**Derivation of Transmission Loss Formula:**

Derivation of Transmission Loss Formula – An accurate method of obtaining a general formula for transmission loss has been given by Kron. This, however, is quite complicated. The aim of this article is to give a simpler derivation by making certain assumptions.

Figure 7.9 (c) depicts the case of two generating plants connected to an arbitrary number of loads through a transmission network. One line within the network is designated as branch p.

Imagine that the total load current I_{D} is supplied by plant 1 only, as in Fig. 7.9a. Let the current in line p be I_{P1}. Define

Similarly, with plant 2 alone supplying the total load current (Fig. 7.9b), we can define

M_{P1} and M_{P2} are called **current distribution factors**. The values of current distribution factors depend upon the impedances of the lines and their interconnection and are independent of the current I_{D}.

When both generators 1 and 2 are supplying current into the network as in Fig. 7.9(c), applying the principle of superposition the current in the line p can be expressed as

where I_{G1} and I_{G2} are the currents supplied by plants 1 and 2, respectively. At this stage let us make certain simplifying assumptions outlined below:

1. All load currents have the same phase angle with respect to a common To understand the implication of this assumption consider the load current at the ith bus. It can be written as

where δ_{i} is the phase angle of the bus voltage and Φ_{i} is the lagging phase angle of the load. Since δ_{i} and Φ_{i} vary only through a narrow range at various buses, it is reasonable to assume that θ_{i} is the same for all load currents at all times.

2. Ratio X/R is the same for all network branches.

These two assumptions lead us to the conclusion that I_{P1} and I_{D} [Fig. 7.9(a)] have the same phase angle and so have I_{P2} and I_{D} [Fig. 7.9(b)], such that the current distribution factors M_{P1} and M_{P2} are real rather than complex.

where σ_{1} and σ_{2} are phase angles of I_{G1} and I_{G2}, respectively with respect to the common reference.

From Eq. (7.35), we can write

Expanding the simplifying the above equation, we get

Now

where P_{G1} and P_{G2} are the three-phase real power outputs of plants 1 and 2 at power factors of cos Φ_{1}, and cos Φ_{2}, and V_{1} and V_{2} are the bus voltages at the plants.

If R_{P} is the resistance of branch p, the total transmission loss is given by

^{}

Substituting for |I_{P}|^{2} from Eq. (7.37), and |I_{G1}| and |I_{G2}| from Eq. (7.38), we obtain

Equation (7.39) can be recognized as

The terms B_{11,} B_{12} and B_{22} are called **loss coefficients** or **B-coefficients**. If voltages are line to line kV with resistances in ohms, the units of B-coefficients are in MW^{-1}. Further, with P_{G1} and P_{G2} expressed in MW, P_{L} will also be in MW.

The above results can be extended to the general case of k plants with transmission loss expressed as

Where

The following assumptions including those mentioned already are necessary, if B-coefficients are to be treated as constants as total load and load sharing between plants vary. These assumptions are:

**All load currents maintain a constant ratio to the total current.****Voltage magnitudes at all plants remain constant.****Ratio of reactive to real power, i.e. power factor at each plant remains****Voltage phase angles at plant buses remain fixed. This is equivalent to assuming that the plant currents maintain constant phase angle with respect to the common reference, since source power factors are assumed constant as per assumption 3 above.**

In spite of the number of assumptions made, it is fortunate that treating B-coefficients as constants, yields reasonably accurate results, when the coefficients are calculated for some average operating conditions. Major system changes require recalculation of the coefficients.

Losses as a function of plant outputs can be expressed by other methods, but the simplicity of loss equations is the chief advantage of the B-coefficients method.

Accounting for transmission losses results in considerable operating economy. Furthermore, this consideration is equally important in future system planning and, in particular, with regard to the location of plants and building of new transmission lines: