Question

Each unit in Problem 12.5 is initially operating at one-half its own rating when the load suddenly increases by $\mathbf{100}\mathbf{MW}$ . Determine (a) the steady state decrease in area frequency, and (b) the MW increase in mechanical power output of each turbine. Assume that the reference power setting of each turbine-generator remains constant. Neglect losses and the dependence of load on frequency.

Short answer

(a) The steady state decrease in the area frequency is$-0.264\text{\hspace{0.17em}Hz}$ .

(b) The value of the mechanical power output for turbine 1, 2 and 3 are $27.33\text{\hspace{0.17em}MVA}$, $31.5\text{\hspace{0.17em}MVA}$and$41\text{\hspace{0.17em}MVA}$ respectively.

Step-by-step solution

Write the given data from the question.

The frequency of the system,$f=60\text{\hspace{0.17em}Hz}$

Write the rating of three turbine generator

$\begin{array}{l}{S}_{1,old}=200\text{\hspace{0.17em}MVA}\\ {\text{S}}_{2,old}=300\text{\hspace{0.17em}MVA}\\ S_{3,old}=500\text{\hspace{0.17em}MVA}\end{array}$

Write the regulation constant of three generator units

$\begin{array}{l}{R}_{1,old}=0.03\text{\hspace{0.17em}pu\hspace{0.17em}}\\ R_{2,old}=\text{0.04\hspace{0.17em}pu}\\ R_{3,old}=0.05\text{\hspace{0.17em}pu}\end{array}$

The base rating, $S_{new}=100\text{\hspace{0.17em}MVA}$

The increase value of load, $P_L=100\text{\hspace{0.17em}MW}$

Change in the reference setting power, $\Delta p_{ref}=0$

Determine the equation to calculate the steady state decrease in the area frequency and MW increase in the mechanical power output.

The equation to calculate the new value of the regulation constant for 1^st unit is given as follows.

${\mathbf{R}}_{\mathbf{1}\mathbf{new}}\mathbf{=}{\mathbf{R}}_{\mathbf{1}\mathbf{old}}\frac{{\mathbf{S}}_{\mathbf{new}}}{{\mathbf{S}}_{\mathbf{1}\mathbf{old}}}$ …… (1)

The equation to calculate the new value of the regulation constant for 2^nd unit is given as follows.

${\mathbf{R}}_{\mathbf{2}\mathbf{new}}\mathbf{=}{\mathbf{R}}_{\mathbf{2}\mathbf{old}}\frac{{\mathbf{S}}_{\mathbf{new}}}{{\mathbf{S}}_{\mathbf{2}\mathbf{old}}}$ …… (2)

The equation to calculate the new value of the regulation constant for 3^rd unit is given as follows.

${\mathbf{R}}_{\mathbf{3}\mathbf{new}}\mathbf{=}{\mathbf{R}}_{\mathbf{3}\mathbf{old}}\frac{{\mathbf{S}}_{\mathbf{new}}}{{\mathbf{S}}_{\mathbf{3}\mathbf{old}}}$ …… (3)

The equation to calculate the unit area frequency response characteristics is given as follows.

$\mathbf{\beta }\mathbf{=}\frac{\mathbf{1}}{{\mathbf{R}}_{\mathbf{1}\mathbf{new}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{R}}_{\mathbf{2}\mathbf{new}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{R}}_{\mathbf{3}\mathbf{new}}}$ …… (4)

The equation to calculate the total change in turbine mechanical power in per unit is given as,

${\mathbf{\Delta p}}_{\mathbf{m}}\mathbf{=}\frac{{\mathbf{P}}_{\mathbf{L}}}{{\mathbf{S}}_{\mathbf{new}}}$ …… (5)

The equation for the steady state response power relation is given as follows.

${\mathbf{\Delta p}}_{\mathbf{m}}\mathbf{=}{\mathbf{\Delta p}}_{\mathbf{ref}}\mathbf{-}\mathbf{\beta \Delta f}$ …… (6)

Here $\mathbf{\Delta f}$ is the steady state increase in the area frequency.

The equation to calculate the steady state decrease in the value of frequency in hertz is given as follows.

${\mathbf{\Delta f}}_{\mathbf{hertz}}\mathbf{=}\mathbf{\Delta f}\mathbf{\times }\mathbf{f}$ …… (7)

The equation to calculate the per unit value of mechanical power is given as follows.

${\mathbf{\Delta p}}_{\mathbf{m}}\mathbf{=}{\mathbf{\Delta p}}_{\mathbf{ref}}\mathbf{-}\frac{\mathbf{\Delta f}}{{\mathbf{R}}_{\mathbf{new}}}$ …… (8)

The equation to calculate the per unit value of mechanical power in MW is given as follows.

${\mathbf{\Delta p}}_{\mathbf{m}\mathbf{,}\mathbf{MW}}\mathbf{=}{\mathbf{\Delta p}}_{\mathbf{m}}{\mathbf{S}}_{\mathbf{new}}$ …… (9)

Calculate the steady state decrease in the area frequency.

(a)

Calculate the new value of the regulation constant for 1^stunit.

Substitute $0.03\text{\hspace{0.17em}pu}$ for${\mathrm{R}}_{1\mathrm{old}}$,$200\text{\hspace{0.17em}MVA}$ for $S_{1old}$ and $100\text{\hspace{0.17em}MVA}$ for $S_{new}$ into equation (1).

$\begin{array}{l}{R}_{1new}=0.03\times \frac{100}{200}\\ R_{1new}=0.015\text{\hspace{0.17em}pu}\end{array}$

Calculate the new value of the regulation constant for 2^nd unit.

Substitute $0.04\text{\hspace{0.17em}pu}$ for$R_{2old}$,$300\text{\hspace{0.17em}MVA}$for $S_{2old}$ and $100\text{\hspace{0.17em}MVA}$for $S_{new}$ into equation (2).

$\begin{array}{l}{R}_{2new}=0.04\times \frac{100}{300}\\ R_{2new}=0.013\text{\hspace{0.17em}pu}\end{array}$

Calculate the new value of the regulation constant for 3^rd unit.

Substitute $0.05\text{\hspace{0.17em}pu}$ for $R_{3old}$,$500\text{\hspace{0.17em}MVA}$for $S_{3old}$ and $100\text{\hspace{0.17em}MVA}$ for $S_{new}$ into equation (3).

$\begin{array}{l}{R}_{3new}=0.05\times \frac{100}{500}\\ R_{3new}=0.01\text{\hspace{0.17em}pu}\end{array}$

Calculate the unit area frequency response characteristics.

Substitute $0.015\text{\hspace{0.17em}pu}$ for $R_{1new}$, $0.013\text{\hspace{0.17em}pu}$for $R_{2new}$and $0.01\text{\hspace{0.17em}pu}$ for$R_{3new}$ into equation (4).

$\begin{array}{l}\beta =\frac{1}{0.015}+\frac{1}{0.013}+\frac{1}{0.01}\\ \beta =66.66+75.92+100\\ \beta =243.58\text{\hspace{0.17em}pu}\end{array}$

Calculate the total change in turbine mechanical power in per unit

Substitute $100\text{\hspace{0.17em}MW}$for $P_L$and $100\text{\hspace{0.17em}MVA}$for $S_{new}$into equation (5).

$\begin{array}{l}\Delta p_m=\frac{100}{100}\\ \Delta p_m=1\text{\hspace{0.17em}pu}\end{array}$

Calculate the steady state decrease in the area frequency

Substitute 0 for $\Delta p_{ref}$ , $1\text{\hspace{0.17em}pu}$for $\Delta p_m$and $243.58\text{\hspace{0.17em}pu}$ for $\beta$ into equation (6).

Calculate the steady state decrease in the area frequency in hertz.

Substitute$-0.0041\text{\hspace{0.17em}pu}$ for $\Delta f$ and $60\text{\hspace{0.17em}Hz}$ for $f$ into equation (7).

$\begin{array}{l}\Delta f_{hertz}=-0.0041\times 60\\ \Delta f_{hertz}=-0.246\text{\hspace{0.17em}Hz}\end{array}$

Hence the steady state decrease in the area frequency is $-0.264\text{\hspace{0.17em}Hz}$.

Calculate the MW increase in the mechanical power output of each turbine.

(b)

Calculate the per unit value of mechanical power output for 1^st unit.

Substitute $0$ for $\Delta p_{ref}$, $-0.0041\text{\hspace{0.17em}pu}$for $\Delta f$and $0.015\text{\hspace{0.17em}pu}$ for $R_{1new}$ into equation (8).

$\begin{array}{l}\Delta p_{m1}=0-\frac{(-0.0041)}{0.015}\\ \Delta p_{m1}=0.273\text{\hspace{0.17em}pu}\end{array}$

Calculate the per unit value of mechanical power output for 1^st unit in MW.

Substitute $0.273\text{\hspace{0.17em}pu}$for $\Delta p_{m1}$and $100\text{\hspace{0.17em}MVA}$for $S_{new}$into equation (9).

$\begin{array}{l}\Delta p_{m1,MW}=\left(0.273\right)\left(100\right)\\ \Delta p_{m1,MW}=27.33\text{\hspace{0.17em}MVA}\end{array}$

Calculate the per unit value of mechanical power output for 2^nd unit.

Substitute $0$ for $\Delta p_{ref}$, $-0.0041\text{\hspace{0.17em}pu}$ for$\Delta f$ and $0.0133\text{\hspace{0.17em}pu}$ for$R_{2new}$ into equation (8).

$\begin{array}{l}\Delta p_{m2}=0-\frac{(-0.0041)}{0.013}\\ \Delta p_{m2}=0.315\text{\hspace{0.17em}pu}\end{array}$

Calculate the per unit value of mechanical power output for 2^nd unit in MW.

Substitute $0.315\text{\hspace{0.17em}pu}$ for $\Delta p_{m2}$ and $100\text{\hspace{0.17em}MVA}$ for $S_{new}$into equation (9).

$\begin{array}{l}\Delta p_{m2,MW}=\left(0.315\right)\left(100\right)\\ \Delta p_{m2,MW}=31.5\text{\hspace{0.17em}MVA}\end{array}$

Calculate the per unit value of mechanical power output for 3^rd unit.

Substitute $0$ for $\Delta p_{ref}$, $-0.0041\text{\hspace{0.17em}pu}$ for$\Delta f$ and$0.01\text{\hspace{0.17em}pu}$ for$R_{3new}$ into equation (8).

$\begin{array}{l}\Delta p_{m2}=0-\frac{(-0.0041\text{\hspace{0.17em}pu})}{0.01}\\ \Delta p_{m2}=0.41\text{\hspace{0.17em}pu}\end{array}$

Calculate the per unit value of mechanical power output for 3^rd unit in MW.

Substitute$0.41\text{\hspace{0.17em}pu}$ for $\Delta p_{m3}$ and $100\text{\hspace{0.17em}MVA}$ for$S_{new}$ into equation (9).

$\begin{array}{l}\Delta p_{m3,MW}=\left(0.41\right)\left(100\right)\\ \Delta p_{m3,MW}=41\text{\hspace{0.17em}MVA}\end{array}$

Hence, the value of the mechanical power output for turbine 1, 2 and 3 are$27.33\text{\hspace{0.17em}MVA}$ ,$31.5\text{\hspace{0.17em}MVA}$ and $41\text{\hspace{0.17em}MVA}$ respectively.