Dam Height Optimization Calculator

Dam Height Optimization Calculator estimates the optimal dam height to maximize net annual revenue from a hydropower plant, balancing power output against construction costs, with a detailed breakdown and step-by-step calculations.

Flow Rate (m³/s, e.g., 2):

Turbine Efficiency (0 to 1, e.g., 0.85):

Operating Days per Year (e.g., 365):

Electricity Price ($/kWh, e.g., 0.12):

Base Construction Cost ($, e.g., 1000000):

Linear Cost Coefficient ($/m, e.g., 50000):

Quadratic Cost Coefficient ($/m², e.g., 1000):

Project Lifetime (years, e.g., 50):

Formulas Used in Dam Height Optimization Calculator

The calculator uses the following formulas to optimize dam height:

Power Output:

\\[ P = \eta \cdot \rho \cdot g \cdot Q \cdot h \\]

Annual Energy Production:

\\[ E = \frac{P}{1000} \cdot 24 \cdot D \\]

Gross Annual Revenue:

\\[ R_{\text{gross}} = E \cdot r \\]

Annualized Construction Cost:

\\[ C = \frac{c_0 + c_1 h + c_2 h^2}{L} \\]

Net Annual Revenue:

\\[ R_{\text{net}} = R_{\text{gross}} – C \\]

Optimal Dam Height:

\\[ h_{\text{opt}} = \frac{\frac{\eta \cdot \rho \cdot g \cdot Q \cdot 24 \cdot D \cdot r}{1000} – \frac{c_1}{L}}{\frac{2 c_2}{L}} \\]

Where:

\$ P \$: Power output (W)
\$ \eta \$: Turbine efficiency (0 to 1)
\$ \rho \$: Water density (1000 kg/m³)
\$ g \$: Gravitational acceleration (9.81 m/s²)
\$ Q \$: Flow rate (m³/s)
\$ h \$: Dam height (gross head, m)
\$ E \$: Energy (kWh/year)
\$ D \$: Operating days per year
\$ R_{\text{gross}} \$: Gross revenue ($/year)
\$ r \$: Electricity price ($/kWh)
\$ C \$: Annualized construction cost ($/year)
\$ c_0 \$: Base construction cost ($)
\$ c_1 \$: Linear cost coefficient ($/m)
\$ c_2 \$: Quadratic cost coefficient ($/m²)
\$ L \$: Project lifetime (years)
\$ R_{\text{net}} \$: Net revenue ($/year)
\$ h_{\text{opt}} \$: Optimal dam height (m)

Example Calculations

Example 1: Small-Scale Dam

Input: Flow Rate = 1 m³/s, Efficiency = 0.8, Days = 365, Price = $0.15/kWh, Base Cost = $500,000, Linear Cost = $20,000/m, Quadratic Cost = $500/m², Lifetime = 50 years

\\[ h_{\text{opt}} = \frac{\frac{0.8 \cdot 1000 \cdot 9.81 \cdot 1 \cdot 24 \cdot 365 \cdot 0.15}{1000} – \frac{20000}{50}}{\frac{2 \cdot 500}{50}} \approx 51.74 \ \text{m} \\] \\[ P = 0.8 \cdot 1000 \cdot 9.81 \cdot 1 \cdot 51.74 \approx 406,390 \ \text{W} \\] \\[ E = \frac{406390}{1000} \cdot 24 \cdot 365 \approx 3,559,796 \ \text{kWh/year} \\] \\[ R_{\text{gross}} = 3559796 \cdot 0.15 \approx 533,969 \ \$/\text{year} \\] \\[ C = \frac{500000 + 20000 \cdot 51.74 + 500 \cdot 51.74^2}{50} \approx 47,387 \ \$/\text{year} \\] \\[ R_{\text{net}} = 533969 – 47387 \approx 486,582 \ \$/\text{year} \\]

Result: Optimal Height = 51.74 m, Power = 406,390 W, Energy = 3,559,796 kWh/year, Gross Revenue = $533,969/year, Cost = $47,387/year, Net Revenue = $486,582/year

Example 2: Medium-Scale Dam

Input: Flow Rate = 2.265 m³/s, Efficiency = 0.85, Days = 200, Price = $0.12/kWh, Base Cost = $1,000,000, Linear Cost = $50,000/m, Quadratic Cost = $1,000/m², Lifetime = 40 years

\\[ h_{\text{opt}} = \frac{\frac{0.85 \cdot 1000 \cdot 9.81 \cdot 2.265 \cdot 24 \cdot 200 \cdot 0.12}{1000} – \frac{50000}{40}}{\frac{2 \cdot 1000}{40}} \approx 21.18 \ \text{m} \\] \\[ P = 0.85 \cdot 1000 \cdot 9.81 \cdot 2.265 \cdot 21.18 \approx 400,159 \ \text{W} \\] \\[ E = \frac{400159}{1000} \cdot 24 \cdot 200 \approx 1,920,763 \ \text{kWh/year} \\] \\[ R_{\text{gross}} = 1920763 \cdot 0.12 \approx 230,492 \ \$/\text{year} \\] \\[ C = \frac{1000000 + 50000 \cdot 21.18 + 1000 \cdot 21.18^2}{40} \approx 62,240 \ \$/\text{year} \\] \\[ R_{\text{net}} = 230492 – 62240 \approx 168,252 \ \$/\text{year} \\]

Result: Optimal Height = 21.18 m, Power = 400,159 W, Energy = 1,920,763 kWh/year, Gross Revenue = $230,492/year, Cost = $62,240/year, Net Revenue = $168,252/year

Example 3: Large-Scale Dam

Input: Flow Rate = 11.992 m³/s, Efficiency = 0.9, Days = 300, Price = $0.08/kWh, Base Cost = $5,000,000, Linear Cost = $100,000/m, Quadratic Cost = $2,000/m², Lifetime = 60 years

\\[ h_{\text{opt}} = \frac{\frac{0.9 \cdot 1000 \cdot 9.81 \cdot 11.992 \cdot 24 \cdot 300 \cdot 0.08}{1000} – \frac{100000}{60}}{\frac{2 \cdot 2000}{60}} \approx 43.66 \ \text{m} \\] \\[ P = 0.9 \cdot 1000 \cdot 9.81 \cdot 11.992 \cdot 43.66 \approx 4,625,685 \ \text{W} \\] \\[ E = \frac{4625685}{1000} \cdot 24 \cdot 300 \approx 33,304,932 \ \text{kWh/year} \\] \\[ R_{\text{gross}} = 33304932 \cdot 0.08 \approx 2,664,395 \ \$/\text{year} \\] \\[ C = \frac{5000000 + 100000 \cdot 43.66 + 2000 \cdot 43.66^2}{60} \approx 152,844 \ \$/\text{year} \\] \\[ R_{\text{net}} = 2664395 – 152844 \approx 2,511,551 \ \$/\text{year} \\]

Result: Optimal Height = 43.66 m, Power = 4,625,685 W, Energy = 33,304,932 kWh/year, Gross Revenue = $2,664,395/year, Cost = $152,844/year, Net Revenue = $2,511,551/year

Formulas Used in Dam Height Optimization Calculator

Example Calculations

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