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# Mathematical Modeling of Energy Efficiency

## 1.

Mathematical Modeling of EnergyEfficiency

Lecture 1 – Introduction. Basic

notions. Graphs. Incidence

matrixes

Lecturer: Masheyeva R.U.

## 2.

Introduction. Basic notions. Graphs. Incidence matrixesFormalized topological methods for the analysis of

electrical circuits

These methods are based on the use of basic concepts of

topology and provide automatic generation of models of

electrical circuits.

The basis of electrical circuits are active two-terminal

networks, which have equivalent resistances and EMF, and

passive ones, which have only resistance, and the EMF is

zero.

Auto two-pole networks are sources of EMF and current,

batteries, generators, electric motors;

Passive two-pole networks are transformers, load

resistance lines;

Lecturer: Masheyeva R.U.

## 3.

Introduction. Basic notions. Graphs. Incidence matrixesThe connection point of two or more electrical circuits is

called a node or a node is the point of connection between two or

more branches.

Links between nodes are called branches.

A branch represents a single element such as a voltage

source or a resistor. In other words, a branch represents

any two-terminal element.

The branches form loops. A loop is any closed path in a circuit.

Lecturer: Masheyeva R.U.

## 4.

Introduction. Basic notions. Graphs. Incidence matrixesA node is usually indicated by a

dot in a circuit. If a short circuit

(a connecting wire) connects two

nodes, the two nodes constitute a

single node. The circuit in Figure 1

has three nodes a, b, and c.

Notice that the three points that

form node b are connected by

perfectly conducting wires and

therefore constitute a single point.

The same is true of the four

points forming node c. We

demonstrate that the circuit in

Fig. 1 has only three nodes by

redrawing the circuit in Fig. 2.

The two circuits in Figs. 1 and

2 are identical.

However, for the sake of

clarity, nodes b and c are

spread

out

with

perfect

conductors as in Fig. 1.

Lecturer: Masheyeva R.U.

## 5.

Introduction. Basic notions. Graphs. Incidence matrixesA loop is a closed path formed by starting at a node, passing

through a set of nodes, and returning to the starting node without

passing through any node more than once. A loop is said to be

independent if it contains at least one branch which is not a part of

any other independent loop. Independent loops or paths result in

independent sets of equations.

It is possible to form an independent set of loops where one of the

loops does not contain such a branch. In Fig. 2, abca with the 2Ω

resistor is independent. A second loop with the 3Ω resistor and the

current source is independent. The third loop could be the one with

the 2Ω resistor in parallel with the 3Ω resistor. This does form an

independent set of loops.

A network with b branches, n nodes, and I independent loops will

satisfy the fundamental theorem of network topology

b= I+n-1

Lecturer: Masheyeva R.U.

## 6.

Introduction. Basic notions. Graphs. Incidence matrixesThe method of connecting branches and nodes of an electrical circuit, that

is, a structural diagram of a circuit, is presented in the form of a directed

graph (Граф), the tops of which correspond to the nodes of the circuit, and

the edges correspond to its branches.

Sources of EMF, current, resistance do not show in this graphs but only

take into account the nodes and their circuits.

For each branch, its orientation (positive direction) is set, in accordance

with which the positive directions of the current and voltage of the branch

are taken. We will take the direction of the current in the branch to the

node as a positive direction. For loop currents, we take the clockwise

direction as positive. Under these conditions, any electrical circuit can be

represented in the form of a graph and P matrix, connections that uniquely

reflects the structural diagram of the graph.

Lecturer: Masheyeva R.U.

## 7.

Introduction. Basic notions. Graphs. Incidence matrixesAn electrical circuit graph is a conditional image of an electrical circuit

diagram, in which the branches of the circuit are represented by segments the branches of the graph, and the nodes by points are the nodes of the

graph. The nodes and branches of the graph correspond to the nodes and

branches of the electrical circuit. A schema subgraph is part of a schema

graph.

Electrical circuit’s (network) and it’s graph

Lecturer: Masheyeva R.U.

## 8.

Introduction. Basic notions. Graphs. Incidence matrixesTree of the graph relate connects all the

nodes of the graphs. It is not form a

closed loop. The edges of the graph that

are not part of the tree are called

chords. (1,3,4) –chords, (2,4,5) – tree of

the graph.

A directed graph can be expressed in a compact matrix form. The

branches are connected with each other with the help of nodes. A

directed branch connected to a node is called incidence. It

represents the orientation of the branches and the number of

branches incident to a node. The number of branches incident to a

node is called degree of node.

Lecturer: Masheyeva R.U.

## 9.

Introduction. Basic notions. Graphs. Incidence matrixesOrder of incidence matrix: If there are 'n’ nodes and 'b’ branches in a

network graph, then incidence matrix have 'n’ rows and 'b’ columns. So

order of the incidence matrix in nxb.

Reduced incidence matrix (A): When any one row is completely

deleted from the matrix then this is called reduced incidence

matrix. The order of this matrix is (n-1)xb. This reduction results from

mathematical manipulation.

Lecturer: Masheyeva R.U.

## 10.

Introduction. Basic notions. Graphs. Incidence matrixesConnection of this two matrix:

n – number of nodes

m – number of branches

L – number of independent loops

Current source matrix is stored relative to nodes:

Resistance matrix of the branches is quadratic

matrix, the numbers of row and columns is

equal to the number of branches :

Vector of the branches’ currents:

Vector of the branches’ EMF :

Lecturer: Masheyeva R.U.