# Application of Dynamic Non-Linear Programming Technique to Non-Convex Short-Term Hydrothermal Scheduling Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Objective Function

_{S}is indicator used for the number of thermal plants. Moreover, ${P}_{i}^{t}$ is power generated by the ith thermal plant at time t. ${a}_{i}$, ${b}_{i}$, and ${c}_{i\text{\hspace{0.17em}}}$ are the cost coefficients of ith thermal plant. Considering multiple steams valves in conventional thermal power plants, it is essential to model the effect of valve-points on fuel cost. Valve-points effect can be modeled by a sinusoidal term, which will be added to the quadratic cost function [27]. ${P}_{i}^{min}$ is minimum power generation of thermal unit i. Moreover, ${e}_{i}$ and ${f}_{i}$ are valve-point coefficients of cost function of thermal unit i.

#### 2.2. Power Balance Constraint

_{h}is the number of hydro units. ${P}_{j}^{t}$ is the generation of hydro units in megawatts (MW). Moreover, ${P}_{D}^{t}$ and ${P}_{L}^{t}$ are load demand and total transmission loss in MW, respectively. ${P}_{L}^{t}$ can be calculated using the Kron’s loss formula known as B-matrix coefficients [28]. Equation (3) calculates power transmission loss utilizing Kron’s loss formula, which is defined as B-matrix coefficients method in this paper as follows:

_{ij}, B

_{io}, and B

_{00}. In such formulation, B

_{mn}is element of matrix B with dimension of $({N}_{S}+{N}_{h})\times ({N}_{S}+{N}_{h})$. In addition, B

_{0n}is vector of the same length as P, and B

_{00}is considered as a constant.

^{3}, and ${C}_{1,j}$, ${C}_{2,j}$, ${C}_{3,j}$, ${C}_{4,j}$, ${C}_{5,j}$, and ${C}_{6,j}$ represent hydro power generation coefficients. Moreover, ${Q}_{j}^{t}$ is the water discharge amount in m

^{3}.

#### 2.3. Limitations of Power Production

#### 2.4. Hydraulic Network Constraints

#### 2.4.1. Water Dynamic Balance

^{th}reservoir. Additionally, τ is time delay of immediate downstream plants.

#### 2.4.2. Reservoir Storage Volume Limits

#### 2.4.3. Water Release Limits

#### 2.4.4. Initial and Final Reservoir Storage Volume

## 3. Solution Methodology

## 4. Case Studies and Simulation Results

#### 4.1. Test System 1

_{i}= 0.002, b

_{i}= 19.2, and c

_{i}= 5000. The lower and upper operation limits of this thermal plant are 500 and 2500 MW, respectively. Data of thermal unit and hydro plants are adopted from [25]. Two different cases including convex and non-convex cost function are studied for this test system.

#### 4.1.1. Test System 1 Case 1: Quadratic Cost without Valve-Point Loading Effect

#### 4.1.2. Test System 1 Case 2: Quadratic Cost Function with Valve-Point Loading

_{i}= 700 and f

_{i}= 0.085. The simulations are provided for case 2 with non-convex fuel cost. The optimal planning of discharge of four hydro units are reported in Table 3. In addition, power generation of hydro units, which is obtained by applying Equation (7), are provided in this table. In addition, power production of thermal power plants are presented in Table 3. It can be observed from Table 3 that the power demand during 24-h scheduling time is satisfied by total power generation of four hydro units and one thermal unit.

#### 4.2. Test System 2

#### 4.2.1. Test System 2, Case 1: Quadratic Cost without Valve-Point Loading Effect

#### 4.2.2. Test System 2 Case 2: Quadratic Cost Function with Valve-Point Loading

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

Indexes | |

t | Time interval of planning |

N_{s} | The number of thermal plants |

N_{h} | The number of hydro units |

Constants | |

${a}_{i}$, ${b}_{i}$ and ${c}_{i\text{\hspace{0.17em}}}$ | Cost coefficients of ith thermal plant |

${e}_{i}$ and ${f}_{i}$ | Valve-point coefficients of cost function of thermal unit i |

${P}_{i}^{\mathrm{min}}$ | Minimum power generation of thermal unit i |

${P}_{i}^{\mathrm{max}}$ | Maximum power generation of thermal unit i |

${V}_{j}^{\mathrm{min}}$ | Lower of operating volume of reservoir of ith hydro unit |

${V}_{j}^{\mathrm{max}}$ | Upper bounds of operating volume of reservoir of ith hydro unit |

${Q}_{j}^{\mathrm{min}}$ | Minimum release of water reservoir of the ith hydro plant |

${Q}_{j}^{\mathrm{max}}$ | Maximum release of water reservoir of the ith hydro plant |

${V}_{j}^{begin}$ | Elementary volume of reservoir |

${V}_{j}^{end}$ | Final volume of reservoir |

${P}_{D}^{t}$ | Load demand at time t |

${C}_{1,j}$, ${C}_{2,j}$, ${C}_{3,j}$, ${C}_{4,j}$, ${C}_{5,j}$, and ${C}_{6,j}$ | Hydro power generation coefficients |

${\varphi}_{j}$ | Set of instant upstream hydro plants of jth |

Variables | |

${P}_{i}^{t}$ | Power generated by the ith thermal plant at time t |

${P}_{j}^{t}$ | Generation of hydro units |

${P}_{L}^{t}$ | Total transmission loss at time t |

${V}_{j}^{t}$ | The storage volume of reservoir |

${Q}_{j}^{t}$ | The water discharge amount |

${I}_{j}^{t}$ | The inflow rate of the reservoir |

Acronyms | |

STHTS | Short-term hydro-thermal scheduling |

DNLP | Dynamic non-linear programming |

MGS | Micro-grids |

DGS | Distributed generations |

DSM | Direct search method |

ED | Economic dispatch |

MABC | Modified artificial bee colony |

DE | Differential evolution |

MDNLPSO | Modified dynamic neighborhood learning based particle swarm optimization |

QADEVT | Quadratic approximation based on differential evolution with valuable trade-off |

PPO | Predator prey optimization |

PSO | Particle swarm optimization |

RCGA | Real coded genetic algorithm |

LR | Lagrangian relaxation |

MIP | Mixed integer programming |

IMO | Improved merit order |

ALHN | Augmented Lagrangian hopfield network |

UC | Unit commitment |

BFPSO | Bacterial foraging oriented by particle swarm optimization |

DNLP | Dynamic non-linear programming |

GAMS | General algebraic modeling system |

LP | Linear programming |

NLP | Nonlinear programming |

MILP | Mixed-integer linear programming |

MINLP | Mixed-integer nonlinear programming |

DNLP | Dynamic nonlinear programming |

QEA | Quantum-inspired evolutionary algorithm |

WDA | Whole distribution algorithm |

SPSO | Small population-based particle swarm optimization |

RQEA | Real-coded quantum-inspired evolutionary algorithm |

MDE | Modified differential evolution |

DRQEA | Differential real-coded quantum-inspired evolutionary algorithm |

DNLPSO | Dynamic neighborhood learning based particle swarm optimization |

HCRO | Hybrid chemical reaction optimization |

MAPSO | Modified adaptive particle swarm optimization |

RCGA-AFSA | Real coded genetic algorithm and artificial fish swarm algorithm |

TLBO | Teaching learning-based optimization |

SOHPSO_TVAC | Self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients |

IDE | Improved differential evolution |

FAPSO | Fuzzy adaptive particle swarm optimization |

ACDE | Adaptive chaotic differential evolution |

CABC | Adaptive chaotic artificial bee colony |

CSA | Clonal selection algorithm |

IQPSO | Improved quantum-behaved particle swarm optimization |

GSO | Group search optimization |

ACDE | Adaptive chaotic differential evolution algorithm |

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**Table 1.**Hourly plant discharges, power outputs and total thermal generation (test system 1, case 1).

Hour | Hydro Plant Discharges (10^{4} m^{3}) | Hydro Power Output (megawatts (MW)) | Thermal Generation (MW) | Total Generation (MW) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Plant 1 | Plant 2 | Plant 3 | Plant 4 | Plant 1 | Plant 2 | Plant3 | Plant 4 | |||

1 | 6.254 | 6.000 | 11.632 | 15.344 | 61.528 | 45.316 | 56.480 | 224.231 | 982.445 | 1370 |

2 | 6.488 | 6.000 | 11.914 | 16.919 | 64.339 | 46.576 | 55.928 | 219.694 | 1003.464 | 1390 |

3 | 6.594 | 6.000 | 12.303 | 18.537 | 65.777 | 47.804 | 56.376 | 209.020 | 981.024 | 1360 |

4 | 6.592 | 6.000 | 12.741 | 20.000 | 66.102 | 49.586 | 57.320 | 189.900 | 927.092 | 1290 |

5 | 6.431 | 6.000 | 13.129 | 20.000 | 64.972 | 51.296 | 57.694 | 306.000 | 810.039 | 1290 |

6 | 6.617 | 6.000 | 13.531 | 20.000 | 66.284 | 52.396 | 58.133 | 306.000 | 927.187 | 1410 |

7 | 7.005 | 6.000 | 13.923 | 20.000 | 69.238 | 52.934 | 58.611 | 306.000 | 1163.217 | 1650 |

8 | 7.545 | 6.000 | 14.254 | 20.000 | 73.259 | 52.934 | 58.728 | 306.000 | 1509.079 | 2000 |

9 | 7.927 | 6.000 | 14.568 | 20.000 | 76.166 | 53.464 | 58.565 | 306.000 | 1745.805 | 2240 |

10 | 8.109 | 6.000 | 14.894 | 20.000 | 77.851 | 54.500 | 58.136 | 306.000 | 1823.512 | 2320 |

11 | 8.087 | 6.000 | 15.277 | 20.000 | 78.367 | 55.994 | 57.612 | 306.000 | 1732.027 | 2230 |

12 | 8.272 | 6.000 | 15.621 | 20.000 | 80.353 | 57.416 | 57.056 | 306.000 | 1809.175 | 2310 |

13 | 8.165 | 6.254 | 16.047 | 20.000 | 79.963 | 60.180 | 56.475 | 306.000 | 1727.382 | 2230 |

14 | 8.124 | 6.613 | 16.494 | 20.000 | 80.131 | 63.512 | 56.112 | 306.000 | 1694.245 | 2200 |

15 | 8.043 | 6.927 | 16.939 | 20.000 | 80.074 | 66.727 | 55.351 | 306.000 | 1621.848 | 2130 |

16 | 7.930 | 7.272 | 17.137 | 20.000 | 79.565 | 69.947 | 54.948 | 306.000 | 1559.540 | 2070 |

17 | 7.950 | 7.670 | 15.694 | 20.000 | 79.858 | 72.868 | 57.561 | 306.000 | 1613.712 | 2130 |

18 | 7.768 | 7.950 | 14.281 | 20.000 | 78.597 | 74.372 | 59.277 | 306.000 | 1621.754 | 2140 |

19 | 7.662 | 8.374 | 12.888 | 20.000 | 77.830 | 76.131 | 60.031 | 306.000 | 1720.008 | 2240 |

20 | 7.452 | 8.751 | 18.733 | 20.000 | 76.216 | 77.728 | 51.940 | 306.000 | 1768.116 | 2280 |

21 | 7.063 | 15.000 | 19.145 | 20.000 | 73.145 | 101.607 | 49.988 | 303.055 | 1712.205 | 2240 |

22 | 11.991 | 15.000 | 19.676 | 20.000 | 101.750 | 98.082 | 47.637 | 298.534 | 1573.998 | 2120 |

23 | 11.931 | 15.000 | 20.368 | 20.000 | 100.691 | 94.269 | 44.320 | 292.356 | 1318.364 | 1850 |

24 | 15.000 | 15.000 | 13.133 | 20.000 | 107.020 | 80.950 | 59.005 | 284.400 | 1058.625 | 1590 |

**Table 2.**Comparisons of simulation results for test system 1, case 1. Employing quantum-inspired evolutionary algorithm (QEA); quantum-inspired evolutionary algorithm (WDA); small population-based particle swarm optimization (SPSO); real-coded genetic algorithm (RCGA); real-coded quantum-inspired evolutionary algorithm (RQEA); modified differential evolution (MDE); differential real-coded quantum-inspired evolutionary algorithm (DRQEA); hybrid chemical reaction optimization-differential evolution (HCRO-DE); modified adaptive particle swarm optimization (MAPSO); real coded genetic algorithm and artificial fish swarm algorithm (RCGA-AFSA); teaching learning-based optimization (TLBO); SPPSO; self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients (SOHPSO_TVAC); particle swarm optimization (PSO); improved differential evolution (IDE); fuzzy adaptive particle swarm optimization (FAPSO); dynamic neighborhood learning based particle swarm optimization (DNLPSO) and; modified dynamic neighborhood learning based particle swarm optimization (MDNLPSO).

Optimization Method | Min. Cost ($) | Max. Cost ($) | Ave. Cost ($) |
---|---|---|---|

QEA [25] | 926,538.29 | 930,484.13 | 928,426.95 |

WDA [32] | 925,618.5 | - | 928,219.8 |

SPSO [33] | 925,308.86 | 923,083.48 | 926,185.32 |

RCGA [34] | 923,966.285 | 924,108.731 | 924,232.072 |

RQEA [25] | 923,634.53 | 926,957.39 | 924,992.46 |

DE [25] | 923,234.56 | 928,395.84 | 925,157.28 |

MDE [33] | 922,556.38 | 923,201.13 | 923,813.99 |

DRQEA [25] | 922,526.73 | 925,871.51 | 923,419.37 |

DNLPSO [15] | 922,498 | 923,580 | 922,837 |

HCRO [35] | 922,444.79 | 922,513.62 | 922,936.17 |

MAPSO [36] | 922,421.66 | 923,508 | 922,544 |

RCGA-AFSA [34] | 922,339.625 | 922,346.323 | 922,362.532 |

TLBO [37] | 922,373.39 | 922,873.81 | 922,462.24 |

SPPSO [33] | 922,336.31 | 922,362.532 | 923,083.48 |

SOHPSO_TVAC [38] | 922,018.24 | - | - |

PSO [39] | 921,920 | - | - |

IDE [40] | 917,237.7 | 917,277.8 | 917,250.1 |

FAPSO [39] | 914,660 | - | - |

Proposed method | 884,733.965 | - | - |

**Table 3.**Hourly plant discharges, power outputs and total thermal generation (test system 1, case 2).

Hour | Hydro Plant Discharges (10^{4} m^{3}) | Hydro Power Output (MW) | Thermal Generation (MW) | Total Generation (MW) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

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

1 | 5.212 | 6.000 | 10.730 | 13.080 | 53.156 | 45.316 | 55.550 | 198.539 | 1017.439 | 1370 |

2 | 5.555 | 6.487 | 10.976 | 16.211 | 57.199 | 49.935 | 55.337 | 210.090 | 1017.439 | 1390 |

3 | 5.148 | 6.000 | 10.252 | 15.327 | 54.329 | 47.508 | 54.789 | 185.936 | 943.519 | 1360 |

4 | 5.023 | 6.000 | 10.000 | 19.571 | 53.613 | 49.302 | 55.220 | 188.347 | 832.639 | 1290 |

5 | 5.052 | 6.000 | 17.571 | 20.000 | 54.126 | 51.024 | 52.771 | 299.440 | 943.519 | 1290 |

6 | 5.150 | 6.000 | 10.000 | 19.615 | 55.150 | 52.133 | 55.526 | 303.672 | 1168.561 | 1410 |

7 | 5.605 | 6.574 | 12.835 | 20.000 | 59.407 | 56.704 | 59.328 | 306.000 | 1586.197 | 1650 |

8 | 5.257 | 6.196 | 12.065 | 12.500 | 56.689 | 53.757 | 59.195 | 244.162 | 1756.637 | 2000 |

9 | 5.613 | 6.661 | 12.694 | 20.000 | 60.242 | 57.427 | 59.693 | 306.000 | 1762.699 | 2240 |

10 | 13.476 | 13.306 | 13.137 | 19.485 | 104.423 | 90.251 | 59.762 | 302.866 | 1719.677 | 2320 |

11 | 11.589 | 6.021 | 10.104 | 19.739 | 98.015 | 51.320 | 56.792 | 304.196 | 1793.597 | 2230 |

12 | 11.752 | 6.122 | 11.916 | 19.739 | 98.732 | 53.678 | 59.620 | 304.373 | 1740.260 | 2310 |

13 | 9.254 | 6.170 | 21.626 | 20.000 | 86.633 | 55.018 | 42.089 | 306.000 | 1756.637 | 2230 |

14 | 8.274 | 6.137 | 27.943 | 19.921 | 80.932 | 55.725 | 1.827 | 304.880 | 1675.480 | 2200 |

15 | 5.000 | 6.000 | 21.954 | 19.476 | 55.068 | 56.152 | 40.489 | 302.810 | 1608.797 | 2130 |

16 | 5.320 | 6.544 | 21.608 | 20.000 | 58.135 | 61.472 | 41.271 | 300.325 | 1645.757 | 2070 |

17 | 5.234 | 8.595 | 14.157 | 19.978 | 57.184 | 75.322 | 60.241 | 291.496 | 1645.757 | 2130 |

18 | 7.821 | 6.640 | 14.810 | 20.000 | 79.151 | 62.077 | 59.593 | 293.422 | 1719.677 | 2140 |

19 | 8.424 | 9.199 | 10.298 | 19.987 | 83.520 | 77.592 | 57.383 | 301.828 | 1793.597 | 2240 |

20 | 10.474 | 11.904 | 27.516 | 19.607 | 96.000 | 88.807 | 6.038145 | 301.595 | 1674.858 | 2280 |

21 | 14.459 | 14.871 | 10.000 | 19.888 | 109.205 | 94.848 | 55.758 | 305.331 | 1645.757 | 2240 |

22 | 8.381 | 14.998 | 27.296 | 19.993 | 82.343 | 91.135 | 0.747 | 300.017 | 1387.038 | 2120 |

23 | 8.425 | 14.996 | 26.778 | 19.399 | 82.203 | 86.813 | 2.930 | 291.016 | 1096.580 | 1850 |

24 | 15.000 | 6.721 | 13.133 | 19.190 | 107.020 | 47.557 | 59.005 | 279.837 | 1058.625 | 1590 |

Optimization Method | Min. Cost ($) |
---|---|

QEA [25] | 930,647.96 |

DE [25] | 928,662.84 |

RCGA-AFSA [34] | 927,899.872 |

RQEA [25] | 926,068.33 |

DRQEA [25] | 925,485.21 |

CRQEA [25] | 925,403.1 |

RCCRO [41] | 925,214.20 |

ACDE [42] | 924,661.53 |

MAPSO [36] | 924,636 |

TLBO [37] | 924,550.78 |

RCGA [34] | 923,966.285 |

RQEA [25] | 923,634.53 |

DE [25] | 923,234.56 |

MDE [33] | 922,556.38 |

DRQEA [25] | 922,526.73 |

HCRO-DE [35] | 922,444.79 |

MAPSO [36] | 922,421.66 |

MDNLPSO [15] | 923,961 |

IDE [40] | 923,016.29 |

TLBO [37] | 922,373.39 |

RCGA-AFSA [34] | 922,339.625 |

SPPSO [33] | 922,336.31 |

SOHPSO_TVAC [38] | 922,018.24 |

PSO [39] | 921,920 |

Improved DE [40] | 917,250.1 |

IDE [40] | 917,237.7 |

FAPSO [39] | 914,660.00 |

Proposed method | 901,191.9735 |

Hour | Hydro Plant Discharges (10^{4} m^{3}) | Hydro Power Output (MW) | Thermal Power Output (MW) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

1 | 5.940 | 6.000 | 11.919 | 15.335 | 59.103 | 45.317 | 54.084 | 224.158 | 102.673 | 124.908 | 139.758 |

2 | 6.699 | 7.766 | 14.024 | 20.000 | 65.989 | 58.072 | 52.592 | 236.005 | 102.673 | 124.908 | 139.760 |

3 | 6.864 | 6.002 | 16.358 | 13.099 | 67.796 | 46.733 | 48.618 | 169.512 | 102.673 | 124.908 | 139.760 |

4 | 5.000 | 6.000 | 20.251 | 12.102 | 53.000 | 48.545 | 34.952 | 146.162 | 102.673 | 124.908 | 139.760 |

5 | 5.000 | 6.000 | 19.354 | 7.486 | 53.361 | 50.298 | 36.816 | 162.185 | 102.673 | 124.908 | 139.760 |

6 | 5.562 | 6.000 | 18.081 | 20.000 | 58.435 | 51.426 | 41.026 | 281.773 | 102.673 | 124.908 | 139.760 |

7 | 8.717 | 7.235 | 10.050 | 9.521 | 81.434 | 60.381 | 48.532 | 189.953 | 175.000 | 165.181 | 229.519 |

8 | 6.793 | 6.000 | 10.000 | 12.002 | 68.402 | 51.295 | 49.484 | 226.483 | 175.000 | 209.816 | 229.520 |

9 | 8.278 | 6.984 | 10.000 | 17.853 | 79.131 | 58.617 | 50.361 | 287.556 | 175.000 | 209.816 | 229.520 |

10 | 9.721 | 6.901 | 10.000 | 15.166 | 87.819 | 58.640 | 51.683 | 267.523 | 175.000 | 209.816 | 229.520 |

11 | 9.980 | 7.651 | 10.000 | 16.174 | 89.468 | 64.626 | 52.759 | 278.812 | 175.000 | 209.816 | 229.519 |

12 | 11.526 | 9.682 | 12.521 | 19.991 | 96.739 | 76.822 | 56.159 | 305.945 | 175.000 | 209.816 | 229.519 |

13 | 8.830 | 8.827 | 18.363 | 19.431 | 83.318 | 71.352 | 48.797 | 292.197 | 175.000 | 209.816 | 229.519 |

14 | 8.337 | 6.894 | 17.750 | 12.235 | 80.723 | 59.110 | 51.359 | 224.472 | 175.000 | 209.816 | 229.519 |

15 | 7.791 | 6.297 | 19.374 | 13.639 | 77.682 | 56.137 | 47.225 | 236.017 | 175.000 | 209.812 | 208.126 |

16 | 5.474 | 7.836 | 10.000 | 17.345 | 59.431 | 67.731 | 56.316 | 262.186 | 175.000 | 209.816 | 229.519 |

17 | 8.555 | 7.625 | 10.000 | 12.243 | 83.821 | 66.503 | 57.468 | 227.928 | 174.999 | 209.816 | 229.465 |

18 | 9.628 | 9.144 | 10.000 | 17.593 | 90.452 | 75.044 | 58.138 | 282.031 | 175.000 | 209.816 | 229.519 |

19 | 8.801 | 8.764 | 10.000 | 12.800 | 85.273 | 71.213 | 58.453 | 240.726 | 175.000 | 209.816 | 229.520 |

20 | 5.000 | 8.951 | 21.305 | 14.379 | 55.099 | 71.229 | 46.410 | 262.927 | 175.000 | 209.816 | 229.520 |

21 | 7.167 | 12.825 | 25.403 | 8.680 | 73.741 | 87.127 | 22.093 | 197.616 | 175.000 | 124.903 | 229.520 |

22 | 10.381 | 13.611 | 26.991 | 19.652 | 94.074 | 86.783 | 8.525 | 303.277 | 102.673 | 124.908 | 139.760 |

23 | 9.956 | 15.000 | 26.187 | 19.731 | 91.469 | 86.389 | 11.133 | 293.668 | 102.673 | 124.908 | 139.760 |

24 | 7.979 | 7.656 | 13.143 | 14.001 | 79.296 | 53.388 | 59.005 | 240.969 | 102.673 | 124.908 | 139.760 |

Optimization Method | Min. Cost | Mean. Cost | Max. Cost |
---|---|---|---|

SA [44] | 45,466 | - | - |

DE [32] | 44,526.11 | - | - |

CABC [43] | 43,362.68 | - | - |

ADE [43] | 43,222.41 | - | - |

RCGA [34] | 42,886.352 | 43,261.912 | 43,032.334 |

DE [25] | 42 801.04 | - | - |

SPPSO [33] | 42,740.23 | 43,622.14 | 44,346.97 |

RQEA [25] | 42 715.69 | - | - |

PSO [45] | 42,474.00 | - | - |

CDE [42] | 42,452.99 | - | - |

CSA [46] | 42,440.574 | - | - |

TLBO [47] | 42,386.13 | 42,407.23 | 42,441.36 |

TLBO [33] | 42,385.88 | 42,407.23 | 42,441.36 |

IQPSO [48] | 42,359.00 | - | - |

GSO [49] | 42,316.39 | 42,339.35 | 42,379.18 |

QTLBO [47] | 42,187.49 | 42,193.46 | 42,202.75 |

QOGSO [49] | 42,120.02 | 42,130.15 | 42,145.37 |

IDE [40] | 41,856.5 | - | - |

ACDE [42] | 41,593.48 | - | - |

RCCRO [41] | 41,497.85 | 41,498.21 | 41,502.36 |

DRQEA [25] | 41,435.76 | - | - |

ACABC [43] | 41,274.42 | - | - |

Proposed method | 41,101.738 | - | - |

**Table 7.**Hourly plant discharges, power outputs and total thermal generation for test system 2 case 2.

Hour | Hydro Plant Discharge, 10^{4} m^{3} | Hydro Plant Generation (MW) | Thermal Plant Generation (MW) | Loss MW | Total Generation, MW | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

1 | 6.224 | 7.064 | 12.991 | 6.797 | 61.232 | 52.277 | 56.939 | 130.234 | 102.511 | 124.322 | 229.518 | 7.033 | 757.033 |

2 | 7.280 | 8.237 | 13.107 | 19.956 | 69.961 | 59.942 | 55.806 | 237.968 | 102.673 | 124.908 | 139.759 | 11.017 | 791.017 |

3 | 5.092 | 6.000 | 12.818 | 5.327 | 53.354 | 45.533 | 55.918 | 188.300 | 102.674 | 122.365 | 139.760 | 7.903 | 707.903 |

4 | 5.000 | 6.000 | 14.572 | 18.111 | 53.119 | 47.349 | 55.882 | 184.839 | 95.539 | 80.854 | 139.759 | 7.341 | 657.341 |

5 | 5.000 | 6.000 | 11.545 | 6.000 | 53.470 | 49.149 | 56.573 | 149.761 | 102.657 | 124.908 | 139.760 | 6.278 | 676.278 |

6 | 5.000 | 6.000 | 15.314 | 16.648 | 53.632 | 50.309 | 56.596 | 280.372 | 102.674 | 124.908 | 145.345 | 13.836 | 813.836 |

7 | 5.393 | 6.000 | 16.445 | 17.907 | 57.420 | 50.877 | 54.608 | 291.657 | 157.940 | 124.907 | 229.519 | 16.928 | 966.928 |

8 | 5.981 | 6.000 | 10.000 | 18.064 | 62.814 | 50.877 | 54.961 | 292.422 | 175.000 | 162.192 | 229.520 | 17.786 | 1027.786 |

9 | 8.635 | 6.213 | 16.133 | 19.707 | 82.533 | 52.949 | 55.354 | 304.242 | 175.000 | 209.815 | 229.520 | 19.413 | 1109.413 |

10 | 8.277 | 6.156 | 17.258 | 18.833 | 80.533 | 53.535 | 51.892 | 298.668 | 175.000 | 209.816 | 229.519 | 18.963 | 1098.963 |

11 | 9.990 | 7.668 | 10.111 | 18.233 | 91.041 | 65.154 | 54.262 | 294.077 | 175.000 | 209.816 | 229.519 | 18.870 | 1118.870 |

12 | 14.072 | 11.849 | 13.399 | 20.000 | 105.690 | 86.772 | 57.662 | 306.000 | 175.000 | 209.815 | 229.520 | 20.458 | 1170.458 |

13 | 9.023 | 6.013 | 20.356 | 13.234 | 85.283 | 52.690 | 42.348 | 246.964 | 175.000 | 294.724 | 229.520 | 16.529 | 1126.529 |

14 | 7.224 | 7.208 | 18.028 | 18.421 | 73.582 | 61.938 | 50.867 | 295.082 | 175.000 | 162.205 | 229.519 | 18.193 | 1048.193 |

15 | 8.031 | 7.651 | 16.110 | 18.682 | 79.963 | 65.732 | 57.051 | 295.698 | 175.000 | 124.908 | 229.519 | 17.872 | 1027.872 |

16 | 8.314 | 8.539 | 19.347 | 20.000 | 82.214 | 71.786 | 50.242 | 294.907 | 175.000 | 209.816 | 194.414 | 18.379 | 1078.379 |

17 | 9.238 | 9.661 | 25.080 | 16.986 | 88.314 | 77.558 | 18.529 | 268.360 | 175.000 | 209.815 | 229.520 | 17.097 | 1067.097 |

18 | 10.245 | 11.054 | 10.689 | 20.000 | 93.953 | 82.465 | 57.043 | 291.376 | 175.000 | 209.816 | 229.519 | 19.172 | 1139.172 |

19 | 8.849 | 9.962 | 10.000 | 15.111 | 85.629 | 74.482 | 56.903 | 254.959 | 175.000 | 209.815 | 229.519 | 16.308 | 1086.308 |

20 | 8.546 | 9.668 | 21.651 | 15.281 | 83.461 | 71.185 | 41.508 | 257.413 | 175.000 | 208.174 | 229.519 | 16.260 | 1066.260 |

21 | 7.029 | 9.978 | 23.588 | 20.000 | 72.399 | 71.584 | 30.674 | 294.843 | 102.674 | 124.908 | 229.520 | 16.601 | 926.601 |

22 | 6.077 | 11.227 | 26.324 | 15.162 | 64.681 | 76.310 | 11.250 | 265.100 | 102.673 | 124.907 | 229.520 | 14.442 | 874.442 |

23 | 11.224 | 13.406 | 24.422 | 20.000 | 97.745 | 81.821 | 23.040 | 295.500 | 102.673 | 124.908 | 139.760 | 15.448 | 865.448 |

24 | 15.000 | 15.000 | 13.133 | 9.594 | 107.020 | 80.950 | 59.005 | 194.843 | 102.674 | 124.907 | 139.757 | 9.157 | 809.157 |

Optimization Method | Min. Cost |
---|---|

QEA [25] | 44,686.31 |

ABC [43] | 43,362.00 |

QOGSO [49] | 43,560.35 |

DE [32] | 42,801.04 |

SPPSO [50] | 42,740.23 |

RQEA [25] | 42,715.69 |

DNLPSO [15] | 42,645 |

PSO [51] | 42,474.00 |

CSA [44] | 42,440.574 |

TLBO [33] | 42,386.13 |

SA-MOCDE [37] | 42,038.00 |

GSA [52] | 42,032.35 |

QOTLBO [33] | 42,187.49 |

MOCA-PSO [53] | 42,001.00 |

SHPSO-TAC [51] | 41,983.00 |

IDE [40] | 41,856.5 |

RCGA-AFSA [34] | 41,818.42 |

QABDEVT [16] | 41,762.00 |

ACDE [54] | 41,593.48 |

Proposed method | 41,350.5574 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hoseynpour, O.; Mohammadi-ivatloo, B.; Nazari-Heris, M.; Asadi, S.
Application of Dynamic Non-Linear Programming Technique to Non-Convex Short-Term Hydrothermal Scheduling Problem. *Energies* **2017**, *10*, 1440.
https://doi.org/10.3390/en10091440

**AMA Style**

Hoseynpour O, Mohammadi-ivatloo B, Nazari-Heris M, Asadi S.
Application of Dynamic Non-Linear Programming Technique to Non-Convex Short-Term Hydrothermal Scheduling Problem. *Energies*. 2017; 10(9):1440.
https://doi.org/10.3390/en10091440

**Chicago/Turabian Style**

Hoseynpour, Omid, Behnam Mohammadi-ivatloo, Morteza Nazari-Heris, and Somayeh Asadi.
2017. "Application of Dynamic Non-Linear Programming Technique to Non-Convex Short-Term Hydrothermal Scheduling Problem" *Energies* 10, no. 9: 1440.
https://doi.org/10.3390/en10091440