# Dynamic Electric Dispatch for Wind Power Plants: A New Automatic Controller System Using Evolutionary Algorithms

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## Abstract

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## 1. Introduction

- Wind turbines: the blades turn a shaft inside the wind unit. Power energy is produced via a rotational generator (which uses the conversion of magnetic energy into electric);
- Offshore substation: the transformer equipment transforms the power for distribution and delivery to the onshore substation and;
- Lines and grid: power transmission and distribution to consumers.

- A new electric dispatch controller with automatic adjustment to solve OPF-like problems in WPP;
- A novel framework to fine-tune the parameters of the C-DEEPSO algorithm. For this, the irace package is coupled to the internal mechanism of C-DEEPSO;
- A new approach to compare stochastic algorithms is performed via an application of post hoc tests using the DSCtool in order to validate the statistical robustness of the results;
- An in-depth performance assessment of C-DEEPSO + irace compared to two different algorithms, DEPPSO and MVMO, when solving the OPF-like problem applied to WPPs;
- Obtained results indicate a competitive performance favoring C-DEEPSO in terms of efficiency and accuracy when applied to the OPF-like problem;
- A projection analysis is conducted indicating reduction of the system energy losses around 2.35 MWh per day.

## 2. Materials and Methods

#### 2.1. Optimal Power Flow Modeling for Reactive Electric Dispatch

#### 2.1.1. Optimal Power Flow Formulation

- Active (in MW) and Reactive (in MVar) power balance$$\begin{array}{c}\hfill {P}_{i}={P}_{i}^{gen}-{P}_{i}^{load}=\sum _{j=1}^{n}{U}_{i}{U}_{j}[{G}_{ij}cos({\theta}_{i}-{\theta}_{j})+{B}_{ij}sin({\theta}_{i}-{\theta}_{j})],\end{array}$$$$\begin{array}{c}\hfill {Q}_{i}={Q}_{i}^{gen}-{Q}_{i}^{load}=\sum _{j=1}^{n}{U}_{i}{U}_{j}[{G}_{ij}sin({\theta}_{i}-{\theta}_{j})-{B}_{ij}cos({\theta}_{i}-{\theta}_{j})],\end{array}$$
- Bus voltage (in kV)$$\begin{array}{c}\hfill {U}_{i}^{min}\le {U}_{i}\le {U}_{i}^{max},\end{array}$$
- Branch apparent power flow (in MVA)$$\begin{array}{c}\hfill {S}_{ij}^{min}\le {S}_{ij}\le {S}_{ij}^{max},\end{array}$$
- Transformer tap$$\begin{array}{c}\hfill {T}_{k}^{min}\le {T}_{k}\le {T}_{k}^{max},\end{array}$$
- Shunt VAR$$\begin{array}{c}\hfill {Q}_{k}^{min}\le {Q}_{k}\le {Q}_{k}^{max},\end{array}$$

#### 2.1.2. Offshore Wind Power Plant (WPP)

#### 2.2. The C-DEEPSO Algorithm

Algorithm 1:C-DEEPSO algorithm. |

#### 2.3. Optimal Adjustment Control Proposed: C-DEEPSO Integrated with irace Package

## 3. Experiments and Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data about the Test Network—IEEE 41-Bus System

Bus | Type | Pd | Qd | Gs | Bs | Area | Vm | Va | BaseKV | Zone | Vmax | Vmin |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 220 | 1 | 1.1 | 0.9 |

2 | 1 | 0 | 0 | 0.5208 | −12.1 | 1 | 1 | 0 | 110 | 1 | 1.1 | 0.9 |

3 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 110 | 1 | 1.1 | 0.9 |

4 | 1 | 0 | 0 | 0.4586 | −8.0667 | 1 | 1 | 0 | 110 | 1 | 1.1 | 0.9 |

5 | 1 | 0 | 0 | 0 | 9.8965 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

6 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

7 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

8 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

9 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

10 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

11 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

12 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

13 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

14 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

15 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

16 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

17 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

18 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

19 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

20 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

21 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

22 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

23 | 1 | 0 | 0 | 0.0049 | 0 | 1 | 1 | 0 | 33 | 1 | 1.1 | 0.9 |

24 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

25 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

26 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

27 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

28 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

29 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

30 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

31 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

32 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

33 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

34 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

35 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

36 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

37 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

38 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

39 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

40 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

41 | 1 | −5 | 0 | 0 | 0 | 1 | 1 | 0 | 0.95 | 1 | 1.05 | 0.95 |

fbus | tbus | r | x | b | rateA | rateB | rateC | R | Angle | Status | ${\mathit{R}}_{\mathbf{max}}$ | ${\mathit{R}}_{\mathit{min}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0016 | 0.0640 | 0 | 200 | 200 | 200 | 1 | 0 | 1 | 1.149 | 0.851 |

2 | 3 | 0.0024 | 0.0104 | 0.0653 | 135 | 135 | 135 | 0 | 0 | 1 | 0 | 0 |

3 | 4 | 0.0136 | 0.0241 | 0.2115 | 135 | 135 | 135 | 0 | 0 | 1 | 0 | 0 |

4 | 5 | 0.0032 | 0.1654 | 0 | 100 | 100 | 100 | 1 | 0 | 1 | 1.13 | 0.87 |

6 | 24 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

7 | 25 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

8 | 26 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

9 | 27 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

10 | 28 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

11 | 29 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

12 | 30 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

13 | 31 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

14 | 32 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

15 | 33 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

16 | 34 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

17 | 35 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

18 | 36 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

19 | 37 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

20 | 38 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

21 | 39 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

22 | 40 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

23 | 41 | 0.0065 | 1.5282 | 0 | 5.5 | 5.5 | 5.5 | 1 | 0 | 1 | 0 | 0 |

5 | 6 | 0.0081 | 0.0279 | 0.0024 | 31.5 | 31.5 | 31.5 | 0 | 0 | 1 | 0 | 0 |

6 | 7 | 0.0023 | 0.0064 | $4.90\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

7 | 8 | 0.0022 | 0.0062 | $4.73\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

8 | 9 | 0.0052 | 0.0080 | $4.03\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

9 | 10 | 0.0046 | 0.0070 | $3.51\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

10 | 11 | 0.0048 | 0.0074 | $3.70\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

5 | 12 | 0.0019 | 0.0067 | $5.76\times {10}^{-4}$ | 31.5 | 31.5 | 31.5 | 0 | 0 | 1 | 0 | 0 |

12 | 13 | 0.0025 | 0.0069 | $5.25\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

13 | 14 | 0.0025 | 0.0069 | $5.25\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

14 | 15 | 0.0046 | 0.0071 | $3.57\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

15 | 16 | 0.0046 | 0.0071 | $3.57\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

16 | 17 | 0.0046 | 0.0071 | $3.57\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

5 | 18 | 0.0056 | 0.0193 | 0.0017 | 31.5 | 31.5 | 31.5 | 0 | 0 | 1 | 0 | 0 |

18 | 19 | 0.0030 | 0.0085 | $6.48\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

19 | 20 | 0.0024 | 0.0068 | $5.17\times {10}^{-4}$ | 28.5 | 28.5 | 28.5 | 0 | 0 | 1 | 0 | 0 |

20 | 21 | 0.0049 | 0.0075 | $3.77\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

21 | 22 | 0.0048 | 0.0074 | $3.70\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

22 | 23 | 0.0048 | 0.0074 | $3.70\times {10}^{-4}$ | 20 | 20 | 20 | 0 | 0 | 1 | 0 | 0 |

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**Figure 2.**WPP control scheme layout inspired on [49].

**Figure 3.**Proposed optimized control: the iterative R Script (1) is responsible for iterating the instances of the WPP problem and calling irace (2). The irace uses three main files: the scenario (2.1), where we set the number of times the algorithm should run, the hypothesis test we want to use, and the confidence we want in our results; the parameters (2.2), where we set what parameters we want to find the best values in the fine-tuning process; and the target runner (2.3), which is responsible for running the algorithm and performing the tests to find out the best parameters values. There is also one Config Instance Matlab script (3) that creates an interface between the irace and the C-DEEPSO (4), running C-DEEPSO for the WPP problem instance (from iterative R script) with the irace parameters. In the end, the irace returns a data structure called irace.Rdata (5) with the results.

Method | Description | Advantages | Disadvantages | Works |
---|---|---|---|---|

Linear | Usage of linear or | Convergence is fast | Inaccuracy due | [11] |

Programming | piecewise linear cost | and guaranteed; | to linearization | [12] |

functions and | less computational | of cost funtions | ||

usage of DC power | effort; easy | and nonlinear | ||

flow instead | handling of | constraints | ||

of AC power flow, | inequality constraints | |||

which provides a | ||||

linear relation between | ||||

injections and | ||||

line flows | ||||

Quadratic | Usage of a quadratic | It does not require | Computation cost | [13] |

Programming | objective function, | linearization of | can be significantly | [14] |

all constraints | functions | high | ||

are linear | ||||

Meta-heuristics | Genetic Algorithms (GA), | No linearization | The solution | [15] |

Particle Swarm | is required; | obtained is not | [16] | |

Optimization (PSO), | uses non-continuos, | guaranteed | [17] | |

Differential Evolution (DE), | non convex, and | to be optimal | [18] | |

Ant Colony | non differentiable | [19] | ||

Optimization (ACO) | functions; easy | |||

are the most | implementation | |||

applied meta-heuristics |

Term | Description |
---|---|

${P}_{loss}$ | Total losses of the energy system (in MWh) |

$NL$ | Total number of lines in system |

${G}_{k}$ | Conductance of line K |

U | Voltage magnitude of lines in (in kV): sending end (i); receiving end (j) |

$\delta $ | Angle of the bus voltages (in degrees) |

${P}_{i}$ | Active power injected (in MW) |

${Q}_{i}$ | Reactive power (in MVar) |

$\theta $ | Voltage angle (in degrees) |

${U}_{i}^{min}$, ${U}_{i}^{max}$ | Bus voltage magnitude in limits minimum and maximum (in kV) |

${S}_{ij}$ | Apparent power flow injection (in MVA) |

${T}_{k}$ | Tap change limits |

${Q}_{k}$ | Number of capacitor/reactor banks |

$NB$ | Number of buses |

IEEE 41-Bus | |||
---|---|---|---|

Test System | |||

Description of test system | |||

Generators | 18 | ||

Linked cable | 1 | ||

Transfomers | Fixed tap: T1 | 33 | |

Fixed tap: T2 | 13 | ||

Description of the optmization problem | Continuous | 18 | |

variables | |||

Optimization | Discrete | 2 | |

variables | variables | ||

Continuous and | 2 | ||

stepwise variables | |||

Constraints system | 123 |

**Table 4.**Comparative result of the C-DEEPSO + irace, DEEPSO and MVMO algorithms. Legend: scenario (Sc.) is the number of each measured interval of energy production (96); significance (S.) with $\alpha =5\%$ on Kolmogorov–Smirnov test (“+” reject null hypothesis of mean difference and “=” not reject the null hypothesis); optimal mutation rate found (Mut.); optimal communication rate found (Com.); mean (mean results of 31 runs); STD (standard deviation of 31 runs).

C-DEEPSO + irace | DEEPSO | MVMO | |||||||
---|---|---|---|---|---|---|---|---|---|

Sc. | S. (5%) | Mut. | Com. | AVR | STD | AVR | STD | AVR | STD |

1 | (+) | 0.70 | 0.50 | 1.2961 | 0.0063 | 1.3004 | 0.0065 | 1.2995 | 0.0021 |

2 | (+) | 0.30 | 0.40 | 1.2670 | 0.0007 | 1.2737 | 0.0068 | 1.2709 | 0.0025 |

3 | (+) | 0.50 | 0.20 | 1.0137 | 0.0032 | 1.0185 | 0.0091 | 1.0180 | 0.0062 |

4 | (+) | 0.90 | 0.50 | 1.2061 | 0.0016 | 1.2104 | 0.0067 | 1.2142 | 0.0125 |

5 | (+) | 0.10 | 0.30 | 1.2745 | 0.0012 | 1.2774 | 0.0049 | 1.2775 | 0.0030 |

6 | (+) | 0.40 | 0.40 | 1.5893 | 0.0013 | 1.5907 | 0.0024 | 1.5950 | 0.0030 |

7 | (+) | 0.60 | 0.90 | 1.4272 | 0.0021 | 1.4306 | 0.0039 | 1.4317 | 0.0034 |

8 | (+) | 0.50 | 0.40 | 1.4864 | 0.0023 | 1.4901 | 0.0027 | 1.4912 | 0.0018 |

9 | (+) | 0.70 | 0.50 | 1.8886 | 0.0067 | 1.9206 | 0.0251 | 1.9127 | 0.0177 |

10 | (+) | 0.60 | 0.70 | 2.0370 | 0.0068 | 2.0599 | 0.0210 | 2.0681 | 0.0248 |

11 | (+) | 0.90 | 0.80 | 1.4493 | 0.0018 | 1.4605 | 0.0192 | 1.4668 | 0.0288 |

12 | (+) | 0.20 | 0.10 | 1.6732 | 0.0021 | 1.6879 | 0.0092 | 1.6967 | 0.0202 |

13 | (+) | 0.10 | 0.10 | 1.7389 | 0.0060 | 1.7545 | 0.0092 | 1.7653 | 0.0197 |

14 | (+) | 0.60 | 0.80 | 2.1052 | 0.0067 | 2.1385 | 0.0232 | 2.1313 | 0.0158 |

15 | (+) | 0.60 | 0.50 | 2.5555 | 0.0096 | 2.5764 | 0.0157 | 2.5835 | 0.0178 |

16 | (+) | 0.70 | 0.30 | 2.4950 | 0.0081 | 2.5230 | 0.0226 | 2.5268 | 0.0157 |

17 | (+) | 0.70 | 0.40 | 2.6245 | 0.0127 | 2.7631 | 0.0632 | 2.7530 | 0.0497 |

18 | (+) | 0.70 | 0.30 | 2.7023 | 0.0155 | 2.8377 | 0.0498 | 2.8328 | 0.0653 |

19 | (+) | 0.60 | 0.40 | 2.7028 | 0.0139 | 2.8389 | 0.0547 | 2.8321 | 0.0596 |

20 | (+) | 0.90 | 0.30 | 2.7010 | 0.0185 | 2.8295 | 0.0629 | 2.8368 | 0.0379 |

21 | (+) | 0.50 | 0.10 | 2.7075 | 0.0225 | 2.8264 | 0.0541 | 2.8243 | 0.0522 |

22 | (+) | 0.70 | 0.30 | 2.7038 | 0.0204 | 2.8400 | 0.0420 | 2.8105 | 0.0433 |

23 | (+) | 0.80 | 0.40 | 2.7057 | 0.0174 | 2.8492 | 0.0521 | 2.8130 | 0.0604 |

24 | (+) | 0.90 | 0.30 | 2.7018 | 0.0162 | 2.8246 | 0.0604 | 2.8306 | 0.0384 |

25 | (+) | 0.50 | 0.70 | 2.2891 | 0.0017 | 2.3205 | 0.0267 | 2.3284 | 0.0260 |

26 | (+) | 0.90 | 0.80 | 2.2363 | 0.0085 | 2.2791 | 0.0185 | 2.2739 | 0.0227 |

27 | (+) | 0.90 | 0.50 | 2.2987 | 0.0008 | 2.3322 | 0.0199 | 2.3268 | 0.0164 |

28 | (+) | 0.70 | 0.50 | 2.5500 | 0.0051 | 2.5840 | 0.0246 | 2.5884 | 0.0252 |

29 | (+) | 0.90 | 0.70 | 2.4046 | 0.0047 | 2.4348 | 0.0256 | 2.4355 | 0.0167 |

30 | (+) | 0.60 | 0.70 | 2.4728 | 0.0041 | 2.5042 | 0.0195 | 2.5105 | 0.0216 |

31 | (+) | 0.90 | 0.90 | 1.8216 | 0.0095 | 1.8371 | 0.0133 | 1.8455 | 0.0319 |

32 | (+) | 0.50 | 0.30 | 1.6190 | 0.0032 | 1.6326 | 0.0126 | 1.6384 | 0.0242 |

33 | (+) | 0.40 | 0.70 | 1.6158 | 0.0020 | 1.6206 | 0.0055 | 1.6215 | 0.0022 |

34 | (=) | 0.10 | 0.80 | 1.6868 | 0.0026 | 1.6888 | 0.0039 | 1.6901 | 0.0024 |

35 | (=) | 0.30 | 0.90 | 1.3567 | 0.0022 | 1.3586 | 0.0032 | 1.3593 | 0.0035 |

36 | (+) | 0.70 | 0.80 | 1.3935 | 0.0016 | 1.3961 | 0.0032 | 1.3979 | 0.0039 |

37 | (+) | 0.90 | 0.90 | 0.9846 | 0.0011 | 0.9891 | 0.0080 | 0.9863 | 0.0023 |

38 | (+) | 0.50 | 0.90 | 0.7589 | 0.0015 | 0.7652 | 0.0154 | 0.7616 | 0.0024 |

39 | (=) | 0.40 | 0.80 | 0.7549 | 0.0015 | 0.7603 | 0.0126 | 0.7578 | 0.0031 |

40 | (=) | 0.30 | 0.90 | 0.8379 | 0.0008 | 0.8402 | 0.0071 | 0.8419 | 0.0053 |

41 | (+) | 0.40 | 0.40 | 0.9269 | 0.0136 | 0.9272 | 0.0073 | 0.9272 | 0.0039 |

42 | (+) | 0.80 | 0.50 | 1.1522 | 0.0003 | 1.1555 | 0.0052 | 1.1555 | 0.0023 |

43 | (+) | 0.90 | 0.70 | 1.2172 | 0.0012 | 1.2209 | 0.0065 | 1.2215 | 0.0044 |

44 | (+) | 0.80 | 0.40 | 1.1964 | 0.0012 | 1.2018 | 0.0099 | 1.1997 | 0.0022 |

45 | (+) | 0.70 | 0.80 | 2.0350 | 0.0061 | 2.0626 | 0.0186 | 2.0708 | 0.0257 |

46 | (+) | 0.60 | 0.10 | 1.6322 | 0.0026 | 1.6450 | 0.0100 | 1.6557 | 0.0256 |

47 | (+) | 0.90 | 0.60 | 1.5231 | 0.0019 | 1.5351 | 0.0109 | 1.5530 | 0.0339 |

48 | (+) | 0.80 | 0.30 | 1.4491 | 0.0012 | 1.4647 | 0.0211 | 1.4668 | 0.0220 |

49 | (+) | 0.50 | 0.80 | 1.5546 | 0.0004 | 1.5732 | 0.0143 | 1.5935 | 0.0382 |

50 | (+) | 0.90 | 0.50 | 1.4189 | 0.0001 | 1.4217 | 0.0040 | 1.4298 | 0.0155 |

51 | (+) | 0.80 | 0.90 | 1.3725 | 0.0001 | 1.3730 | 0.0006 | 1.3795 | 0.0097 |

52 | (=) | 0.80 | 0.90 | 1.2804 | 0.0000 | 1.2803 | 0.0000 | 1.2804 | 0.0001 |

53 | (+) | 0.70 | 0.80 | 1.2160 | 0.0000 | 1.2160 | 0.0001 | 1.2161 | 0.0001 |

54 | (=) | 0.70 | 0.90 | 1.2579 | 0.0000 | 1.2578 | 0.0000 | 1.2578 | 0.0000 |

55 | (=) | 0.60 | 0.90 | 1.2610 | 0.0000 | 1.2611 | 0.0000 | 1.2611 | 0.0000 |

56 | (+) | 0.60 | 0.70 | 1.4257 | 0.0001 | 1.4279 | 0.0029 | 1.4352 | 0.0164 |

57 | (+) | 0.80 | 0.50 | 1.0896 | 0.0013 | 1.1006 | 0.0192 | 1.1134 | 0.0098 |

58 | (+) | 0.80 | 0.70 | 1.0940 | 0.0057 | 1.1071 | 0.0241 | 1.1287 | 0.0379 |

59 | (+) | 0.40 | 0.30 | 1.1506 | 0.0027 | 1.1626 | 0.0252 | 1.1852 | 0.0330 |

60 | (+) | 0.90 | 0.50 | 0.9422 | 0.0012 | 0.9676 | 0.0411 | 0.9699 | 0.0126 |

61 | (+) | 0.90 | 0.40 | 0.7590 | 0.0009 | 0.7683 | 0.0195 | 0.7642 | 0.0098 |

62 | (+) | 0.90 | 0.60 | 0.7825 | 0.0079 | 0.7859 | 0.0103 | 0.7883 | 0.0133 |

63 | (+) | 0.90 | 0.50 | 0.7726 | 0.0010 | 0.7820 | 0.0185 | 0.7800 | 0.0126 |

64 | (+) | 0.90 | 0.50 | 0.9194 | 0.0007 | 0.9249 | 0.0168 | 0.9238 | 0.0095 |

65 | (+) | 0.90 | 0.90 | 1.1182 | 0.0007 | 1.1215 | 0.0043 | 1.1202 | 0.0024 |

66 | (+) | 0.90 | 0.50 | 1.3421 | 0.0012 | 1.3444 | 0.0022 | 1.3445 | 0.0014 |

67 | (+) | 0.40 | 0.90 | 1.2360 | 0.0013 | 1.2396 | 0.0048 | 1.2393 | 0.0013 |

68 | (+) | 0.40 | 0.70 | 0.8317 | 0.0009 | 0.8387 | 0.0145 | 0.8341 | 0.0036 |

69 | (+) | 0.50 | 0.70 | 0.8642 | 0.0044 | 0.8690 | 0.0106 | 0.8660 | 0.0021 |

70 | (+) | 0.50 | 0.80 | 1.0739 | 0.0006 | 1.0766 | 0.0045 | 1.0757 | 0.0015 |

71 | (+) | 0.50 | 0.70 | 1.1317 | 0.0005 | 1.1339 | 0.0031 | 1.1344 | 0.0041 |

72 | (+) | 0.70 | 0.70 | 1.1299 | 0.0004 | 1.1329 | 0.0034 | 1.1324 | 0.0028 |

73 | (=) | 0.50 | 0.10 | 1.4195 | 0.0016 | 1.4244 | 0.0035 | 1.4223 | 0.0021 |

74 | (+) | 0.90 | 0.60 | 1.7804 | 0.0041 | 1.7891 | 0.0250 | 1.7833 | 0.0038 |

75 | (+) | 0.80 | 0.80 | 2.0113 | 0.0003 | 1.2987 | 0.0018 | 2.0223 | 0.0030 |

76 | (+) | 0.90 | 0.50 | 2.6373 | 0.0000 | 1.2964 | 0.0010 | 2.6373 | 0.0000 |

77 | (=) | 0.20 | 0.20 | 2.6373 | 0.0000 | 1.2972 | 0.0018 | 2.6373 | 0.0000 |

78 | (+) | 0.30 | 0.40 | 2.7373 | 0.0000 | 1.2963 | 0.0010 | 2.6373 | 0.0000 |

79 | (=) | 0.80 | 0.30 | 2.6373 | 0.0000 | 1.2974 | 0.0020 | 2.6373 | 0.0000 |

80 | (=) | 0.30 | 0.40 | 2.6373 | 0.0000 | 2.6373 | 0.0000 | 2.6373 | 0.0000 |

81 | (+) | 0.70 | 0.20 | 2.4610 | 0.0012 | 2.4738 | 0.0373 | 2.4637 | 0.0070 |

82 | (+) | 0.80 | 0.80 | 1.5201 | 0.0017 | 1.5227 | 0.0030 | 1.5248 | 0.0022 |

83 | (+) | 0.40 | 0.50 | 1.5445 | 0.0017 | 1.5465 | 0.0030 | 1.5493 | 0.0020 |

84 | (+) | 0.60 | 0.90 | 1.4070 | 0.0022 | 1.4095 | 0.0026 | 1.4096 | 0.0013 |

85 | (+) | 0.10 | 0.90 | 1.2795 | 0.0019 | 1.2822 | 0.0035 | 1.2819 | 0.0012 |

86 | (+) | 0.60 | 0.90 | 1.3203 | 0.0028 | 1.3238 | 0.0033 | 1.3229 | 0.0018 |

87 | (+) | 0.10 | 0.50 | 1.5585 | 0.0014 | 1.5608 | 0.0025 | 1.5635 | 0.0014 |

88 | (+) | 0.10 | 0.90 | 1.5526 | 0.0018 | 1.5546 | 0.0022 | 1.5574 | 0.0020 |

89 | (+) | 0.40 | 0.50 | 1.4724 | 0.0012 | 1.4757 | 0.0034 | 1.4769 | 0.0017 |

90 | (+) | 0.20 | 0.70 | 2.0066 | 0.0026 | 2.0119 | 0.0068 | 2.0221 | 0.0063 |

91 | (+) | 0.80 | 0.70 | 2.1807 | 0.0004 | 2.1924 | 0.0126 | 2.1981 | 0.0053 |

92 | (=) | 0.30 | 0.60 | 1.6123 | 0.0018 | 1.6172 | 0.0024 | 1.6223 | 0.0032 |

93 | (+) | 0.80 | 0.90 | 2.0554 | 0.0059 | 2.0642 | 0.0102 | 2.0719 | 0.0052 |

94 | (+) | 0.60 | 0.90 | 2.4899 | 0.0043 | 2.5161 | 0.0236 | 2.5123 | 0.0071 |

95 | (+) | 0.60 | 0.80 | 1.7829 | 0.0015 | 1.7849 | 0.0036 | 1.7923 | 0.0050 |

96 | (+) | 0.80 | 0.40 | 1.4976 | 0.0016 | 1.5001 | 0.0022 | 1.5030 | 0.0028 |

Total | 38.01 (MWh) | 38.80 (MWh) | 40.35 (MWh) |

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**MDPI and ACS Style**

Marcelino, C.G.; Avancini, J.V.C.; Delgado, C.A.D.M.; Wanner, E.F.; Jiménez-Fernández, S.; Salcedo-Sanz, S.
Dynamic Electric Dispatch for Wind Power Plants: A New Automatic Controller System Using Evolutionary Algorithms. *Sustainability* **2021**, *13*, 11924.
https://doi.org/10.3390/su132111924

**AMA Style**

Marcelino CG, Avancini JVC, Delgado CADM, Wanner EF, Jiménez-Fernández S, Salcedo-Sanz S.
Dynamic Electric Dispatch for Wind Power Plants: A New Automatic Controller System Using Evolutionary Algorithms. *Sustainability*. 2021; 13(21):11924.
https://doi.org/10.3390/su132111924

**Chicago/Turabian Style**

Marcelino, Carolina G., João V. C. Avancini, Carla A. D. M. Delgado, Elizabeth F. Wanner, Silvia Jiménez-Fernández, and Sancho Salcedo-Sanz.
2021. "Dynamic Electric Dispatch for Wind Power Plants: A New Automatic Controller System Using Evolutionary Algorithms" *Sustainability* 13, no. 21: 11924.
https://doi.org/10.3390/su132111924