**Network Model Formulation:**

Network Model Formulation – The load flow problem has, in fact, been already introduced in Chapter 5 with the help of a fundamental system, i.e. a two-bus problem *(see *Example 5.8). For a load flow study of a real life power system comprising a large number of buses, it is necessary to proceed systematically by first formulating the network model of the system.

A power system comprises several buses which are interconnected by means of transmission lines. Power is injected into a bus from generators, while the loads are tapped from it. Of course, there may be buses with only generators and no-loads, and there may be others with only loads and no generators. Further, VAR generators may also be connected to some buses. The surplus power at some of the buses is transported via transmission lines to buses deficient in power. Figure 6.1a shows the one-line diagram of a four-bus system with generators and loads at each bus. To arrive at the network model of a power system, it is sufficiently accurate to represent a short line by a series impedance and a long line by a nominal- π model (equivalent-π may be used for very long lines). Often, line resistance may be neglected with a small loss in accuracy but a great deal of saving in computation time.

For systematic analysis, it is convenient to regard loads as negative generators and lump together the generator and load powers at the buses. Thus at the ith bus, the net complex power injected into the bus is given by

where the complex power supplied by the generators is

and the complex power drawn by the loads is

The real and reactive powers injected into the ith bus are then

Figure 6.1b shows the network model of the sample power system prepared on the above lines. The equivalent power source at each bus is represented by a shaded circle. The equivalent power source at the ith bus injects current *J _{i}* into the bus. It may be observed that the structure of a power system is such that all the sources are always connected to a

*common ground node.*

The network model of Fig. 6.1b has been redrawn in Fig. 6.1c after lumping the shunt admittances at the buses. Besides the ground node, it has four other nodes (buses) at which the current from the sources is injected into the network. The line admittance between nodes *i *and *k *is depicted by y_{ik}* = *y_{ki}*. *Further, the mutual admittance between lines is assumed to be zero.

Applying Kirchhoff’s current law (KCL) at nodes 1, 2, 3 and 4, respectively, we get the following four equations:

Rearranging and writing in matrix form, we get

Equation (6.3) can be recognized to be of the standard form

Comparing Eqs. (6.3) and (6.4), we can write

Each admittance y_{ii}* (i *= 1, 2, 3, 4) is called the *self admittance (or driving **point admittance) *of node *i *and equals the algebraic sum of all the admittances terminating on the node. Each off-diagonal term y_{ik} *(i, k = *1, 2, 3, 4) is the *mutual admittance (transfer admittance) *between nodes *i *and *k *and equals the negative of the sum of all admittances connected directly between these nodes. Further, y_{ik}* =* y_{ki}

Using index notation, Eq. (6.4) can be written in compact form as

or, in matrix form

where Y_{BUS} denotes the matrix of bus admittance and is known as *bus **admittance matrix. *The dimension of the Y_{BUS} matrix is *(n *x *n) *where *n *is the number of buses. [The total number of nodes are *m = n + *1 including the ground (reference) node.]

As seen above,Y_{BUS} is a symmetric matrix, except when phase shifting transformers are involved , so that only n(n+1)/2* *terms are to be stored for an n-bus system. Furthermore, Y_{ik} = 0 if buses *i *and *k *are not connected (e.g. Y_{14} = 0). Since in a power network each bus is connected only to a few other buses (usually to two or three buses), the Y_{BUS} of a large network is very sparse, i.e. it has a large number of zero elements. Though this property is not evident in a small system like the sample system under consideration, in a system containing hundreds of buses, the sparsity may be as high as 90%. Tinney and associates [22] at Bonnevile Power Authority were the first to exploit the sparsity feature of Y_{BUS} in greatly reducing numerical computations in load flow studies and in minimizing the memory required as only non-zero terms need be stored.

Equation (6.6) can also be written in the form

for a network of four buses (four independent nodes)

Symmetric Y_{BUS} yields symmetric Z_{BUS} S. The diagonal elements of Z_{BUS} are called *driving point impedances *of the nodes, and the off-diagonal elements are called *transfer impedances *of the nodes. Z_{BUS} need not be obtained by inverting Y_{BUS}. While Y_{BUS} is a sparse matrix, Z_{BUS} is a full matrix. i.e., zero elements of Y_{BUS} become non-zero in the corresponding Z_{BUS} elements.

It is to be stressed here that Y_{BUS}/Z_{BUS} constitute models of the passive portions of the power network.

Bus admittance matrix is often used in solving load flow problem. It has gained widespread application owing to its simplicity of data preparation and the ease with which the bus admittance matrix can be formed and modified for network changes addition of lines, regulating transformers, etc. *(see *Examples 6.2 and 6.7). Of course, sparsity is one of its greatest advantages as it heavily reduces computer memory and time requirements. In contrast to this, the formation of a bus impedance matrix requires either matrix inversion^{*} or use of involved algorithms Furthermore, the impedance matrix is a full matrix.

*Note: *In the sample system of Fig. 6.1, the buses are numbered in an arbitrary manner, although in more sophisticated studies of large power systems, it has been shown that certain ordering of nodes produces faster convergence and solutions. Appendix C deals with the topics of *sparsity and optimal ordering.*