Let's solve the problem step by step. --- ## **Given Data** - **Wind source rated:** 80 MW - **Central station rated:** 500 MW - **Capital cost of wind:** $500/kW - **Capital cost of central station:** $100/kW - **Wind maintenance cost:** 1 ¢/kWh (no fuel cost) - **Central station fuel & maintenance cost:** 3.2 ¢/kWh - **Time period:** 5 years - **Load factor:** Varying from to 1 (compute EUC as a function of load factor) --- ## **Step 1: Convert Power Ratings to kW** - Wind: \( 80\,\text{MW} = 80,000\,\text{kW} \) - Central station: \( 500\,\text{MW} = 500,000\,\text{kW} \) --- ## **Step 2: Calculate Annual Energy Output for Given Load Factor (\( LF \))** For any source: \[ \text{Annual Energy (kWh)} = \text{Rated Power (kW)} \times 876 \times LF \] where 876 is the number of hours in a year. --- ## **Step 3: Compute Capital Cost per Year** Assume straight-line depreciation over 5 years (ignore interest for simplicity): \[ \text{Annual Capital Cost} = \frac{\text{Total Capital Cost}}{5} \] ### **Wind:** \[ \text{Total Capital Cost} = 80,000 \times \$500 = \$40,000,000 \] \[ \text{Annual Capital Cost} = \frac{\$40,000,000}{5} = \$8,000,000 \] ### **Central Station:** \[ \text{Total Capital Cost} = 500,000 \times \$100 = \$50,000,000 \] \[ \text{Annual Capital Cost} = \frac{\$50,000,000}{5} = \$10,000,000 \] --- ## **Step 4: Compute Annual Operating Cost** ### **Wind:** - Maintenance = 1 ¢/kWh = $.01/kWh Annual Operating Cost = \( \text{Annual Energy} \times .01 \) ### **Central Station:** - Fuel + Maintenance = 3.2 ¢/kWh = $.032/kWh Annual Operating Cost = \( \text{Annual Energy} \times .032 \) --- ## **Step 5: Compute EUC (Equivalent Unit Cost) for Each Source** \[ \text{EUC} = \frac{\text{Annual Capital Cost} + \text{Annual Operating Cost}}{\text{Annual Energy Output (kWh)}} \] --- ### **For Wind:** \[ \text{Annual Energy} = 80,000 \times 876 \times LF = 700,800,000 \times LF \] \[ \text{Annual Operating Cost} = 700,800,000 \times LF \times .01 = 7,008,000 \times LF \] \[ \text{EUC}_\text{wind} = \frac{8,000,000 + 7,008,000 \times LF}{700,800,000 \times LF} \] --- ### **For Central Station:** \[ \text{Annual Energy} = 500,000 \times 876 \times LF = 4,380,000,000 \times LF \] \[ \text{Annual Operating Cost} = 4,380,000,000 \times LF \times .032 = 140,160,000 \times LF \] \[ \text{EUC}_\text{central} = \frac{10,000,000 + 140,160,000 \times LF}{4,380,000,000 \times LF} \] --- ## **Step 6: Simplify EUC Expressions** ### **Wind:** \[ \text{EUC}_\text{wind} = \frac{8,000,000}{700,800,000 \times LF} + \frac{7,008,000 \times LF}{700,800,000 \times LF} \] \[ = \frac{8,000,000}{700,800,000 \times LF} + .01 \] \[ = \frac{.01142}{LF} + .01 \qquad \text{(rounded to 5 decimals)} \] --- ### **Central Station:** \[ \text{EUC}_\text{central} = \frac{10,000,000}{4,380,000,000 \times LF} + \frac{140,160,000 \times LF}{4,380,000,000 \times LF} \] \[ = \frac{10,000,000}{4,380,000,000 \times LF} + .032 \] \[ = \frac{.00228}{LF} + .032 \qquad \text{(rounded to 5 decimals)} \] --- ## **Step 7: Compute EUC at Key Load Factors** | Load Factor (LF) | EUC (wind) (\$/kWh) | EUC (central) (\$/kWh) | |:----------------:|:-------------------:|:----------------------:| | .1 | .1142 + .01 = .1242 | .0228 + .032 = .0548 | | .2 | .0571 + .01 = .0671 | .0114 + .032 = .0434 | | .4 | .0286 + .01 = .0386 | .0057 + .032 = .0377 | | .6 | .019 + .01 = .029 | .0038 + .032 = .0358 | | .8 | .0143 + .01 = .0243 | .00285 + .032 = .03485 | | 1. | .0114 + .01 = .0214 | .00228 + .032 = .03428 | --- ## **Step 8: **Summary Table** | Load Factor (LF) | EUC (wind) [¢/kWh] | EUC (central) [¢/kWh] | |:----------------:|:------------------:|:---------------------:| | .1 | 12.42 | 5.48 | | .2 | 6.71 | 4.34 | | .4 | 3.86 | 3.77 | | .6 | 2.90 | 3.58 | | .8 | 2.43 | 3.49 | | 1. | 2.14 | 3.43 | --- ## **Final Answer and Explanation** - **The EUC for wind and central generation decreases as load factor increases, because the capital cost is spread over more kWh.** - **For wind, EUC drops below central station cost at higher load factors.** - **At low load factor, wind is much more expensive due to high capital cost.** - **At full utilization (LF = 1):** - Wind: **2.14 ¢/kWh** - Central: **3.43 ¢/kWh** - **At low utilization (LF = .1):** - Wind: **12.42 ¢/kWh** - Central: **5.48 ¢/kWh** You can plot these values as a function of load factor for a figure as requested. --- ### **Summary Formulae** \[ \boxed{ \begin{align*} \text{EUC}_\text{wind} &= \frac{.01142}{LF} + .01 \ (\$/kWh) \\ \text{EUC}_\text{central} &= \frac{.00228}{LF} + .032 \ (\$/kWh) \end{align*} } \] --- Let me know if you'd like the plot or further explanation!