Question

The power flow at any point on a transmission line can be calculated in terms of the ABCDparameters. By letting$\mathbf{A}\mathbf{=}\mathbf{\left|}\mathbf{A}\mathbf{\right|}\mathbf{\angle }\mathfrak{a}\mathbf{,}\mathbf{}\mathbf{B}\mathbf{=}\mathbf{\left|}\mathbf{B}\mathbf{\right|}\mathbf{\angle }\mathbf{\beta }\mathbf{}\mathbf{,}{\mathbf{V}}_{\mathbf{R}}\mathbf{=}\mathbf{\left|}{\mathbf{V}}_{\mathbf{R}}\mathbf{\right|}\mathbf{\angle }{\mathbf{0}}^{\mathbf{0}}$and ${\mathbf{V}}_{\mathbf{S}}\mathbf{=}\left|V_S\right|\mathbf{\angle }\mathbf{\delta }$, the complex power at the receiving end can be shown to be

${\mathbf{P}}_{\mathbf{R}}\mathbf{+}{\mathbf{jQ}}_{\mathbf{R}}\mathbf{=}\frac{\left|V_S\right|\left|V_R\right|}{\left|B\right|}\mathbf{\angle }\mathbf{\beta }\mathbf{-}\mathfrak{a}\mathbf{-}\frac{\left|\delta \right|\left|V_R^2\right|}{\left|B\right|}\mathbf{\angle }\mathbf{\beta }\mathbf{-}\mathfrak{a}$

(a) Draw a phasor diagram corresponding to the above equation. Let it be represented by a triangle O’OA with O’ as the origin and OA representing${\mathbf{P}}_{\mathbf{R}}\mathbf{+}{\mathbf{jQ}}_{\mathbf{R}}$ .

(b) By shifting the origin from O’ to O, turn the result of part (a) into a power diagram, redrawing the phasor diagram. For a given fixed value of $\left|V_R\right|$and a set of values for $\left|V_S\right|$, draw the loci of point A , thereby showing the so-called receiving-end circles.

(c) From the result of part (b) for a given load with a lagging power factor angle${\mathbf{\theta }}_{\mathbf{R}}$ , determine the amount of reactive power that must be supplied to the receiving end to maintain a constant receiving-end voltage if the sending-end voltage magnitude decreases from $\left|V_{S1}\right|$ to$\left|V_{S2}\right|$ .

Short answer

(a) The phasor diagram is,

(b) The modified phasor diagram is,

(c) The line representing the reactive power in below diagram:

The amount of reactive power to be supplied is determine by the line AB.

Step-by-step solution

Given data.

The and parameters are given as $\mathrm{A}=\left|\mathrm{A}\right|\angle \mathfrak{a},\mathrm{and}\mathrm{B}=\left|\mathrm{B}\right|\angle \mathrm{\beta }$.

The receiving end and sending end voltage is ${\mathrm{V}}_{\mathrm{R}}=\left|{\mathrm{V}}_{\mathrm{R}}\right|\angle 0^0\mathrm{and}{\mathrm{V}}_{\mathrm{S}}=\left|{\mathrm{V}}_{\mathrm{S}}\right|\angle \mathrm{\delta }$and .

Determine the formulas of complex power transferred to load.

Write the formula of complex power.

${\mathbf{P}}_{\mathbf{R}}\mathbf{+}{\mathbf{jQ}}_{\mathbf{R}}\mathbf{=}\frac{\mathbf{\left|}{\mathbf{V}}_{\mathbf{S}}\mathbf{\right|}\mathbf{\left|}{\mathbf{V}}_{\mathbf{R}}\mathbf{\right|}}{\mathbf{\left|}\mathbf{B}\mathbf{\right|}}\mathbf{\angle }\mathbf{\beta }\mathbf{-}\mathbf{\alpha }\mathbf{-}\frac{\mathbf{\left|}\mathbf{\delta }\mathbf{\right|}\mathbf{\left|}{{\mathbf{V}}^{\mathbf{2}}}_{\mathbf{R}}\mathbf{\right|}}{\mathbf{\left|}\mathbf{B}\mathbf{\right|}}\mathbf{\angle }\mathbf{\beta }\mathbf{-}\mathbf{\alpha }$ ……. (1)

Here, ${\mathrm{P}}_{\mathrm{R}}$ is real power, ${\mathrm{Q}}_{\mathrm{R}}$is reactive power, ${\mathrm{V}}_{\mathrm{S}}$is sending end power, ${\mathrm{V}}_{\mathrm{R}}$ is receiving end power, $\left|\mathrm{B}\right|$ is magnitude and $\mathrm{\beta }$ is direction of parameter B , $\left|\mathrm{A}\right|$is magnitude and $\mathfrak{a}$ is direction of parameter A and $\mathrm{\delta }$is a load angle.

Determine the phasor diagram.

(a)

Draw the phasor diagram for equation (1).

In above diagram, O'OA is a triangle, O' is a origin, OA representing complex power ${\mathrm{P}}_{\mathrm{R}}+{\mathrm{jQ}}_{\mathrm{R}}$.

Determine the phasor diagram.

(b)

Shift the origin O' to point O , and redraw the phasor diagram below:

In the above diagram, O' A is a radius, and representing $\frac{\left|{\mathrm{V}}_{\mathrm{S}}\right|\left|{\mathrm{V}}_{\mathrm{R}}\right|}{\left|\mathrm{B}\right|}$.

Now, make ${\mathrm{V}}_{\mathrm{R}}$ constant and change the value of${\mathrm{V}}_{\mathrm{S}}$ .

Hence, the power diagram is drawn below with locus of point A .

Determine the phasor diagram.

(c) For lagging power factor, let ${\mathrm{\theta }}_{\mathrm{R}}$is a power angle.

The receiving end voltage is kept constant and sending end voltage is decreased from ${\mathrm{V}}_{\mathrm{S}1}$to ${\mathrm{V}}_{\mathrm{S}2}$.

Draw the load line below.

The reactive power is determined by the line AB .

Therefore, the diagram is drawn.