An Evolutionary Computational Approach for Designing Micro Hydro Power Plants
Abstract
:1. Introduction
1.1. Micro-Hydro Power Plants
1.2. Traditional Design Procedure of MHPPs in Rural Emplacements
- Measurement of available flow rate, Q.
- Measurement of available height, .
- Decision-making of dam and powerhouse location.
- Estimation of power generation, P.
- Sizing of the equipment.
Contributions
2. Related Work
3. Problem Statement
3.1. Model of the System
3.1.1. Generated Power
3.1.2. Cost Function
3.1.3. Model of the MHPP Layout
- Each represents the placement of an elbow in point .
- The minimal index i that makes represents the location of the powerhouse.
- The maximum index i that makes represents the location of the water intake (dam).
3.2. Problem Formulation
3.2.1. Power Constraint
3.2.2. Flow Constraint
3.2.3. Feasibility Constraints
- The pipe can be disposed at a certain height from the terrain (see points 4 and 5 in Figure 4), where the use of supports is assumed, only if this height remains under a maximum value, .
- The pipe can be disposed under a certain depth from the terrain (see points 8 and 9 in Figure 4), where excavations are assumed as part of the civil works, only if this depth remains under a maximum value, .
4. Evolutionary Computational Approach
4.1. Single-Objective Optimization Problem
Algorithm 1: GA mupluslambda. |
4.1.1. Individual Representation
4.1.2. Fitness Function
4.1.3. Genetic Operators
4.2. Multi-Objective Optimization Problem
Algorithm 2: GA based on NSGA-II. |
4.2.1. Individual Representation
4.2.2. Fitness Function
4.2.3. Genetic Operators
5. Simulation Results
5.1. Example Scenario Settings
5.2. Genetic Algorithm Settings
5.3. Case-Studies Proposal
- Case-study 1: Fixed diameter, without considering elbows. For this first case, a fixed pipe diameter is considered. The problem is formulated in the form of a cost-minimization problem in single-objective mode. The objective function is formulated as minimizing the length of the penstock, L.
- Case-study 2: Fixed diameter, considering an equivalent cost of the elbows. This second case consists in the problem proposed in case-study 1 with the additional consideration of an equivalent cost for the pipe elbows, . Thus, the objective function is formulated as minimizing the cost C.
- Case-study 3: Variable diameter, single-objective. In this third case, the problem in case 2 is extended by considering a range of available discrete values for the pipe diameter , this being considered as an additional optimization variable (see Figure 5a). The problem is then formulated in the form of a cost-minimization problem (single-objective mode), with the objective function formulated as minimizing the cost of the penstock, defined in terms of its length L, number of elbows and diameter .
- Case-study 4: Variable diameter, multi-objective. The last case is formulated on the same basis of case-study 2, but considering a multi-objective mode, where the cost of the plant is minimized and the generated power P is maximized.
5.4. Results
5.4.1. Case-Study 1: Fixed Diameter, without Considering Elbows
5.4.2. Case-Study 2: Fixed Diameter, Considering an Equivalent Cost of the Elbows
5.4.3. Case-Study 3: Variable Diameter, Single-Objective
5.4.4. Case-Study 4: Variable Diameter, Multi-Objective
5.4.5. Comparison with Other Algorithms
5.4.6. Application to a Real Scenario
5.5. Discussion of the Results
- An example scenario domain has been introduced first, in order to propose a total of four case-studies that have been addressed in order to validate the benefits of the GA to optimize an MHPP layout.
- When the objective function is assumed to be the minimization of the length of the penstock, the GA provides approximately the same solution than the BBA, with a slight improvement (about 0.055%) in the length, and reaching also a lower number of elbows, although it is not considered a design variable in the minimization function.
- The minimization of the equivalent cost leads to noticeabley better solutions, reaching an improvement of 11.78% in the cost, reducing both the length (about 0.05%) and the number of elbows (20%) of the penstock.
- The GA has been proven to be especially effective addressing the optimization problem when considering the penstock diameter as an optimization variable, leading to a cost reduction of 70.67%. Note that this problem can not be addressed by BBA-based due to its complexity, which implies nonlinearities in its formulation.
- In all the studied cases, the convergence is reached in the first 50 generations, with small improvements until the 100th generation.
- The problem has been addressed in a multi-objective mode, minimizing the cost and maximizing the generated power. The solutions have been used to determine the Pareto front, and, from this, the marginal rate of substitution, which allows for studying how variances in the cost can affect the performance of the plant.
- The problem has also been solved using Random Search and SA algorithms. Although the results of the random search algorithm and SA are poor and far from the optimal solutions obtained with the GA, the implementation of the proposed mutation scheme in the SA provides a noticeable improvement in its performance. Nevertheless, it reaches a reduction of 99.66% of the cost obtained with the SA. This noticeable difference is a consequence of the high number of elbows obtained by the SA algorithm (20 elbows obtained from SA facing five elbows obtained using GA), while the difference in length between the solutions is not that high (481.3 m using SA facing 175 m from GA).
- The GA has been successfully used to solve a real scenario problem, proving a good robustness when dealing with a low quality sampled domain.
6. Conclusions and Further Work
Author Contributions
Funding
Conflicts of Interest
Abbreviations
BBA | Branch and Bound Algorithm |
CCS | Carbon Capture and Storage |
GA | Genetic Algorithm |
GHG | Greenhouse Gas |
HBMO | Honey Bee Mating Optimization |
MHPP | Micro Hydro Power Plants |
RES | Renewable Energy Sources |
SA | Simulated Annealing |
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Parameter | Value | Unit |
---|---|---|
8 | kW | |
70 | L/s | |
0.5 | - | |
50 | m | |
1.5 | m | |
1.5 | m |
Parameter | Value |
---|---|
2000 | |
2000 | |
Individuals (multi-objective) | 2000 |
Generations | 100 |
Selection | Tournament size = 3 (single-objective) NSGA-II (multi-objective) |
Crossover | Two-point scheme |
Mutation | Modified bitflip , , |
Number of trials | 30 |
Case | Objective Function | Pipe Diameter |
---|---|---|
Case-study 1 | min L (13) | Fixed |
Case-study 2 | min C (9) | Fixed |
Case-study 3 | min C (9) | Variable |
Case-study 4 | min C (9), max P (8) | Variable |
Genetic Algorithm | BBA [29] | |||
---|---|---|---|---|
0.80 | 0.70 | 0.60 | ||
0.20 | 0.30 | 0.40 | ||
Gross height (m) | 66.648 | 66.648 | 66.648 | 66.648 |
Flow rate (L/s) | 13.718 | 13.718 | 13.718 | 13.718 |
Power (kW) | 8.064 | 8.064 | 8.064 | 8.064 |
Number of elbows | 4 | 4 | 4 | 7 |
Length (m) | 174.903 | 174.903 | 174.903 | 174.999 |
Genetic Algorithm | BBA [29] | |||
---|---|---|---|---|
0.80 | 0.70 | 0.60 | ||
0.20 | 0.30 | 0.40 | ||
Gross height (m) | 66.648 | 66.648 | 66.648 | 66.648 |
Flow rate (L/s) | 13.718 | 13.718 | 13.718 | 13.718 |
Power (kW) | 8.039 | 8.039 | 8.039 | 8.039 |
Length (m) | 174.9240 | 174.9240 | 174.9240 | 175.005 |
Number of elbows | 4 | 4 | 4 | 5 |
Cost (m) | 14.997 | 14.997 | 14.997 | 17.000 |
Genetic Algorithm | BBA [29] | |||
---|---|---|---|---|
0.80 | 0.70 | 0.60 | ||
0.20 | 0.30 | 0.40 | ||
Gross height (m) | 78.919 | 115.642 | 115.642 | 66.648 |
Flow rate (L/s) | 13.7863 | 13.7127 | 13.7128 | 13.718 |
Power (kW) | 8.160 | 8.030 | 8.030 | 8.039 |
Length (m) | 310.754 | 429.114 | 429.103 | 175.005 |
Number of elbows | 5 | 7 | 7 | 5 |
Pipe diameter (cm) | 10 | 8 | 8 | 20 |
Cost (c.u.) | 5.608 | 4.986 | 4.986 | 17.000 |
Random Search | Simulated Annealing | Reference [29] | ||
---|---|---|---|---|
Bit Flip | Custom Mutation | |||
Gross height (m) | 226.192 | 216.005 | 123.513 | 66.648 |
Flow rate (L/s) | 14.601 | 14.491 | 13.819 | 13.718 |
Power (kW) | 9.695 | 9.477 | 8.217 | 8.039 |
Length (m) | 1159.8 | 1106.8 | 481.3 | 175.005 |
Number of elbows | 99 | 63 | 20 | 5 |
Pipe diameter (cm) | 8 | 8 | 8 | 20 |
Cost (c.u.) | 6109.8 | 4256.8 | 1481.3 | 17.000 |
Parameter | Value | Unit |
---|---|---|
8 | kW | |
50 | L/s | |
0.5 | - | |
50 | m | |
1.5 | m | |
1.5 | m |
Genetic Algorithm | |||
---|---|---|---|
0.80 | 0.70 | 0.60 | |
0.20 | 0.30 | 0.40 | |
Gross height (m) | 86.664 | 77.756 | 77.756 |
Flow rate (L/s) | 15.445 | 14.654 | 14.654 |
Power (kW) | 11.473 | 9.800 | 9.800 |
Length (m) | 536.247 | 471.740 | 471.740 |
Number of elbows | 8 | 5 | 5 |
Pipe diameter (cm) | 16 | 16 | 16 |
Cost (c.u.) | 23.968 | 18.4765 | 18.4765 |
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Tapia Córdoba, A.; Gutiérrez Reina, D.; Millán Gata, P. An Evolutionary Computational Approach for Designing Micro Hydro Power Plants. Energies 2019, 12, 878. https://doi.org/10.3390/en12050878
Tapia Córdoba A, Gutiérrez Reina D, Millán Gata P. An Evolutionary Computational Approach for Designing Micro Hydro Power Plants. Energies. 2019; 12(5):878. https://doi.org/10.3390/en12050878
Chicago/Turabian StyleTapia Córdoba, Alejandro, Daniel Gutiérrez Reina, and Pablo Millán Gata. 2019. "An Evolutionary Computational Approach for Designing Micro Hydro Power Plants" Energies 12, no. 5: 878. https://doi.org/10.3390/en12050878