"10.1201/9780429325311-6","DOIUrl":"https://doi.org/10.1201/9780429325311-6","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131388915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
{"title":"Differential Equations with Regular Pencil","authors":"F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas","doi":"10.1201/9780429325311-3","DOIUrl":"https://doi.org/10.1201/9780429325311-3","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123698162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-22DOI: 10.1201/9780429325311-11
F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
{"title":"Systems with Constant Delays","authors":"F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas","doi":"10.1201/9780429325311-11","DOIUrl":"https://doi.org/10.1201/9780429325311-11","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"306 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125763918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
{"title":"Mathematical Outlines","authors":"F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas","doi":"10.1201/9780429325311-2","DOIUrl":"https://doi.org/10.1201/9780429325311-2","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133604267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-22DOI: 10.1201/9780429325311-12
F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas
{"title":"Systems with Time-Varying Delays","authors":"F. Milano, I. Dassios, Muyang Liu, Georgios Tzounas","doi":"10.1201/9780429325311-12","DOIUrl":"https://doi.org/10.1201/9780429325311-12","url":null,"abstract":"","PeriodicalId":121538,"journal":{"name":"Eigenvalue Problems in Power Systems","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131302390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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