Mathematical Modeling Of Electrical System

[Catherin Bergan]

Manyimportant engineering problems may be solved and the behaviour of many electrical systems may be understood by using mathematical modeling. Thus the electrical systems may often be described, with sufficient accuracy for engineering purposes, by a set of ideal lumped elements which represent essential electrical phenomena.

This requirement fits into the framework of the quasi-static field theory which implies that the rate of change of magnetic field normal to the region of the path on the subsystem energy boundary must be negligible. If it is necessary to do work on a positive charge (i.e., to apply a force in the direction of motion) as it moves from point A to point B, then point B is said to be at a higher potential than point A and the voltage uBA is considered positive.

Since it is often not known a priori which of two points will be at the higher potential or voltage, it is necessary to establish an algebraic reference convention for voltage. Usually, the assumed positive orientation of voltage is denoted by an empty-head arrow pointing in the direction of the from the point of a lower to the point of a higher potential.

where ta and tb are the beginning and end of the time interval during which power flows. If energy is delivered to an electrical system through more than one pair of terminals, the total energy supplied is the sum of the energies supplied at all the terminals.The law of conservation of energy, or the first law of thermodynamics, states that the energy (of all forms) which is delivered to a system must either be stored in the system or transferred out of the system.

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. The elemental equation for an ideal inductor is:

The control systems can be represented with a set of mathematical equations known as mathematical model. These models are useful for analysis and design of control systems. Analysis of control system means finding the output when we know the input and mathematical model. Design of control system means finding the mathematical model when we know the input and the output.

Consider the following electrical system as shown in the following figure. This circuit consists of resistor, inductor and capacitor. All these electrical elements are connected in series. The input voltage applied to this circuit is $v_i$ and the voltage across the capacitor is the output voltage $v_o$.

Transfer function model is an s-domain mathematical model of control systems. The Transfer function of a Linear Time Invariant (LTI) system is defined as the ratio of Laplace transform of output and Laplace transform of input by assuming all the initial conditions are zero.

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Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.

Abstract: This paper presents a mathematical modeling approach by which to solve the power flow and state estimation problems in electric power systems through a mathematical programming language (AMPL). The main purpose of this work is to show the advantages of representing these problems through mathematical optimization models in AMPL, which is a modeling language extensively used in a wide range of research applications. The proposed mathematical optimization models allow for dealing with particular issues in that they are not usually considered in the classical approach for power flow and state estimation, such as solving the power flow problem considering reactive power limits in generation buses, as well as the treatment of errors in state estimation analysis. Furthermore, the linearized mathematical optimization models for both problems at hand are also presented and discussed. Several tests were carried out to validate the proposed optimization models, evidencing the applicability of the proposed approach. Keywords: AMPL modeling language; power flow; state estimation; power systems

Ruiz Florez, Hugo A., Gloria P. Lpez, lvaro Jaramillo-Duque, Jess M. Lpez-Lezama, and Nicols Muoz-Galeano. 2022. "A Mathematical Modeling Approach for Power Flow and State Estimation Analysis in Electric Power Systems through AMPL" Electronics 11, no. 21: 3566.

While the previous page (System Elements) introduced the fundamental elements of thermal systems, as well as their mathematical models, no systems were discussed. This page discusses how the system elements can be included in larger systems, and how a system model can be developed. The actual solution of such models is discussedelsewhere.

To develop a mathematical model of a thermal system we use the concept of an energy balance. The energy balance equation simply states that at any given location, or node, in a system, the heat into that node is equal to the heat out of the node plus any heat that is stored (heat is stored as increased temperature in thermal capacitances).

Consider a situation in which we have an internal temperature, θi, and an ambient temperature, θa with two resistances between them. An example of such a situation is your body. There is a (nearly) constant internal temperature, there is a thermal resistance between your core and your skin (at θs), and there is a thermal resistance between the skin and ambient. We will call the resistance between the internal temperature and the skin temperature Ris, and the temperature between skin and ambient Rsa.

a) Draw a thermal model of the system showing all relevant quantities.

b) Draw an electrical equivalent

c) Develop a mathematical model (i.e., an energy balance).

d) Solve for the temperature of the skin if θi, =37C, θa =9C, Ris=0.75/W; for a patch of skin and Rsa= 2.25/W for that same patch.

Consider a building with a single room. The resistance of the walls between the room and the ambient is Rra, and the thermal capacitance of the room is Cr, the heat into the room is qi, the temperature of the room is θr, and the external temperature is a constant, θa.

Consider the room from the previous example. Repeat parts a, b, and c if the temperature outside is no longer constant but varies. Call the external temperature θe(t) (this will be the temperature relative to the ambient temperature). We will also change the name of the resistance of the walls to Rre to denote the fact that the external temperature is no longer the ambient temperature.

Consider a building that consists of two adjacent rooms, labeled 1 and 2. The resistance of the walls room 1 and ambient is R1a, between room 2 and ambient is R2a and between room 1 and room 2 is R12. The capacitance of rooms 1 and 2 are C1 and C2, with temperatures θ1 and θ2, respectively. A heater in in room 1 generates a heat qin. The temperaturexternal temperature is a constant, θa.

Consider a block of metal (capacitance=Cm, temperature=θm). It is placed in a well mixed tank (at termperature θt, with capacitance Ct). Fluid flows into the tank at temperature θin with mass flow rate Gin, and specific heat cp. The fluid flows out at the same rate There is a thermal resistance to between the metal block and the fluid of the tank, Rmt, and between the tank and the ambient Rta. Write an energy balance for this system.

To model this system with an electrical analog, we can represent the fluid flow as a voltage source at θin, with a resistance equal to 1/(Gincp). If you sum currents at the nodes θt and θm you can show that this circuit is equivalent to the thermal system above.

Thus far we have only developed the differential equations that represent a system. To solve the system, the model must be put into a more useful mathematical representation such as transfer function or state space. Details about developing the mathematical representation arehere.

Prerequisites: Math ACTE score of 24 or higher, or a grade of C or higher in MATH 1113, or MATH 1914, or MATH 1203, or consent of the instructor.

An introductory lecture/lab course to acquaint students with the fundamental techniques in the field of electrical engineering. Topics include technical aspects of electrical engineering including an introduction to computational techniques/software, basic introduction to computer-aided drafting (CAD), an introduction to programming, and basic circuit prototyping.

$25 per credit hour curriculum content fee.

Prerequisite: MATH 2924 with a grade of C or better.

An introduction to circuit theory and electrical devices. Topics include resistive circuits, independent and dependent sources; analysis methods, network theorems; RC and RL first order circuits, and RLC second order circuits.

$25 per credit hour curriculum content fee.

Co-requisite: ELEG 2134 or consent of instructor

Laboratory must be taken during the same semester as the lecture, ELEG 2134. A study of basic digital logic circuit design and implementation. Circuit schematic development utilizing computerized automated design tools. Computer modeling and simulation of digital systems. Emphasis will be placed on proper laboratory techniques, including data collection, data reduction, and report preparation.

Laboratory three hours. $40 laboratory fee.

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