STEM Problem Detail

Problem Statement

Prompt

please provide the answer of the Example 13.1

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## 1. Machine Parameters and Initial Steady-State

Using equations (11.7.3) to (11.7.5), the induction machine parameters are determined:
\[ X' = 0.2297, \quad X = 3.867 \text{ pu}, \quad T'_o = 0.85 \text{ s} \]

Verification of steady-state initial conditions ($$\frac{dE'}{dt} = 0$$) with $$V_T = 1.0723 + j0.220$$:
\[ V_r = 0.9314 - (0.013)(0.7974) + (0.2297)(0.6586) = 1.0723 \]
\[ V_i = 0.4117 - (0.013)(0.6586) - (0.2297)(0.7974) = 0.2200 \]
\[ \frac{dE'_r}{dt} = 2\pi60(-0.0111)(0.4117) - \frac{1}{0.85}(0.9314 - (3.637)(0.6586)) = 0.0 \]

## 2. Source Current and Internal Flux

With $$I = 1.0$$ and impedance $$j0.12$$ to the infinite bus:
\[ V_T = 1.0 + (1.0)(j0.22) = 1.0 + j0.22 = 1.0239 \angle 12.41^\circ \]
\[ I_{sorc} = 1.00 + \frac{1.0 + j0.220}{j0.8} = 1.275 - j1.25 \]
\[ I_p + jI_q = (1.275 - j1.25) \times (1 / \angle 12.41^\circ) = 0.977 - j1.495 \]
\[ E_q = -(-1.495)(0.8) = 1.196 \]

## 3. Excitation and Fault Response

Initial conditions for steady-state excitation:
\[ V_r = (K_e)(E_{fd}) = (1.0)(2.9135) = 2.9135 \]
Reference voltage $$V_{ref}$$ with gain $$K_a = 100$$:
\[ V_{ref} = \frac{V_r}{K_a} + V_t + V_f = \frac{2.9135}{100} + 1.0946 = 1.1237 \]

Reactive power injection during a fault ($$V_t = 0.5$$):
\[ Q_{net} = (V_t)(1.8125) - \frac{V_t^2}{0.8} = 0.593 \text{ pu} = 59.3 \text{ Mvar} \]

## 4. System Frequency Regulation

Composite regulation constant $$\beta$$ on a 1000 MVA base:
\[ \beta = \sum \frac{1}{R_i} = \frac{1}{0.05} + \frac{1}{0.06667} + \frac{1}{0.10} = 45.0 \text{ pu} \]
Frequency drop for a 200 MW (0.20 pu) load increase:
\[ \Delta f = \left( \frac{-1}{45} \right)(0.20) = -4.444 \times 10^{-3} \text{ pu} = -0.2667 \text{ Hz} \]

\section*{5. Economic Dispatch and Losses}
Optimal dispatch condition with penalty factors $L_i$:
\[
\lambda = \frac{dC_i}{dP_i} \left( \frac{1}{1 - \frac{\partial P_L}{\partial P_i}} \right) = \frac{dC_i}{dP_i} L_i
\]
For $\lambda = 16.00$ \$/MWh:
\[
P_1 = 282 \text{ MW}, \quad P_2 = 417 \text{ MW}, \quad P_L = 19.5 \text{ MW}
\]
\[
P_T = 679.5 \text{ MW}, \quad C_T = 8357 \text{ \$/h}
\]

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Problem #67