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Application of Jacobian in Engineering. What is Jacobian ? A determinant which is defined for a finite number of functions of the same number of variables and in which each row consists of the first partial derivatives of the same function with respect to each of the variables ..

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Jacobian :. If u v and are functions of two independent variables x y and having first order partial derivatives, then the Jacobian determinant or the Jacobian of u and v with respect to x y and is denoted by ∂(x , y)/​∂(u ,v)= |∂x/∂u ​∂x/∂ v | | ∂y/​∂u ∂y/∂ u |.

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Application of Jacobian in Engineering. In the field of control engineering the use of Jacobian matrices allows the local (approximate) linearisation of non-linear systems around a given equilibrium point and so allows the use of linear systems techniques, such as the calculation of eigenvalues (and thus allows an indication of the type of the equilibrium point)..

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Jacobians are also used in the estimation of the internal states of non-linear systems in the construction of the extended Kalman filter, and also if the extended Kalman filter is to be used to provide joint state and parameter estimates for a linear system (since this is a non-linear system analysis due to the products of what are then effectively inputs and outputs of the system)..

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The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral..

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Despite a tremendous development in optimal power flow (OPF), owing to the obvious complexity, non-linearity and unwieldy size of the large interconnected power systems, several problems remain unanswered in the existing methods of OPF. Seizing specific topics for maximizing voltage stability margin and its implementation, a detailed literature survey discussing the existing methods of solution and their drawbacks are presented in this research ...

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The phenomenon of voltage collapse in power systems, methods to investigate voltage collapse, and methods related to voltage stability are briefly surveyed. Finally, the study presents a statistical method for analyzing a power system through eigenvalue analysis in relation to the singular values of the load flow Jacobian . Future study may focus on changes in theories in conjunction with large power systems.