QuestionCalculate the terminal voltage (in p.u.) of a synchronous machine with a load of $50 \mathrm{MW}$ at $0.8 \mathrm{pf}$ lagging, given transformer reactances of $0.10 \mathrm{p.u.}$ and $0.12 \mathrm{p.u.}$ , and a line impedance of $j100 \, \Omega$ . Confirm the sub-transient reactance and impedance units.

Studdy Solution

STEP 1

Assumptions1. The load is $50 \mathrm{MW}$ at $0.8 \mathrm{pf}$ lagging. The substation voltage is $33 \mathrm{kV}$ and is to be maintained at $30 \mathrm{kV}$
3. The reactances of transformer1 and transformer are $0.10 \mathrm{p.u.}$ and $0.12 \mathrm{p.u.}$ respectively4. The impedance of the line is $j100 \, \Omega$
5. The base power is $100 \mathrm{MVA}$
6. The sub-transient reactance of the generator is not provided7. The line and transformers can be represented by series reactances

STEP 2

First, we need to convert the load power from MW to MVA. The power factor is used for this conversion. The formula is $= / pf$ where- $is the apparent power (in MVA) -$ is the real power (in MW) - $pf$ is the power factor

STEP 3

Now, plug in the given values for the real power and the power factor to calculate the apparent power.
$=50 /0.8$

STEP 4

Calculate the apparent power.
$=50 /0.8 =62. \, \mathrm{MVA}$

STEP 5

Now, we need to convert the apparent power to per unit. The formula is $_{p.u.} = /_{base}$ where- $_{p.u.}$ is the apparent power in per unit- $$ is the apparent power (in MVA) - $_{base}$ is the base power (in MVA)

STEP 6

Now, plug in the given values for the apparent power and the base power to calculate the apparent power in per unit.
$_{p.u.} =62.5 /100$

STEP 7

Calculate the apparent power in per unit.
$_{p.u.} =62.5 /100 =0.625 \, \mathrm{p.u.}$

STEP 8

Now, we need to calculate the total reactance of the system. The total reactance is the sum of the reactances of the transformers and the line. The line impedance is given in ohms, so we need to convert it to per unit. The formula for the conversion is $Z_{line_{p.u.}} = Z_{line} / Z_{base}$ where- $Z_{line_{p.u.}}$ is the line impedance in per unit- $Z_{line}$ is the line impedance (in ohms) - $Z_{base}$ is the base impedance (in ohms)

STEP 9

The base impedance can be calculated using the formula $Z_{base} = V_{base}^2 /_{base}$ where- $V_{base}$ is the base voltage (in kV) - $_{base}$ is the base power (in MVA)

STEP 10

Now, plug in the given values for the base voltage and the base power to calculate the base impedance.
$Z_{base} = (33)^2 /100$

STEP 11

Calculate the base impedance.
$Z_{base} = (33)^ /100 =10.89 \, \Omega$

STEP 12

Now, plug in the given values for the line impedance and the base impedance to calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89$

STEP 13

Calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89 = j9.18 \, \mathrm{p.u.}$

STEP 14

Now, calculate the total reactance of the system.
$X_{total} = X_{} + X_{line} + X_{2}$ where- $X_{total}$ is the total reactance of the system- $X_{}$ is the reactance of transformer- $X_{line}$ is the line reactance- $X_{2}$ is the reactance of transformer2

STEP 15

Now, plug in the given values for the reactances of the transformers and the line to calculate the total reactance.
$X_{total} =0.10 + j9.18 +0.12$

STEP 16

Calculate the total reactance of the system.
$X_{total} =0.10 + j9.18 +0.12 =0.22 + j9.18 \, \mathrm{p.u.}$

STEP 17

Now, we need to calculate the terminal voltage of the synchronous machine. The formula is $V_{terminal} = V_{substation} -_{p.u.} \times X_{total}$ where- $V_{terminal}$ is the terminal voltage of the synchronous machine- $V_{substation}$ is the substation voltage- $_{p.u.}$ is the apparent power in per unit- $X_{total}$ is the total reactance of the system

STEP 18

Now, plug in the given values for the substation voltage, the apparent power in per unit, and the total reactance to calculate the terminal voltage.
$V_{terminal} = -0.625 \times (0.22 + j.18)$

STEP 19

Calculate the terminal voltage of the synchronous machine.
$V_{terminal} =1 -.625 \times (.22 + j9.18) =.8625 - j5.7375 \, \mathrm{p.u.}$ The terminal voltage of the synchronous machine is $.8625 - j5.7375 \, \mathrm{p.u.}$ .

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Studdy Solution

STEP 1

STEP 2

First, we need to convert the load power from MW to MVA. The power factor is used for this conversion. The formula is $= / pf$ where- $is the apparent power (in MVA) -$ is the real power (in MW) - $pf$ is the power factor

STEP 3

Now, plug in the given values for the real power and the power factor to calculate the apparent power.
$=50 /0.8$

STEP 4

Calculate the apparent power.
$=50 /0.8 =62. \, \mathrm{MVA}$

STEP 5

Now, we need to convert the apparent power to per unit. The formula is $_{p.u.} = /_{base}$ where- $_{p.u.}$ is the apparent power in per unit- $$ is the apparent power (in MVA) - $_{base}$ is the base power (in MVA)

STEP 6

Now, plug in the given values for the apparent power and the base power to calculate the apparent power in per unit.
$_{p.u.} =62.5 /100$

STEP 7

Calculate the apparent power in per unit.
$_{p.u.} =62.5 /100 =0.625 \, \mathrm{p.u.}$

STEP 8

STEP 9

The base impedance can be calculated using the formula $Z_{base} = V_{base}^2 /_{base}$ where- $V_{base}$ is the base voltage (in kV) - $_{base}$ is the base power (in MVA)

STEP 10

Now, plug in the given values for the base voltage and the base power to calculate the base impedance.
$Z_{base} = (33)^2 /100$

STEP 11

Calculate the base impedance.
$Z_{base} = (33)^ /100 =10.89 \, \Omega$

STEP 12

Now, plug in the given values for the line impedance and the base impedance to calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89$

STEP 13

Calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89 = j9.18 \, \mathrm{p.u.}$

STEP 14

Now, calculate the total reactance of the system.
$X_{total} = X_{} + X_{line} + X_{2}$ where- $X_{total}$ is the total reactance of the system- $X_{}$ is the reactance of transformer- $X_{line}$ is the line reactance- $X_{2}$ is the reactance of transformer2

STEP 15

Now, plug in the given values for the reactances of the transformers and the line to calculate the total reactance.
$X_{total} =0.10 + j9.18 +0.12$

STEP 16

Calculate the total reactance of the system.
$X_{total} =0.10 + j9.18 +0.12 =0.22 + j9.18 \, \mathrm{p.u.}$

STEP 17

STEP 18

Now, plug in the given values for the substation voltage, the apparent power in per unit, and the total reactance to calculate the terminal voltage.
$V_{terminal} = -0.625 \times (0.22 + j.18)$

STEP 19

Calculate the terminal voltage of the synchronous machine.
$V_{terminal} =1 -.625 \times (.22 + j9.18) =.8625 - j5.7375 \, \mathrm{p.u.}$ The terminal voltage of the synchronous machine is $.8625 - j5.7375 \, \mathrm{p.u.}$ .

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Studdy Solution

STEP 1

STEP 2

First, we need to convert the load power from MW to MVA. The power factor is used for this conversion. The formula is=/pf = / pf=/pfwhere- istheapparentpower(inMVA)− is the apparent power (in MVA) - istheapparentpower(inMVA)− is the real power (in MW) - pfpfpf is the power factor

STEP 3

Now, plug in the given values for the real power and the power factor to calculate the apparent power.=50/0.8 =50 /0.8=50/0.8

STEP 4

Calculate the apparent power.=50/0.8=62. MVA =50 /0.8 =62. \, \mathrm{MVA}=50/0.8=62.MVA

STEP 5

Now, we need to convert the apparent power to per unit. The formula isp.u.=/base_{p.u.} = /_{base}p.u.​=/base​where- p.u._{p.u.}p.u.​ is the apparent power in per unit- $$ is the apparent power (in MVA) - $_{base}$ is the base power (in MVA)

STEP 6

Now, plug in the given values for the apparent power and the base power to calculate the apparent power in per unit.p.u.=62.5/100_{p.u.} =62.5 /100p.u.​=62.5/100

STEP 7

Calculate the apparent power in per unit.p.u.=62.5/100=0.625 p.u._{p.u.} =62.5 /100 =0.625 \, \mathrm{p.u.}p.u.​=62.5/100=0.625p.u.

STEP 8

STEP 9

The base impedance can be calculated using the formulaZbase=Vbase2/baseZ_{base} = V_{base}^2 /_{base}Zbase​=Vbase2​/base​where- VbaseV_{base}Vbase​ is the base voltage (in kV) - base_{base}base​ is the base power (in MVA)

STEP 10

Now, plug in the given values for the base voltage and the base power to calculate the base impedance.Zbase=(33)2/100Z_{base} = (33)^2 /100Zbase​=(33)2/100

STEP 11

Calculate the base impedance.Zbase=(33)/100=10.89 ΩZ_{base} = (33)^ /100 =10.89 \, \OmegaZbase​=(33)/100=10.89Ω

STEP 12

Now, plug in the given values for the line impedance and the base impedance to calculate the line impedance in per unit.Zlinep.u.=j100/10.89Z_{line_{p.u.}} = j100 /10.89Zlinep.u.​​=j100/10.89

STEP 13

Calculate the line impedance in per unit.Zlinep.u.=j100/10.89=j9.18 p.u.Z_{line_{p.u.}} = j100 /10.89 = j9.18 \, \mathrm{p.u.}Zlinep.u.​​=j100/10.89=j9.18p.u.

STEP 14

STEP 15

Now, plug in the given values for the reactances of the transformers and the line to calculate the total reactance.Xtotal=0.10+j9.18+0.12X_{total} =0.10 + j9.18 +0.12Xtotal​=0.10+j9.18+0.12

STEP 16

Calculate the total reactance of the system.Xtotal=0.10+j9.18+0.12=0.22+j9.18 p.u.X_{total} =0.10 + j9.18 +0.12 =0.22 + j9.18 \, \mathrm{p.u.}Xtotal​=0.10+j9.18+0.12=0.22+j9.18p.u.

STEP 17

STEP 18

Now, plug in the given values for the substation voltage, the apparent power in per unit, and the total reactance to calculate the terminal voltage.Vterminal=−0.625×(0.22+j.18)V_{terminal} = -0.625 \times (0.22 + j.18)Vterminal​=−0.625×(0.22+j.18)

STEP 19

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First, we need to convert the load power from MW to MVA. The power factor is used for this conversion. The formula is $= / pf$ where- $is the apparent power (in MVA) -$ is the real power (in MW) - $pf$ is the power factor

Now, plug in the given values for the real power and the power factor to calculate the apparent power.
$=50 /0.8$

Calculate the apparent power.
$=50 /0.8 =62. \, \mathrm{MVA}$

Now, we need to convert the apparent power to per unit. The formula is $_{p.u.} = /_{base}$ where- $_{p.u.}$ is the apparent power in per unit- $$ is the apparent power (in MVA) - $_{base}$ is the base power (in MVA)

Now, plug in the given values for the apparent power and the base power to calculate the apparent power in per unit.
$_{p.u.} =62.5 /100$

Calculate the apparent power in per unit.
$_{p.u.} =62.5 /100 =0.625 \, \mathrm{p.u.}$

The base impedance can be calculated using the formula $Z_{base} = V_{base}^2 /_{base}$ where- $V_{base}$ is the base voltage (in kV) - $_{base}$ is the base power (in MVA)

Now, plug in the given values for the base voltage and the base power to calculate the base impedance.
$Z_{base} = (33)^2 /100$

Calculate the base impedance.
$Z_{base} = (33)^ /100 =10.89 \, \Omega$

Now, plug in the given values for the line impedance and the base impedance to calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89$

Calculate the line impedance in per unit.
$Z_{line_{p.u.}} = j100 /10.89 = j9.18 \, \mathrm{p.u.}$

Now, plug in the given values for the reactances of the transformers and the line to calculate the total reactance.
$X_{total} =0.10 + j9.18 +0.12$

Calculate the total reactance of the system.
$X_{total} =0.10 + j9.18 +0.12 =0.22 + j9.18 \, \mathrm{p.u.}$

Now, plug in the given values for the substation voltage, the apparent power in per unit, and the total reactance to calculate the terminal voltage.
$V_{terminal} = -0.625 \times (0.22 + j.18)$