# LCOE-Based Optimization for the Design of Small Run-of-River Hydropower Plants

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods: Description of the Developed Model

#### 2.1. The Design Procedure

- Penstocks are buried only if a minimum of rock excavation is required [41]. However, since excavation leads to additional costs, this study assumes that penstocks are installed over the ground, hence leading to a positive suction head;
- The plant is connected to an existing electrical network and, hence, all the electricity produced is actually sold;
- The power plant is always in operation: no fault occurs, and as multiple turbines are installed, it is assumed that the maintenance plan can be performed without interruptions;
- Line losses are neglected;
- In order to generalize the model, the penstock configuration and its singularities are understood as sufficiently close to those of the 21 power plants that allowed Singhal MK and Arun Kumar [23] to develop a relationship between total head losses and friction losses: this formula allows one to calculate total head losses from friction losses, penstock length, and gross head and was found to be usable for hydropower projects due to the correlation coefficient of the relation obtained being 0.837, which is within the authorized limit;
- The deviation of the watercourse is made from a weir that does not allow considerable storage, as the role of the retaining structure is only to keep the level constant so as to ensure that the water intake and penstock are always supplied [41]. Additionally, that it is set up with what is necessary for the normal operation of the power plant, for example, it should be able to prevent the entry of suspended sediments;
- The topography of the site is appropriate for the production of hydroelectricity;
- The gross head is constant and that when dealing with low head schemes (2–30 m), since this assumption is no longer verified, an average value of the gross head measured is used;
- The numerical model focuses on small run-of-river hydropower plants with a capacity between 500 kW and 10 MW, therefore having low- and high-head projects (3–20 m and above 100 m) and for a configuration where the intake is directly linked to the penstock, as shown in Figure 1.

- Design of the penstock: calculation of its diameter, its thickness, and its aeration pipe diameter;
- Determination of the optimal number of turbines;
- Optimal selection of turbine: turbine type, admissible suction head, design flow rate, rated head, rated power, rotational speed, specific speed, and main dimensions of the turbine wheels;
- Choice of generator: type of generator, frequency, number of pairs of poles, rated power, and rotation speed;
- Choice of whether or not to use a speed increaser between the turbine and the generator;
- Estimate of the annual energy produced;
- Estimation of the investment, operating, and maintenance costs of the project over its lifetime;
- Calculation of economic criteria: LCC, LCOE, Net Present Value, and return on investment time;
- Choice of the technical solution with the best LCOE;
- Simulation of the operating mode of the sized run-of-river plant equipment (penstocks, turbines, and generators), thus making it possible to estimate the average daily flow rates driving each turbine; the number of turbines used each day of the year; the average daily speeds of water in the penstock; friction; singular and total average daily head losses in the penstock; the average daily net heads; and average daily energies produced.

#### 2.2. Definition of the Flow Rate of the Equipment

#### 2.3. Selection of Turbine and Generator Technology

#### 2.3.1. Turbine Case

_{QE}the specific speed of the turbine, g the acceleration of gravity in m/s

^{2}, H

_{rated}the rated head in m, and Q

_{design}the design flow rate of the turbine in m

^{3}/s.

_{QE}and the maximum admissible suction head H

_{s}, in the case of reaction turbines, can be calculated using Equations (2) and (3) [41]:

**Pelton turbines**

_{jet}the number of nozzles, H

_{rated}the rated head in m, and g the gravity constant in m/s

^{2}.

_{1}is defined as the diameter of the circle describing the bucket’s center line in m. B

_{2}is the bucket width in m, which is mainly dependent on the flow rate and the number of nozzles. D

_{e}is the nozzle diameter in m.

_{1}/B

_{2}>2.7 should always hold; otherwise, a new calculation with a lower rotational speed or a higher number of nozzles will have to be carried out [41].

**Francis turbines**

_{QE}> 0.164 [41]:

_{QE}: D

_{1}= D

_{2}.

**Kaplan turbines**

_{turbine}: Rated power of the turbine (kW);

_{turbine}: Rated efficiency of the turbine (-);

_{rated}: Rated head of the turbine (m);

_{min}: Minimal net head of the turbine (m);

_{gross}: Gross head of the turbine, which is the difference between the upstream and the downstream level for reaction turbines and also the difference between the upstream and the downstream level minus the height of the nozzle jet above the tail race, which is fixed at 1 m here for impulse turbines [15] (m);

_{design}: Design flow rate of the turbine (m

^{3}/s);

_{equipement}: Flow rate of equipment of the plant (m

^{3}/s);

_{t}: Number of turbines (-).

#### 2.3.2. Generator Case

_{generator}is the rated efficiency of the generator depending on its rated power, as can be seen in Table 5, and η

_{increaser}is the efficiency of the speed increaser set at 0.97 (as according to ESHA [41], its efficiency varies from 0.96 to 0.98) and η

_{turbine}is the rated efficiency of the turbine.

#### 2.4. Design of the Penstock

#### 2.4.1. Calculation of the Penstock Diameter

#### 2.4.2. Calculation of the Penstock Thickness

- Proposal of a first estimate of e
- Calculate c, h
_{max}_{,}and e_{min} - Compare e and e
_{min}: if e < e_{min}, then start again with a larger value of e and e > e_{min}. Attempt to make e equal to the available wall thickness that is closest to e_{min}, above or below, and repeat the calculation. - Repeat the two previous steps until the minimum available wall thickness is made to be greater than e
_{min} - e should be increased by 1.5 mm while dealing with mild steel pipes in order to take into consideration corrosion effects.

^{3}/s);

^{2});

^{2});

^{2});

^{2});

#### 2.4.3. Calculation of the Air Vent Pipe Diameter

^{2}):

^{2}.

#### 2.5. Turbine Operation Model

- In the first case (1), no turbine is in operation;
- In case (2), a turbine is in operation, and this is all the exploitable flow that is used;
- In case (3), (x + 1) turbines operate such that $Q=y.\left(x+1\right).{Q}_{max}$ with 0 $\le $ y $\le 1$ and (x + 1), which is the natural number directly greater than $Q/{Q}_{max}$ ($x\le Q/{Q}_{max}\le x+1$), if ${Q}_{min}/{Q}_{max}\le y$ or x turbines operate such that $Q=x.{Q}_{max}$ if $y<{Q}_{min}/{Q}_{max}$;
- In case (4), all the turbines are in rated operation, each of them exploiting the maximum flow that it can turbinate and the rest of the water flow ${Q}_{turbinable}-{n}_{t}{Q}_{max}$ being lost;
- In case (5), no turbine is operating.

^{3}/s, ${Q}_{exploited}$ is the exploited flow rate in m

^{3}/s, ${Q}_{available}$ is the available flow rate in m

^{3}/s, and ${Q}_{interannual}$ is the average daily flow rate available over the year in m

^{3}/s.

^{2});

^{2}/s);

^{3}/s).

#### 2.6. Annual Energy Production Estimation

#### 2.7. Economic Analysis

#### 2.7.1. Cost Estimate Model

- For a low-head project (3–20 m) [33]

^{3}/s, L is the length of the penstock in m, $\mathsf{\u011a}$ is Indian rupee/USD rate exchange and Prix

_{penstock}is the cost of the penstock in USD/ton.

- For a high-head project (more than 100 m) [32]

- The price for structural steel (including fabrication, transportation to site, and erection) is 75,000 Indian Rupee/MT;
- The price for M20 grade concrete work in plain cement concrete as well as in reinforced cement concrete, including shuttering, mixing, placing in position, compacting, and curing is 3640 Indian Rupee/m
^{3}; - The price for reinforcement steel bars of iron 500 grade, including cutting, bending, binding, and placing in position is 55,000 Indian Rupee/MT;
- The price for earthwork in excavation with all leads and lifts in ordinary soil is 265 Indian Rupee/m
^{3}; - The price for earthwork in excavation with all leads and lifts in soft rock, where blasting is not required is 330 Indian Rupee/m
^{3}; - The price for earthwork in excavation with all leads and lifts in hard rock, including blasting is 550 Indian Rupee/m
^{3}.

#### 2.7.2. Economic Criteria Calculation

- Life Cycle Cost (LCC)

- Levelized Cost of Energy (LCOE)

- Net Present Value (NPV)

- Return on investment time

#### 2.8. Optimization Method

## 3. Case Study

^{3}/s continuously, every day of the month). On the other hand, during the months of February, March, and April we have the lowest flows of the year, especially during March when the flows always remain continuously below 45 m

^{3}/s, every day of the month.

## 4. Results

#### 4.1. Optimal Design of the System

^{3}/s, a rated head of 4.8 m, a rated power of 1.68 MW, and a rated rotational speed of 210 rpm, each coupled to a 8 poles asynchronous generator with a rated power of 1.58 MW. Hence, leading to a total installed capacity of 6.32 MW. Since the speed of rotation of the turbine is different from that of the generator (which is 750 rpm), a speed increaser must be coupled between the turbine and the generator (Table 10 and Table 11).

^{3}/s and 309.17 m

^{3}/s (Table 13).

#### 4.2. Optimal System Operation over the Year

^{3}/s, thus making the exploitation of this watercourse particularly difficult during these months.

^{3}/s), is beyond which the safety of the plant is no longer guaranteed.

^{3}/s and 153.41 m

^{3}/s, respectively.

#### 4.3. Analysis of the Energy Efficiency of the Optimal System

^{3}/s per year (or hydraulic energy of about 11.05 GWh per year).

#### 4.4. Economic Analysis of the Optimal System

## 5. Conclusions and Future Scope

^{3}/s and 325.75 m

^{3}/s. Moreover, these results show that the proposed configuration is also very cost effective as its LCOE is around 0.05 USD/kWh (which is the lowest limit for the interval of LCOE for small hydropower plants as presented in the report of the IPCC). Its Net Present Value is USD 11.3 billion, or about 6.1 billion CFA Francs, assuming an energy price of 0.1 USD/kWh (or about 55 F CFA/ kWh), and its payback time is about 5 years 2 months—when it must be less than 7 years to be considered profitable according to ESHA— assuming a lifespan of 50 years.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Turbine application chart [46].

**Figure 7.**(

**a**) Average monthly available flow rates histogram, (

**b**) average daily available flow rates curve, and (

**c**) flow duration curve of the river Nyong (1998–2013).

**Figure 8.**(

**a**) Average monthly exploitable flow rates histogram and (

**b**) average daily exploitable flow rates curve.

**Figure 9.**(

**a**) Average monthly exploited flow rates histogram and (

**b**) average daily exploited flow rates curve.

**Figure 11.**(

**a**) Average monthly electrical power histogram and (

**b**) average daily electrical power curve.

**Figure 12.**(

**a**) Average monthly electrical energy histogram and (

**b**) average daily electrical energy curve.

**Figure 13.**(

**a**) Average monthly lost flow rates histogram and (

**b**) average daily lost flow rates curve.

**Figure 14.**(

**a**) Average monthly turbine efficiency histogram and (

**b**,

**c**) average daily turbine efficiency curve.

**Table 1.**Flow and head variation acceptance [41].

Turbine Type | Acceptance of Flow Variation | Acceptance of Head Variation |
---|---|---|

Pelton | High | Low |

Francis | Medium | Low |

Kaplan double regulated | High | High |

Kaplan single regulated | High | Medium |

Propeller | Low | Low |

**Table 2.**Ranges of the specific speed of each type of turbine [41].

Type of Turbine | Range |
---|---|

Pelton one nozzle | 0.005 ≤ ${n}_{QE}$ ≤ 0.025 |

Pelton n nozzles | 0.005 n^{0.5} ≤ ${n}_{QE}$ ≤ 0.025 n^{0.5} |

Francis | 0.05 ≤ ${n}_{QE}$ ≤ 0.33 |

Kaplan, propellers, bulbs | 0.19 ≤ ${n}_{QE}$ ≤ 1.55 |

Number of Poles | Speed for a Frequency of 50 Hz (rpm) |
---|---|

2 | 3000 |

4 | 1500 |

6 | 1000 |

8 | 750 |

10 | 600 |

12 | 500 |

14 | 428 |

16 | 375 |

18 | 333 |

20 | 300 |

22 | 272 |

24 | 250 |

26 | 231 |

28 | 214 |

**Table 4.**Typical efficiencies of small turbines [41].

Turbine Type | Best Efficiency |
---|---|

Kaplan single regulated | 0.91 |

Kaplan double regulated | 0.93 |

Francis | 0.94 |

Pelton n nozzles | 0.9 |

Pelton 1 nozzle | 0.89 |

Turgo | 0.85 |

**Table 5.**Typical efficiencies of small generators [41].

Rated Power | Best Efficiency |
---|---|

10 | 0.91 |

50 | 0.94 |

100 | 0.95 |

250 | 0.955 |

500 | 0.96 |

1000 | 0.97 |

**Table 6.**Maximum speed in penstock according to the head [47].

Head (m) | Maximum Speed (m/s) |
---|---|

Low head (50 < H) | 2–3 |

Medium head (50 ≤ H ≤ 250) | 3–4 |

High head (H > 250) | 4–5 |

**Table 7.**Correlations of civil works for run-of-river small hydropower plant scheme [51].

No. | Civil Works Components | Items | ||||
---|---|---|---|---|---|---|

Earth Work in Excavation (m^{3}), E/W | Concreting (m^{3}), Conc. | Reinforcement Steel (MT), RS | Structural Steel/Material (MT), SS | |||

1 | Diversion weir | $47.00{P}^{1.10}$ ${H}^{-0.99}$ | $38.55{P}^{1.17}$ ${H}^{-1.16}$ | $2.59{P}^{1.18}$ ${H}^{-1.15}$ | $1.51{P}^{0.71}$ ${H}^{-0.67}$ | |

2 | Penstock (per meter) | PVC pipe | $0.42{P}^{0.83}$ ${H}^{-0.98}$ | $0.31{P}^{0.84}$ ${H}^{-0.98}$ | $0.03{P}^{0.83}$ ${H}^{-0.97}$ | $0.04{P}^{1.5}$ ${H}^{-0.81}$ |

GRP pipe | $0.01{P}^{1.5}$ ${H}^{-0.85}$ | |||||

HDPE pipe | $0.08{P}^{1.5}$ ${H}^{-0.80}$ | |||||

Steel pipe | $0.05{P}^{0.83}$ ${H}^{-0.95}$ | |||||

3 | Tail race channel (per meter) | $0.91{P}^{0.83}$ ${H}^{-0.90}$ | $1.82{P}^{0.87}$ ${H}^{-0.91}$ | $0.01{P}^{0.80}$ ${H}^{-0.79}$ | - | |

4 | Powerhouse building | Pelton/Turgo impulse | $16.09{P}^{2.46}$ ${H}^{-1.75}$ | $0.00052{P}^{2.54}$ ${H}^{-0.42}$ | $0.00022{P}^{4.02}$ ${H}^{-2.83}$ | $0.09{P}^{4.55}$ ${H}^{-3.50}$ |

Francis | $0.08{P}^{2.33}$ ${H}^{-1.33}$ | $0.03{P}^{2.29}$ ${H}^{-1.32}$ | $0.05{P}^{2.36}$ ${H}^{-1.34}$ | $0.02{P}^{2.19}$ ${H}^{-1.26}$ |

**Table 8.**Coefficients in cost correlation for electromechanical equipment with different types [51].

No. | Type of Equipment | Coefficients in Cost Correlation | |||
---|---|---|---|---|---|

${\mathit{a}}_{1}$ | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | |||

1 | Turbine with governing system (TG) | Pelton | 117,313 | −0.03 | −0.39 |

Turgo impulse | 145,121 | −0.12 | −0.24 | ||

Francis | 125,354 | −0.01 | −0.38 | ||

2 | Generator with excitation system or capacitor bank (GE). | Induction | 130,262 | −0.19 | −0.22 |

Synchronous | 143,660 | −0.18 | −0.21 | ||

3 | Auxiliaries | 21,846 | −0.19 | −0.22 | |

4 | Transformer | 221 | 0.11 | 0.01 | |

5 | Switchyard | 1.82 | 0.17 | 0.93 |

Economic and Technical Parameters | Symbol | Value |
---|---|---|

Roughness height for welded steel | $\epsilon $ | 0.6 mm [41] |

Density of welded steel | ${\rho}_{a}$ | 7.9 ton/m^{3} (7.7–8 ton/m^{3}) [55] |

Kinematic viscosity of water (at 26 °C, average temperature of Mbalmayo over the year) | $\nu $ | 0.878 × 10^{−6} m^{2}/s [56] |

Density of water | $\rho $ | 1000 kg/m^{3} |

Gravitational acceleration | $g$ | 9.81 m/s^{2} |

Flow velocity of water at the outlet of the turbine | $V$ | 2 m/s [41] |

Bulk modulus of water | $k$ | 2.1 × 10^{9} N/mm^{2} [41] |

Young’s Modulus of Elasticity for welded steel | $E$ | 2.06 × 10^{11} N/m^{2} [41] |

Ultimate tensile strength of welded steel | $\sigma $ | 400 × 10^{6} N/mm^{2} [41] |

Energy price | $pric{e}_{energy}$ | 0.1 USD/kWh |

Operation and Maintenance cost coefficient | $Coe{f}_{O\&M}$ | 2.5 % [53] |

Indian rupee/ USD exchange rate | $\mathsf{\u011a}$ | 0.0136333 [57] |

Percentage of non-operated interannual flow | ${Q}_{residual}$ | 10%${Q}_{interannual}$ [41] |

Project lifespan | $T$ | 50 years |

Lifespan of civil works components | ${T}_{civil}$ | 50 years |

Lifespan of electromechanical equipment | ${T}_{electrom\xe9ca}$ | 25 years |

Generator efficiency | ${\eta}_{generator}$ | Table 5 [41] |

Average efficiency of the generator | $\eta {^}_{generator}$ | 0.9 |

Turbine efficiency | ${\eta}_{turbine}$ | Table 4 [41] |

Speed increaser efficiency | ${\eta}_{increaser}$ | 0.97 [41] |

Transformer efficiency | ${\eta}_{transformer}$ | 0.98 [49] |

Discount rate | $i$ | 12.5 % [58] |

Atmospheric pressure | ${H}_{a}$ | 10.3 mCE (101 000 Pa) [41] |

Vapor pressure of water (at 26 °C, average temperature of Mbalmayo over the year) | ${H}_{v}$ | 0.34 mCE (3 360 Pa) [59] |

Characteristics of the Turbine | |
---|---|

Turbine type | KAPLAN double regulated |

Design flow rate (m^{3}/s) | 38.35 |

Specific speed (-) | 1.21 |

Rotational speed (rpm) | 210 |

Rated head (m) | 4.8 |

Maximal suction head (m) | 0.34 |

The runner outer diameter_{t} (m) | 2.4 |

The runner hub diameter_{t} (m) | 0.79 |

Best efficiency (-) | 0.93 |

Rated power (MW) | 1.68 |

Number of turbines (-) | 4 |

Minimal flow rate (m^{3}/s) | 5.75 |

Maximal flow rate (m^{3}/s) | 38.35 |

Characteristics of the Generator | |
---|---|

Generator type | Asynchronous |

Number of poles | 8 |

Frequency (Hz) | 50 |

Rotational speed (rpm) | 750 |

Best efficiency (-) | 0.97 |

Rated power (MW) | 1.58 |

Characteristics of the Penstock | |
---|---|

Optimal diameter of the penstock (m) | 4.1 |

Minimal thickness of the penstock (mm) | 19.04 |

Length of the penstock (m) | 97.62 |

Air vent pipe diameter of the penstock (cm) | 55.36 |

Maximal total head losses in the penstock (m) | 0.2 |

Maximal friction losses in the penstock (m) | 0.133 |

Maximal singular losses in the penstock (m) | 0.067 |

Maximal speed of water in the penstock (m/s) | 2.9 |

Average speed of water in the penstock (m/s) | 2.4 |

Surge head in the penstock (m) | 353.88 |

Total head in the penstock (m) | 358.88 |

Wave speed (m/s) | 1447.46 |

Other Characteristics of the Project | |
---|---|

Flow rate of equipment (m^{3}/s) | 153.41 |

Gross head (m) | 5 |

Average available flow rate over the year (m^{3}/s) | 130.13 |

Residual flow rate (m^{3}/s) | 13.01 |

Average exploitable flow rate over the year (m^{3}/s) | 117.11 |

Safety flow rate (m^{3}/s) | 309.17 |

Energy Criteria | |
---|---|

Total installed capacity (MW) | 6.32 |

Productible (GWh) | 28.42 |

Water exploitation Index (%) | 78.04 |

Energy production Index (%) | 56.47 |

Average efficiency of the turbines over the year (-) | 0.88 |

Load Index (%) | 51.32 |

Economic Criteria | |
---|---|

LCC (MUSD) | 11.37 |

LCOE (USD/kWh) | 0.0502 |

Net Present Value (MUSD) | 11.30 |

Payback period (years) | 5 years 2 months |

Cost of civil works (MUSD) | 2.13 |

Cost of electromechanical equipment (MUSD) | 3.13 |

Miscellaneous and indirect cost (MUSD) | 1.09 |

Initial investment cost (MUSD) | 5.94 |

Replacement cost over the lifetime project (MUSD) | 3.54 |

Total investment cost over the lifetime project (MUSD) | 9.48 |

Operation and Maintenance cost per year (MUSD) | 0.237 |

**Table 16.**General values of LCOE for hydropower projects [60].

No. | Investment Cost (IC) (USD_{2005}/kW) | O&M Cost (% of IC) | Capacity Factor (%) | Lifetime (Years) | Discount Rate (%) | LCOE (cents/kWh) | Comments |
---|---|---|---|---|---|---|---|

1 | ˂500–6200 Median 1650 90% below 3250 | 41–61 | 2155 Projects in USA 43,000 MW in total Annual Capacity factor (except Rhode Island) | ||||

2 | ˂500–4500 Median 1000 90% below 1700 | 55–60 | 250 Projects for commissioning 2002–2020 Total Capacity 202,000 MW Worldwide but mostly Asia and Europe | ||||

3 | 1000–3500 700–8000 | 35–60 20–90 | 2–10 2–12 | Large Hydro Small Hydro (˂10 MW) (Not explicitly stated as levelized cost in report) | |||

4 | 2184 | 2.5 | 45 | 40 | 10 | 7.1 | |

5 | 1000–5500 2500–7000 | 2.2–3 | 10 10 | 3–12 5.6–14 | Large Hydro Small Hydro | ||

6 | 2880 in 2010 | 4 | 45 | 40 | 10 | 10.4 | |

7 | 2440 | 6 | 7.3 | Study applies to Germany only | |||

8 | 1000–5500 | 4 | 33 | 30 | 9.8 | Indicative average LCOE year 2000 | |

9 | 750–19,000 in 2010 (1278 average) | 51 | 80 80 | 2.3–45.9 4.8 | Range for 13 projects from 0.3 to 18,000 MW Weighted average for all projects | ||

10 | 5–12 3–5 5–40 | Small Hydro (˂10 MW) Large Hydro (˃10 MW) Off-Grid (˂1 MW) |

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**MDPI and ACS Style**

Amougou, C.B.; Tsuanyo, D.; Fioriti, D.; Kenfack, J.; Aziz, A.; Elé Abiama, P.
LCOE-Based Optimization for the Design of Small Run-of-River Hydropower Plants. *Energies* **2022**, *15*, 7507.
https://doi.org/10.3390/en15207507

**AMA Style**

Amougou CB, Tsuanyo D, Fioriti D, Kenfack J, Aziz A, Elé Abiama P.
LCOE-Based Optimization for the Design of Small Run-of-River Hydropower Plants. *Energies*. 2022; 15(20):7507.
https://doi.org/10.3390/en15207507

**Chicago/Turabian Style**

Amougou, Claude Boris, David Tsuanyo, Davide Fioriti, Joseph Kenfack, Abdoul Aziz, and Patrice Elé Abiama.
2022. "LCOE-Based Optimization for the Design of Small Run-of-River Hydropower Plants" *Energies* 15, no. 20: 7507.
https://doi.org/10.3390/en15207507