# Direct Power Control Optimization for Doubly Fed Induction Generator Based Wind Turbine Systems

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## Abstract

**:**

## 1. Introduction

## 2. Modelling of the DFIG Based Wind Energy Converter

#### 2.1. Modelling of the Wind Turbine

#### 2.2. Modelling of the DFIG

#### 2.3. Modelling of the GSC and the DC-Link

## 3. Vector Control of the DFIG-Based Wind Energy Converter

#### 3.1. Control of the RSC

#### 3.2. Control of the GSC

## 4. PI Controllers Tuning Problem Formulation

## 5. Thermal Exchange Optimization Algorithm

**Step 1.**Randomly initialize the temperature for all objects ${T}_{i}^{0}$, $i=1,2,\dots ,{N}_{pop}$.**Step 2.**Calculate the fitness of each search object.**Step 3.**Save some $T$ best vectors and their related cost values in the TM.**Step 4.**Add the saved solutions and remove the same numbers of the worst objects.**Step 5.**Arrange the objects according to their related fitness in an ascending order.**Step 6.**Divide the objects into two equal groups: environment and cooling objects.**Step 7.**Calculate the parameters $\eta $ and $t$.**Step 8.**Change the environment temperatures by Equation (25).**Step 9.**Update the temperatures according to Equations (26) and (27).**Step 10.**Check the termination criterion and repeat the iterations.

## 6. Simulation Results and Discussions

#### 6.1. Execution of the Metaheuristic Algorithms

- -
- 14.3% step change in the reference of DC-link voltage at time $t=0.5\mathrm{sec}$;
- -
- step change in the reference stator reactive power at time $t=0.8\mathrm{sec}$.

#### 6.2. Statistical Analysis and Comparison

#### 6.3. Computational Time Efficiency

#### 6.4. Sensitivity Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CD | Critical Difference |

CTE | Computational Time Efficiency |

ET | Elapsed Time |

DFIGs | Doubly Fed Induction Generators |

HSA | Harmony Search Algorithm |

IAE | Integral Absolute Error |

IGBTs | Insulated-Gate Bipolar Transistors |

ISE | Integral Square Error |

ITAE | Integral Time-weighted Absolute Error |

ITSE | Integral Time-weighted Square Error |

GA | Genetic Algorithm |

GOA | Grasshopper Optimization Algorithm |

GSC | Grid Side Converter |

MPPT | Maximum Power Point Tracking |

PI | Proportional-Integral |

PSO | Particle Swarm Optimization |

RSC | Rotor Side Converter |

SFO | Stator Flux Orientation |

SPWM | Sinusoidal Pulse Width Modulation |

STD | Standard Deviation |

TEO | Thermal Exchange Optimization |

THD | Total Harmonic Distortion |

TM | Thermal Memory |

VOC | Voltage Oriented Control |

WECS | Wind Energy Converter System |

WCA | Water Cycle Algorithm |

WT | Wind Turbine |

WTs | Wind Turbines |

## Notations

${C}_{dc}$ | DC-link capacitance |

${C}_{f}$ | Filter capacitance |

${C}_{p}$ | Power conversion coefficient |

${e}_{g(d,q)}$ | d-q axis grid voltages |

${i}_{g(d,q)}$ | d-q axis grid currents |

${i}_{r(d,q)}$ | d-q axis rotor currents |

${i}_{s(d,q)}$ | d-q axis stator currents |

${L}_{g}$ | Filter grid side inductance |

${L}_{i}$ | Filter converter side inductance |

${L}_{m}$ | Magnetizing inductance |

${L}_{r},{L}_{s}$ | Rotor and stator inductances |

${L}_{T}$ | Filter total inductance |

${P}_{g},{P}_{s}$ | Grid and stator active powers |

${P}_{m}$ | Mechanical turbine power |

${Q}_{g},{Q}_{s}$ | Grid and stator reactive powers |

$R$ | Turbine radius |

${R}_{d}$ | Filter damping resistance |

${R}_{r},{R}_{s}$ | Rotor and stator resistances |

${V}_{dc}$ | DC-link voltage |

${V}_{f(d,q)}$ | d-q axis grid converter voltage sides |

${V}_{r\text{}(d,q)}$ | d-q axis rotor converter voltage sides |

${V}_{\mathrm{s}\text{}(d,q)}$ | d-q axis stator voltages |

${V}_{w}$ | Wind speed |

$\lambda $ | Tip speed ratio |

${\omega}_{r},{\omega}_{s}$ | Rotor and stator angular frequencies |

${\mathsf{\Omega}}_{t}$ | Mechanical rotational speed |

$\rho $ | Air density |

${\phi}_{dr},{\phi}_{qr}$ | d-q axis rotor fluxes |

${\phi}_{ds},{\phi}_{qs}$ | d-q axis stator fluxes |

## Appendix A. Decision Variables of Problem (19) Relative to the Optimization Mean Case

Indices | Algorithms | PI Controllers’ Gains | |||||
---|---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{p}{\mathit{P}}_{\mathit{s}}}$ | ${\mathit{K}}_{\mathit{i}{\mathit{P}}_{\mathit{s}}}$ | ${\mathit{K}}_{\mathit{p}{\mathit{Q}}_{\mathit{s}}}$ | ${\mathit{K}}_{\mathit{i}{\mathit{Q}}_{\mathit{s}}}$ | ${\mathit{K}}_{\mathit{p}\mathit{d}\mathit{c}}$ | ${\mathit{K}}_{\mathit{i}\mathit{d}\mathit{c}}$ | ||

IAE | PSO | 14.16 | 139.37 | 376.43 | 381.82 | 9 | 31.32 |

GA | 16.92 | 394.62 | 18.22 | 147.46 | 24.36 | 62.17 | |

HSA | 7.89 | 83.96 | 151.27 | 227.72 | 28.03 | 86.11 | |

WCA | 2.28 | 329.93 | 122.55 | 400 | 26.11 | 41.53 | |

GOA | 16.91 | 394.62 | 18.22 | 147.47 | 26.36 | 59.85 | |

TEO | 4.23 | 58.43 | 10.37 | 15.73 | 89.10 | 192.71 | |

ISE | PSO | 11.23 | 207.70 | 169.31 | 353 | 69.65 | 272.40 |

GA | 2.25 | 313.46 | 187.62 | 398.82 | 79.95 | 345.49 | |

HSA | 1 × 10^{−5} | 1.87 | 118.84 | 360.16 | 33.29 | 39.65 | |

WCA | 12.88 | 143.34 | 360.15 | 400 | 191.11 | 293 | |

GOA | 12.42 | 388.97 | 1.93 | 371.52 | 148.46 | 220.78 | |

TEO | 35.94 | 165.25 | 15.44 | 71.59 | 1.48 | 3 | |

ITSE | PSO | 0.1 | 1 × 10^{−5} | 238.93 | 351.81 | 0.99 | 25.03 |

GA | 1 × 10^{−5} | 1 × 10^{−5} | 7.27 | 214.42 | 24.34 | 62.20 | |

HSA | 303.71 | 320.52 | 1.03 | 6.25 | 27.66 | 45.22 | |

WCA | 4.13 | 248.98 | 10.79 | 24.45 | 27.89 | 79.04 | |

GOA | 9.82 | 108.24 | 333.25 | 27.36 | 131.74 | 205.91 | |

TEO | 8.56 | 5.04 | 67.72 | 69.44 | 10.71 | 31.96 | |

ITAE | PSO | 250.16 | 289.53 | 346.50 | 142.34 | 73.74 | 238.53 |

GA | 1 × 10^{−5} | 1 × 10^{−5} | 45.02 | 85.22 | 130.57 | 26.43 | |

HSA | 8.16 | 4.83 | 323.56 | 384.19 | 33.33 | 47.41 | |

WCA | 9.54 | 63.73 | 329.42 | 395.36 | 143.52 | 400 | |

GOA | 13.71 | 399.92 | 302.9 | 265.4 | 97.91 | 196 | |

TEO | 4.8 × 10^{−6} | 11.28 | 1.6 × 10^{−6} | 8.3 × 10^{−6} | 10.24 | 68.66 |

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**Figure 1.**Configuration of the Doubly Fed Induction Generator (DFIG)-based Wind Energy Converter System (WECS).

**Figure 6.**Flowchart of the proposed Thermal Exchange Optimization (TEO)-tuned PI controllers’ parameters.

**Figure 7.**Convergence rates comparison: (

**a**) IAE criterion; (

**b**) ISE criterion; (

**c**) ITAE criterion; (

**d**) ITSE criterion.

**Figure 8.**Box-and-Whisker plot of optimization problem (19): (

**a**) IAE criterion; (

**b**) ISE criterion; (

**c**) ITAE criterion; (

**d**) ITSE criterion.

**Figure 10.**Quadrature grid current responses against a step change for different-tuned PI controllers.

**Figure 11.**Bode plot of the open loop grid current loop: with and without passive damping based technical optimum tuning.

**Figure 14.**Performance comparison of the controlled reactive power under different PI controllers’ tuning methods.

**Figure 15.**Time-domain variations of the TEO-based control of the direct and quadrature rotor currents.

Equipment | Parameter | Value | Unit |
---|---|---|---|

DFIG | Rated power | 1500 | kW |

RMS grid line voltage | 575 | V | |

Slip range | 0.2 | - | |

Rated electrical frequency | 50 | Hz | |

Stator resistance | 0.023 | pu | |

Stator leakage inductance | 0.18 | pu | |

Rotor resistance | 0.016 | pu | |

Rotor leakage inductance | 0.16 | pu | |

Magnetizing inductance | 2.9 | pu | |

Power converters | Rated power | 300 | kW |

Switching frequency at ${f}_{sw,RSC}$ and ${f}_{sw,GSC}$ | 2700 | Hz | |

DC-link voltage | 1050 | V | |

DC-link capacitance | 10000 | $\mathsf{\mu}\mathrm{F}$ | |

L-filter | L-filter grid inductance | 0.1 | pu |

LCL-filter | LCL-filter grid side inductance | 0.018 | pu |

LCL-filter capacitance | 0.104 | pu | |

LCL-filter converter side inductance | 0.077 | pu | |

Passive damping resistor | 0.124 | pu |

Algorithms | Parameters Setting |
---|---|

PSO | Cognitive and social coeffs. ${c}_{1}={c}_{2}=2$, weights ${w}_{\mathrm{max}}=0.9$, ${w}_{\mathrm{min}}=0.2$ [10]. |

GA | Roulette wheel, crossover ${\mathcal{P}}_{cross}=1$, mutation ${\mathcal{P}}_{mut}=0.01$ [11]. |

HSA | Harmony memory rate HMCR = 0.9, pitch adjusting rate PAR = 0.3 [25]. |

WCA | Summation number of rivers ${N}_{sr}=8$ and ${d}_{max}=1\times {10}^{-3}$ [18]. |

GOA | ${c}_{\mathrm{max}}=1$, ${c}_{\mathrm{min}}=0.00001$ [26]. |

TEO | Thermal memory $TM=10$, $pro=0.50$, ${c}_{1}=1,{c}_{2}=1$ [3]. |

Indices | PSO | GA | HSA | WCA | GOA | TEO | |
---|---|---|---|---|---|---|---|

IAE | Best | 1.57 | 1.62 | 1.57 | 1.52 | 1.60 | 1.55 |

Mean | 1.73 | 1.67 | 1.60 | 1.59 | 2.01 | 1.58 | |

Worst | 2.69 | 1.73 | 1.64 | 1.78 | 3.20 | 1.60 | |

STD | 3.4 × 10^{−1} | 3.2 × 10^{−2} | 2.3 × 10^{−2} | 7.6 × 10^{−2} | 6.3 × 10^{−1} | 1.9 × 10^{−2} | |

ET (sec) | 29022 | 28720 | 25480 | 32900 | 20400 | 24640 | |

ISE | Best | 33.17 | 34.86 | 32.64 | 32.66 | 34.06 | 31.45 |

Mean | 41.42 | 37.25 | 35.51 | 45.88 | 45.06 | 33.46 | |

Worst | 61.80 | 40.53 | 38.78 | 61.24 | 64.50 | 36.79 | |

STD | 8.27 | 1.92 | 1.80 | 10.77 | 9.08 | 1.64 | |

ET (sec) | 28140 | 27420 | 19083 | 26900 | 19200 | 23420 | |

ITSE | Best | 17.47 | 17.49 | 18.45 | 17.14 | 17.86 | 17.79 |

Mean | 18.46 | 17.96 | 23.18 | 18.32 | 19.83 | 18.24 | |

Worst | 20.45 | 18.21 | 26.77 | 19.75 | 28.11 | 19.20 | |

STD | 0.921 | 0.207 | 3.616 | 8.1E-1 | 2.96 | 4.1E-1 | |

ET (sec) | 21560 | 26280 | 18720 | 27540 | 19500 | 23380 | |

ITAE | Best | 0.650 | 0.632 | 0.646 | 0.640 | 0.644 | 0.642 |

Mean | 0.683 | 0.650 | 0.683 | 0.659 | 0.688 | 0.658 | |

Worst | 0.707 | 0.62 | 0.718 | 0.736 | 0.742 | 0.675 | |

STD | 0.018 | 0.019 | 0.021 | 0.029 | 0.040 | 0.014 | |

ET (sec) | 20060 | 20480 | 16880 | 25880 | 20840 | 19680 |

PI Tuning Methods | Unit Step Change Response | ||||
---|---|---|---|---|---|

${\mathit{t}}_{\mathit{r}}\text{}\left(\mathbf{sec}\right)$ | ${\mathit{t}}_{\mathit{s}}\text{}\left(\mathbf{sec}\right)$ | ${\mathit{t}}_{\mathit{p}}\text{}\left(\mathbf{sec}\right)$ | $\mathit{\delta}\text{}(\mathbf{\%})$ | ${\mathit{E}}_{\mathit{s}\mathit{s}}$ | |

Frequency response | 0.0441 | 4.3813 | 4.1382 | 1.168 | 0.3355 |

Pole placement | 0.0163 | 4.1111 | 4.0441 | 2.006 | 0.2813 |

Symmetrical optimum | 0.0185 | 4.1324 | 4.0459 | 2.768 | 0.3568 |

Ziegler–Nichols | 0.0037 | 4.0252 | 4.0080 | 1.209 | 0.2843 |

Tyreus–Luyben | 0.0060 | 4.0249 | 4.0182 | 0.299 | 0.2977 |

TEO | 0.0029 | 4.0193 | 4.0076 | 2.537 | 0.2539 |

**Table 5.**Time-domain performances for controlled quadrature grid current under a step change scenario.

PI Tuning Methods | Unit Step Change Response | |||
---|---|---|---|---|

${\mathit{t}}_{\mathit{r}}\text{}\left(\mathbf{sec}\right)$ | ${\mathit{t}}_{\mathit{p}}\text{}\left(\mathbf{sec}\right)$ | $\mathit{\delta}\text{}(\mathbf{\%})$ | ${\mathit{E}}_{\mathit{s}\mathit{s}}$ | |

Frequency response | 0.0171 | 2.0387 | 28.1802 | 0.0035 |

Pole placement | 0.0027 | 2.0096 | 21.95 | 0.0045 |

Technical optimum | 0.0146 | 2.0383 | 8.2931 | 0.0055 |

Ziegler–Nichols | 3.582 × 10^{−4} | 2.0026 | 10.7400 | 0.0051 |

Tyreus–Luyben | 0.0020 | 2.0155 | 7.1657 | 0.0061 |

Algorithms | Indices | Average Rank | |||||||
---|---|---|---|---|---|---|---|---|---|

IAE | ISE | ITSE | ITAE | ||||||

Score | Rank | Score | Rank | Score | Rank | Score | Rank | ||

PSO | 1.73 | 5 | 41.42 | 4 | 18.46 | 4 | 0.683 | 5 | 4.25 |

GA | 1.67 | 4 | 37.25 | 3 | 17.96 | 1 | 0.650 | 1 | 2.25 |

HSA | 1.60 | 3 | 35.51 | 2 | 23.18 | 6 | 0.683 | 4 | 3.75 |

WCA | 1.59 | 2 | 45.88 | 6 | 18.32 | 3 | 0.659 | 3 | 3.75 |

GOA | 2.01 | 6 | 44.06 | 5 | 19.83 | 5 | 0.688 | 6 | 5.5 |

TEO | 1.58 | 1 | 33.46 | 1 | 18.24 | 2 | 0.658 | 2 | 1.5 |

Index | Algorithms | |||||
---|---|---|---|---|---|---|

PSO | GA | HSA | WCA | GOA | TEO | |

IAE | 18.01% | 17.82% | 15.81% | 20.41% | 12.66% | 15.29% |

ISE | 19.52% | 19.02% | 13.24% | 18.66% | 13.32% | 16.25% |

ITSE | 15.74% | 19.19% | 13.67% | 20.11% | 14.24% | 17.07% |

ITAE | 16.20% | 16.54% | 13.63% | 20.90% | 16.83% | 15.89% |

Optimization Scenario | $\mathit{T}\mathit{M}=10,\text{}\mathit{p}\mathit{r}\mathit{o}=0.5\text{}\mathbf{and}\text{}{\mathit{N}}_{\mathit{i}\mathit{t}\mathit{e}\mathit{r}}=100$ | |||
---|---|---|---|---|

Worst | Mean | Best | STD | |

${c}_{1}=0,{c}_{2}=0$ | 1.669 | 1.652 | 1.638 | 1.9 × 10^{−2} |

${c}_{1}=1,{c}_{2}=0$ | 1.664 | 1.640 | 1.622 | 1.4 × 10^{−2} |

${c}_{1}=0,{c}_{2}=1$ | 1.654 | 1.640 | 1.631 | 7.5 × 10^{−3} |

${c}_{1}=1,{c}_{2}=1$ | 1.608 | 1.582 | 1.553 | 1.9 × 10^{−2} |

Optimization Scenario | ${\mathit{c}}_{1}=1,\text{}{\mathit{c}}_{2}=1,\text{}\mathit{p}\mathit{r}\mathit{o}=0.5\text{}\mathbf{and}\text{}{\mathit{N}}_{\mathit{i}\mathit{t}\mathit{e}\mathit{r}}=100$ | |||
---|---|---|---|---|

Worst | Mean | Best | STD | |

$TM=4$ | 1.699 | 1.694 | 1.689 | 3.5 × 10^{−3} |

$TM=8$ | 1.654 | 1.640 | 1.631 | 8.4 × 10^{−2} |

$TM=10$ | 1.608 | 1.582 | 1.553 | 1.9 × 10^{−2} |

Optimization Scenario | ${\mathit{c}}_{1}=1,\text{}{\mathit{c}}_{2}=1,\text{}\mathit{T}\mathit{M}=10\text{}\mathbf{and}\text{}{\mathit{N}}_{\mathit{i}\mathit{t}\mathit{e}\mathit{r}}=100$ | |||
---|---|---|---|---|

Worst | Mean | Best | STD | |

$pro=0.20$ | 1.693 | 1.654 | 1.633 | 1.7 × 10^{−2} |

$pro=0.35$ | 1.678 | 1.617 | 1.575 | 5.2 × 10^{−2} |

$pro=0.50$ | 1.608 | 1.582 | 1.553 | 1.9 × 10^{−2} |

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## Share and Cite

**MDPI and ACS Style**

Alhato, M.M.; Bouallègue, S. Direct Power Control Optimization for Doubly Fed Induction Generator Based Wind Turbine Systems. *Math. Comput. Appl.* **2019**, *24*, 77.
https://doi.org/10.3390/mca24030077

**AMA Style**

Alhato MM, Bouallègue S. Direct Power Control Optimization for Doubly Fed Induction Generator Based Wind Turbine Systems. *Mathematical and Computational Applications*. 2019; 24(3):77.
https://doi.org/10.3390/mca24030077

**Chicago/Turabian Style**

Alhato, Mohammed Mazen, and Soufiene Bouallègue. 2019. "Direct Power Control Optimization for Doubly Fed Induction Generator Based Wind Turbine Systems" *Mathematical and Computational Applications* 24, no. 3: 77.
https://doi.org/10.3390/mca24030077