# A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Mathematical Model of the Hydrothermal System

_{H(i,j)}) is a rate function of water release and is given by:

#### 2.2. Cuckoo Search Algorithm and Lévy Flights

- Concept 1. Individually all cuckoos produce one single egg at a time that regards a proposed solution and randomly throws its egg up into the wanted nest among the set number of possible host nests.
- Concept 2. The egg of high quality thrown up in the best nests regards a better solution transferred to the subsequent generation.
- Concept 3. The possible number of host nests is constant, and the probability that the host bird can find a nest is indicated by the probability constant, Pa, with range [0,1]. Hence, it may either discard the egg or leave this nest and then build a new nest entirely in a different place.

#### 2.3. Rigid Cuckoo Search Algorithm

^{th}cuckoo, the Lévy flight shown in Equation (10) is achieved.

#### 2.4. Implementation of RCSA on a Hydrothermal System

- Step-1.
- In the StHS problem, the influential variables such as the release rate of water for the whole plants for several hours and thermal unit production for the entire period are chosen irregularly within the operating limitations. The storage capacity of every reservoir has been estimated using Equation (4), the generation of hydro plants has been calculated using Equation (3). Subsequently, the thermal power generation has been computed by applying Equation (2). The population of the host nest (N
_{E}) has been explained as:$$Y={\left[{Y}_{1},{Y}_{2},\dots ,{Y}_{{N}_{E}}\right]}^{L}$$_{i}is expressed as:$$Y=\left[\begin{array}{cccccccccc}{q}_{1,1}^{i}& \cdots & {q}_{1,j}^{i}& \cdots & {q}_{1,Nh}^{i}& {P}_{{T}_{1,1}}^{i}& \cdots & {P}_{{T}_{1,j}}^{i}& \cdots & {P}_{{T}_{1,Nh}}^{i}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {q}_{k,1}^{i}& \cdots & {q}_{k,j}^{i}& \cdots & {q}_{k,Nh}^{i}& {P}_{{T}_{k,1}}^{i}& \cdots & {P}_{{T}_{k,j}}^{i}& \cdots & {P}_{{T}_{1,Nh}}^{i}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {q}_{Nh,1}^{i}& \cdots & {q}_{Nh,j}^{i}& \cdots & {q}_{Nh,nh}^{i}& {P}_{{T}_{Nh,1}}^{i}& \cdots & {P}_{{T}_{Nh,j}}^{i}& \cdots & {P}_{{T}_{Nh,nh}}^{i}\end{array}\right]$$ - Step-2.
- Set the production number.
- Step-3.
- Compute the objective function using Equation (17). With the equation of restraints, many restrictions irregularly have been limited. Then, enhanced fuel cost has been calculated as Equation (19).$${F}^{**}={F}^{*}+{\displaystyle \sum}_{k=1}^{{N}_{C}}\left({\lambda}_{k}\times Vi{o}_{k}^{2}\right)$$
- Step-4.
- The modern solution has been created by using Levy flights. The new solution’s computation has been built in the preceding best nest by using Levy flights. For this technique, the optimal way for levy flights has been computed by Yang XS’s contribution, Deb S [18]. The new solution has been presented in Equation (20)$${Y}_{i,new}={Y}_{i,best}+\left(\alpha \times ran{d}_{2}\times \Delta {Y}_{i,new}\right)$$
_{2}is a typical number of allocated stochastic and ΔY_{i,new}has been computed, see Equations (21) and (22):$$\Delta {Y}_{i,new}=\epsilon \times \frac{{\rho}_{1}\left(\psi \right)}{{\rho}_{2}\left(\psi \right)}\times \left({Y}_{i,best}-{G}_{best}\right)$$$$\epsilon =\frac{ran{d}_{A}}{{\left|ran{d}_{B}\right|}^{\frac{1}{\psi}}}$$_{A}and rand_{B}are a couple of commonly allocated stochastic variables with a standard deviation ${\rho}_{1}\left(\psi \right)$ and ${\rho}_{2}\left(\psi \right)$ that has been determined by Equations (23) and (24), respectively.$${\rho}_{1}\left(\psi \right)={\left[\frac{\Phi \left(1+\psi \right)\times sin\left(\frac{\pi \psi}{2}\right)}{\Phi \left(\frac{1+\psi}{2}\right)\times \psi \times {2}^{\left(\frac{\psi -1}{2}\right)}}\right]}^{\frac{1}{\psi}}$$$${\rho}_{2}\left(\psi \right)=1$$ - Step-5.
- The effect of the detection of an alien egg in a nest of a host bird with Pa’s possibility produces a new solution for the problem comparable with the Levy flights. The new solution has been computed as following Equations (25)–(27):$$\Delta {Y}_{i,\text{}dis}={Y}_{i,best}+\left(k\times \Delta {Y}_{i,\text{}dis}\right)$$$$k=\{\begin{array}{ll}1& ifran{d}_{3}{P}_{a}\\ 0& otherwise\end{array}$$The increment value of $\Delta {Y}_{i,\text{}dis}$ has been defined by$$\Delta {Y}_{i,dis}=ran{d}_{3}\times \left[ran{d}_{{p}_{1}}\left({Y}_{d,best}\right)-ran{d}_{{p}_{2}}\left({Y}_{d,best}\right)\right]$$
- Step-6.
- The technique ends if modern production gives the maximum production number.

## 3. Results and Discussion

#### 3.1. Parameter Selection

_{E}) and the possibility of an alien egg to be found out, Pa. In contrast, the number of maximum iterations should have undeviatingly an effect on the optimal solution. Secondly, a couple of other parameters that influence joining the exploration aspect and exploitation aspects are considered. These should be satisfied with the upper and lower limitations and can be adjusted via the Lévy flights power. The obliged with the best solution provides the RCSA technique to enhance its performance and convergence speed. On the other side, the three major parameters from the authentic CSA technique, a couple of others in the justification, were effortless to be selected because they had been clarified by the previously limit equations. After several number of runs with various values of RCSA control parameter, the key control parameter chosen are population (Np) = 100, maximum iteration = 500 and value of probability (Pa) = 0.7.

#### 3.2. Obtained Results

#### 3.3. Proposed System Validation

^{7}, which is then starting to reduce until it gets to $2.5 × 10

^{7}. Additionally, the conventional CSA was employed in the same system but with various parameters and different specialized system constraints and fitness function (for example, the fixed-head). Here, the cost convergence characteristic was restricted from $2 × 10

^{11}to 0.1 × 10

^{11}[27]. T.T Nguyen used the modified CSA and D. N Vo in ref [29]; it was observed that the fitness function’s cost convergence started from $3.9 × 10

^{5}and got settled at $3.7 × 10

^{5}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

F | Gross cost of production |

F_{i}(P_{Tj}) | Production cost for P_{Tj} |

P_{Tj} | Production of power for the thermal unit at period j |

P_{H(i, j)} | Production of power for hydro unit i at period j |

P_{dj} | System load demand at period j |

q_{ij} | Water release rate of hydro unit i at period j |

r_{ij} | Stream rate into the storage reservoir of the hydro plant at a period |

s_{ij} | Spillage of the reservoir at a period |

X_{ij} | Storage volume of hydro plant i at period j |

X_{i}^{0} | Initial reservoir storage of hydro plant |

i | Number of units |

n_{h}, n_{p} | Maximum number of unit hydro/thermal |

j | Number of scheduling intervals |

N_{H}, N_{P} | Maximum number of scheduling periods of hydro/thermal |

N_{C} | Total number of constraints |

R_{j} | Resource constraint |

I_{T} | Total iteration numbers |

I_{C} | Current iterations |

λ_{k} | Penalty value for k^{th} |

Vio_{k} | Violation amount of k^{th} constraint |

ψ | Distribution factor |

Φ | Gamma distribution function |

m | Time period |

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**Figure 1.**Flow chart showing the steps used in the rigid cuckoo search algorithm (RCSA) approach for solving the short-term hydrothermal scheduling (StHS) problem.

P_{a} | Min Cost ($) | Avg. Cost ($) | Max Cost ($) | Std. dev. ($) | CPU (s) |
---|---|---|---|---|---|

0.1 | 709,932.115 | 709,995.586 | 710,655.745 | 327.189 | 17.9 |

0.2 | 709,922.44 | 710,768.874 | 711,384.563 | 599.382 | 17.9 |

0.3 | 709,911.445 | 709,994.412 | 710,989.321 | 489.733 | 17.8 |

0.4 | 709,866.727 | 709,745.236 | 709,999.741 | 103.937 | 18.1 |

0.5 | 709,886.651 | 709,887.698 | 709,988.258 | 47.653 | 18.6 |

0.6 | 709,862.129 | 709,899.987 | 709,991.951 | 54.512 | 18.5 |

0.7 | 709,862.027 | 709,878.852 | 709,996.159 | 59.661 | 18.2 |

0.8 | 709,901.478 | 709,910.357 | 710,920.357 | 478.225 | 18.3 |

0.9 | 709,902.685 | 709,901.753 | 710,901.753 | 471.185 | 18.7 |

m | P_{Dm} | V_{m}(acre-ft) | q_{m}(acre-ft/hr) | P_{sm}(MW) | P_{hm}(MW) |
---|---|---|---|---|---|

1 | 1199 | 101,897 | 1832 | 892 | 300 |

2 | 1497 | 85,959 | 3328 | 892 | 600 |

3 | 1098 | 93,847 | 1340 | 892 | 200 |

4 | 1795 | 59,998 | 4817 | 892 | 900 |

5 | 948 | 70,428 | 1124 | 783 | 158 |

6 | 1289 | 59,998 | 2863 | 783 | 509 |

Hours | Water Discharge (×10^{4} m^{3}) | Ps (MW) | |||
---|---|---|---|---|---|

Plant 1 | Plant 2 | Plant 3 | Plant 4 | ||

1 | 9.4884 | 6.1377 | 26.3477 | 13.1009 | 1036.249 |

2 | 9.4025 | 6.3104 | 25.2088 | 13.1024 | 1066.243 |

3 | 9.197 | 6.256 | 24.504 | 13.1035 | 1049.837 |

4 | 8.8096 | 6.3038 | 23.8832 | 13.1003 | 996.409 |

5 | 8.5269 | 6.2552 | 22.7287 | 13.1009 | 975.264 |

6 | 8.3314 | 6.4278 | 21.9238 | 13.1007 | 1076.828 |

7 | 8.4101 | 6.821 | 206226 | 13.1005 | 1294.982 |

8 | 8.6081 | 7.2309 | 19.2428 | 13.1023 | 1623.787 |

9 | 8.8072 | 7.3387 | 18.5775 | 13.1077 | 1849.655 |

10 | 8.465 | 7.5783 | 17.8006 | 13.1163 | 1917.3778 |

11 | 8.3429 | 7.7186 | 17.0931 | 13 | 1816.169 |

12 | 8.4121 | 7.8032 | 16.9969 | 13.4302 | 1885.493 |

13 | 8.2444 | 7.8624 | 16.3144 | 14.6742 | 1786.317 |

14 | 8.1224 | 8.0001 | 15.439 | 15.9371 | 1737.617 |

15 | 7.9417 | 8.1501 | 14.3596 | 17.2956 | 1652.815 |

16 | 7.8102 | 8.5518 | 12.9066 | 18.4244 | 1580.968 |

17 | 7.8153 | 8.9344 | 11.4353 | 20.0065 | 1630.883 |

18 | 7.6626 | 9.3193 | 10.1008 | 21.4157 | 1639.432 |

19 | 7.672 | 9.9655 | 10.1316 | 23.0205 | 1739.29 |

20 | 7.5945 | 10.5252 | 10.1043 | 24.2843 | 1787.795 |

21 | 7.3908 | 11.1875 | 12.1618 | 25.0985 | 1761.32 |

22 | 7.586 | 12.2327 | 12.747 | 25.0636 | 1659.735 |

23 | 7.5513 | 13.1386 | 13.2403 | 25.0994 | 1411.135 |

24 | 7.4103 | 14.2533 | 13.6662 | 25.0972 | 1177.636 |

**Table 4.**Comparison of the results achieved by the suggested RCSA technique with others for test system 1.

Techniques | Min Cost ($) | Average Cost ($) | Max Cost ($) | CPU Time (s) |
---|---|---|---|---|

GA [4] | 942,600 | 946,609.1 | 951,087 | 1920 |

EGA [5] | 934,727.00 | 936,058.00 | 937,339.00 | --- |

FEP [6] | 930,267.92 | 930,897.44 | 931,396.81 | 1911.2 |

CEP [6] | 930,166.25 | 930,373.23 | 930,927.01 | 2292.1 |

IFEP [6] | 930,129.82 | 930,290.13 | 930,881.92 | 1033.2 |

PSO [5] | 928,878.00 | 933,085.00 | 938,012.00 | --- |

CSA-Lévy [26] | 927,934.23 | 927,980.45 | 928,000.66 | 79.08 |

CSA Cauchy [26] | 927,967.66 | 927,981.49 | 927,992.53 | 81.30 |

CSA Gauss [26] | 927,957.26 | 927,978.911 | 928,003.23 | 85.75 |

APSO [7] | 926,151.54 | --- | --- | --- |

EPSO [5] | 922,904.00 | 923,527.00 | 924,808.00 | --- |

MDE [8] | 922,555.44 | --- | --- | --- |

IPSO [9] | 922,553.49 | --- | --- | --- |

MAPSO [7] | 922,421.66 | 922,544.00 | 923,508.00 | --- |

TLBO [10] | 922,373.39 | 922,462.24 | 922,873.81 | --- |

CSA [20] | 913,945.87 | 917,624.024 | 921,994.25 | --- |

RIFEP [12] | 709,862.05 | --- | --- | --- |

GS [13] | 709,877.38 | --- | --- | --- |

SA [14] | 709,874.36 | --- | --- | 901 |

ORCSA–Lévy flight [11] | 709,862.048 | --- | --- | 18 |

ORCSA–Cauchy [11] | 709,862.048 | --- | --- | 18 |

Proposed RCSA | 709,862.027 | --- | --- | 17 |

**Table 5.**Comparison of the results achieved by the suggested RCSA technique with others for test system 2.

Techniques | Min Cost ($) | Average Cost ($) | Max Cost ($) | CPU Time (s) |
---|---|---|---|---|

SA [27] | 47,306 | -- | -- | -- |

CEP [27] | 45,466 | -- | -- | -- |

CEP-IFS [25] | 45,036.00 | -- | -- | -- |

PSO [27] | 44,740 | -- | -- | -- |

DE [28] | 44,526.1 | -- | -- | 200 |

MDE [28] | 42,611.14 | -- | -- | 125 |

CSR [15] | 42,440 | -- | -- | 109 |

TLBO [10] | 42,385.88 | 42,407.23 | 42,441.36 | -- |

HDE [28] | 42,337.3 | -- | -- | 48 |

SQP [17] | 42,120.02 | -- | -- | 625.07 |

KHA [16] | 41,926 | -- | -- | -- |

MHDE [28] | 41,856.5 | -- | -- | 31 |

CSA [25] | 41,046.897 | -- | -- | 94.4 |

Proposed RCSA | 41,013.09 | 41,401.5 | 41,789.9 | 17 |

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## Share and Cite

**MDPI and ACS Style**

Zheyuan, C.; Hammid, A.T.; Kareem, A.N.; Jiang, M.; Mohammed, M.N.; Kumar, N.M.
A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem. *Sustainability* **2021**, *13*, 4277.
https://doi.org/10.3390/su13084277

**AMA Style**

Zheyuan C, Hammid AT, Kareem AN, Jiang M, Mohammed MN, Kumar NM.
A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem. *Sustainability*. 2021; 13(8):4277.
https://doi.org/10.3390/su13084277

**Chicago/Turabian Style**

Zheyuan, Cui, Ali Thaeer Hammid, Ali Noori Kareem, Mingxin Jiang, Muamer N. Mohammed, and Nallapaneni Manoj Kumar.
2021. "A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem" *Sustainability* 13, no. 8: 4277.
https://doi.org/10.3390/su13084277