Question

Expand the summations in (6.12.14) for N=2, and verify the formula for $\partial P{o}_L/P_i$given by (6.12.15). Assume $B_{ij}=B_{ji}$.

Short answer

The value of $\partial P_L/\partial P_i$is $2\sum _{j=1}^2{P}_j{B}_{ij}$.

Step-by-step solution

Write the given data from the question.

The value of N=2 for given summations in reference (6.12.14).

Assume $B_{ij}=B_{ji}$

Determine the formula for transmission line losses and derivatives.

Write the formula of total transmission losses.

$P_L=\sum _{i=1}^N\sum _{j=1}^N{P}_i{B}_{ij}{P}_j$ …… (1)

Here, $P_i$is real power output,$B_{ij}$is a loss coefficient and $P_j$is imaginary power output.

Write the formula of total derivative with respect to$P_1$.

$\frac{\partial P_L}{\partial P_1}=2P_1{B}_{11}+2P_2{B}_{12}$ …… (2)

Here, $P_1$is real power output,$B_{11}$and $B_{12}$ is a loss coefficient and $P_2$is imaginary power output.

Determine the partial derivative ∂PoL/Pi. 

The total transmission losses are expressed as a quadratic function of unit output power is,

Substitute N=2 into equation (1).

$$\begin{gathered} P_L=P_1{B}_{11}{P}_1+P_1{B}_{12}{P}_2+P_2{B}_{21}{P}_1+P_2{B}_{22}{P}_2 \\ =P_1^2{B}_{11}+P_1{P}_2{B}_{12}+P_2{P}_1{B}_{21}+P_2^2{B}_{22} \end{gathered}$$

Consider the expression.

$B_{ij}=B_{ji}$

Therefore,

$B_{12}=B_{21}$

Then,

$P_L=P_1^2{B}_{11}+2P_1{P}_2{B}_{12}+P_2^2{B}_{22}$ …… (3)

Here, Here, $P_1$is real power output,$B_{11}$and $B_{12}$ is a loss coefficient and $P_2$is imaginary power output.

Taking partial derivative of equation (3) with respect to$P_1$.

$$\begin{gathered} \frac{\partial P_L}{\partial P_1}=2P_1{B}_{11}+2P_2{B}_{12} \\ =2\left(P_1{B}_{11}+P_2{B}_{12}\right) \\ =2\sum _{j=1}^2{P}_j{B}_{ij} \end{gathered}$$

Hence, it is verified.