# A Dynamic Economic Dispatch Model Incorporating Wind Power Based on Chance Constrained Programming

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## Abstract

**:**

## 1. Introduction

## 2. DED Problem Formulation

#### 2.1. Objective Function

- ${f}_{\text{cos}t}$ is the total generation cost over the whole time horizon;
- $T$ is the number of periods;
- $I$ is the number of thermal units;
- ${p}_{i,t}$ is the power output (MW) of the $i$th unit corresponding to time period $t$;
- ${C}_{i}({p}_{i,t})$ is the generation cost of the $i$th unit corresponding to time period $t$;
- ${E}_{i}({p}_{i,t})$ is the valve point loading effect of the $i$th unit corresponding to time period $t$;

#### 2.2. System and Unit Constraints

#### 2.2.1. Power Balance Constraints

_{d,t}in all time period:

_{w,t}is the scheduled wind power of wind farm at time t.

#### 2.2.2. Generation Limits of Thermal Units and Wind Farm

_{max}is the installed capacity of wind farm.

#### 2.2.3. Chance Constraint on Wind Power

_{t}is a random variable representing the wind power generation at time t. ρ is the confidence level. Equation (7) defines the probability that the scheduled wind power can be realized is greater than or equal to ρ. Furthermore, Equation (7) sets a reasonable upper bound for wind power generation, and the probability that this upper bound can be realized is no less than ρ. The larger the confidence level is, the less stochastic the wind power is, and hence the more reliable the power system is. Especially, when ρ = 1, there is no wind power in system, and the DED problem is changed into a deterministic one.

#### 2.2.4. Ramp Rate Limits of Thermal Units

_{60}is the operating period, i.e., 1 h.

#### 2.2.5. Spinning Reserve Constraints

_{10}is 10 min. ${p}_{i,t}^{\text{max}}$ and ${p}_{i,t}^{\text{min}}$ are upper and lower generation limits of unit i including ramp rate limits at time t, and ${p}_{i,t}^{\text{max}}=\text{min}({p}_{i\text{max}},{p}_{i,t-1}+{\Delta}_{i,u})$, ${p}_{i,t}^{\text{min}}=\text{max}({p}_{i\text{min}},{p}_{i,t-1}-{\Delta}_{i,d})$. ${r}_{w,t}^{u}$ and ${r}_{w,t}^{d}$ are the URFW and the DRRW to follow the sudden decrease and increase in wind power at time t.

## 3. Equivalent Transformation of the DED Model

#### 3.1. Beta Distribution of Wind Power

#### 3.2. Equivalent Transformation

## 4. Improved PSO Approach

- J is the population size;
- $\text{\omega}(k)$ is the dynamic inertia weight factor, and can be dynamically set with the following equation [6]:

#### 4.1. Feasible Region Adjustment Strategy

_{mt}is the output of the mth generator at time t. The row vectors represent the output of the individual generator each hour one day.

#### 4.2. Hill Climbing Search Operation

#### 4.3. DED Constraints Handling Using PSO-HCSO

#### 4.4. The Procedure of Improved PSO Approach

## 5. Simulation Results and Discussions

Period | Expected Value (MW) | Standard Deviation (MW) | α | β | Period | Expected Value (MW) | Standard Deviation (MW) | α | β |
---|---|---|---|---|---|---|---|---|---|

1 | 70.4 | 17.25 | 10.38 | 18.81 | 13 | 133.24 | 33.25 | 4.58 | 2.23 |

2 | 55.50 | 13.87 | 11.24 | 28.87 | 14 | 129.50 | 32.37 | 4.88 | 2.58 |

3 | 34.50 | 9.63 | 10.43 | 49.45 | 15 | 147.15 | 36.75 | 3.37 | 1.17 |

4 | 28.3 | 10.23 | 6.42 | 38.47 | 16 | 140.7 | 35.40 | 3.86 | 1.57 |

5 | 42.1 | 10.54 | 12.35 | 45.73 | 17 | 133.4 | 33.25 | 4.58 | 2.22 |

6 | 59.50 | 14.87 | 10.90 | 25.37 | 18 | 108.5 | 27.13 | 6.68 | 5.51 |

7 | 70.56 | 17.17 | 10.51 | 18.99 | 19 | 84.7 | 21.06 | 8.83 | 11.81 |

8 | 80.50 | 20.12 | 9.09 | 13.27 | 20 | 77.3 | 19.25 | 9.44 | 14.74 |

9 | 94.50 | 23.62 | 7.89 | 8.64 | 21 | 66.5 | 16.62 | 10.30 | 20.36 |

10 | 112.4 | 28.84 | 5.99 | 4.57 | 22 | 42.2 | 10.5 | 12.50 | 46.14 |

11 | 126.7 | 31.53 | 5.17 | 2.91 | 23 | 35.3 | 8.75 | 13.20 | 60.82 |

12 | 130.5 | 32.62 | 4.80 | 2.48 | 24 | 63.8 | 15.75 | 10.80 | 22.72 |

J | K | H | hcount | ε |
---|---|---|---|---|

40 | 200 | 200 | 4 | h/(H+0.00001) |

#### 5.1. Comparisons Among the Three Cases of the DED Model

- Case (1): the DED model without considering wind power;
- Case (2): the DED model without considering wind effect in the reserve constraint;
- Case (3): the proposed DED model in this paper.

Cases | Confidence Level | Average Generation Cost ($) | ||
---|---|---|---|---|

System 1 | System 2 | System 3 | ||

Case (1) | 1 | 278,903.3311 | 736,673.0726 | 952,079.3252 |

Case (2) | 0.9 | 264,895.0451 | 723,729.9923 | 939,455.4055 |

0.5 | 258,950.8628 | 719,280.7467 | 928,252.2498 | |

0.1 | 253,792.2061 | 717,075.4436 | 921,874.4037 | |

Case (3) | 0.9 | 265,158.6006 | 724,077.6842 | 937,747.6542 |

0.5 | 260,537.2281 | 720,665.7621 | 930,092.0212 | |

0.1 | 257,194.5871 | 717,358.1034 | 920,580.2708 |

Unit Index | Output/MW (t = 1) | Output/MW (t = 5) | Output/MW (t = 9) | Output/MW (t = 13) | Output/MW (t = 17) | Output/MW (t = 21) |
---|---|---|---|---|---|---|

1 | 6.3027 | 2.4145 | 3.8440 | 3.1906 | 7.1055 | 6.0383 |

2 | 2.4001 | 3.8585 | 3.0919 | 3.8967 | 4.2176 | 4.7125 |

3 | 5.0852 | 2.4041 | 2.4051 | 5.3315 | 6.9680 | 5.5158 |

4 | 2.4005 | 2.4001 | 3.7132 | 2.4245 | 9.1081 | 5.4245 |

5 | 2.4008 | 2.4000 | 2.4001 | 7.7075 | 2.7151 | 2.4124 |

6 | 4.0328 | 9.1759 | 4.0017 | 7.5375 | 14.5461 | 7.5095 |

7 | 7.0616 | 4.0334 | 13.6143 | 8.2265 | 16.3442 | 6.4664 |

8 | 11.4758 | 4.0024 | 15.0108 | 13.5385 | 4.1028 | 8.2579 |

9 | 4.0005 | 4.0094 | 4.0025 | 13.9355 | 14.7954 | 4.0497 |

10 | 19.2252 | 47.0962 | 75.9820 | 75.9848 | 75.9389 | 52.5536 |

11 | 18.3423 | 55.2031 | 61.0610 | 53.5547 | 75.6070 | 31.5227 |

12 | 75.8896 | 15.4675 | 73.2103 | 75.9948 | 75.6298 | 53.2694 |

13 | 44.3981 | 62.5945 | 21.5289 | 40.1115 | 62.9386 | 75.9777 |

14 | 29.0682 | 78.5853 | 67.5916 | 75.0436 | 93.0661 | 91.2458 |

15 | 25.1687 | 25.0787 | 99.7008 | 58.6896 | 80.7636 | 69.7929 |

16 | 66.9136 | 25.1137 | 90.0044 | 60.3383 | 78.6772 | 97.6501 |

17 | 111.8911 | 54.4301 | 150.9160 | 129.1319 | 151.3966 | 102.5665 |

18 | 127.2128 | 91.5275 | 119.8840 | 105.7696 | 155.0000 | 148.3436 |

19 | 82.4911 | 81.7769 | 142.5084 | 101.8499 | 149.4214 | 104.2932 |

20 | 103.2164 | 84.7880 | 154.9953 | 152.8019 | 136.8211 | 102.9801 |

21 | 149.4256 | 79.7549 | 94.8661 | 176.8049 | 163.8401 | 167.1249 |

22 | 131.0308 | 142.1999 | 108.1115 | 171.1631 | 182.4127 | 84.5785 |

23 | 127.8570 | 125.8213 | 135.2337 | 154.7178 | 163.3203 | 153.6023 |

24 | 276.5693 | 158.4972 | 306.4523 | 290.2500 | 284.4019 | 287.2902 |

25 | 239.6986 | 270.5251 | 379.2529 | 328.7933 | 398.3654 | 331.9537 |

26 | 359.0842 | 372.9581 | 320.0320 | 329.1593 | 378.8603 | 275.4983 |

WF | 19.3572 | 19.3837 | 54.5853 | 62.0524 | 63.6360 | 27.8695 |

#### 5.2. Wind Penetration as a Function of the Confidence Level Under Two Systems

#### 5.3. Discussion about the Optimal Reserve Allocation

**Figure 10.**The optimal USR allocation and URR plus reserve for load forecast error in system 1 with ρ = 0.9.

**Figure 12.**The optimal USR allocation and URR plus reserve for load forecast error for 24 h in system 2 with ρ = 0.9.

#### 5.4. Comparisons between the PSO-HCSO and PSO without HCSO

H | Average Generation Cost ($) | Average Time (s) |
---|---|---|

100 | 269,280.7296 | 328.2577 |

200 | 265,158.6006 | 386.8114 |

300 | 262,533.7753 | 436.2356 |

400 | 261,153.0924 | 538.3722 |

Approach | Confidence Level | Average Generation Cost ($) | Average Time (s) | ||
---|---|---|---|---|---|

PSO-HCSO | PSO without HCSO | PSO-HCSO | PSO without HCSO | ||

System 1 | 0.9 | 265,158.6006 | 273,039.6325 | 386.8114 | 336.6071 |

0.5 | 260,537.2281 | 263,514.4852 | 359.3236 | 308.3647 | |

0.1 | 257,194.5871 | 262,368.0024 | 373.6618 | 317.1135 | |

System 2 | 0.9 | 724,077.6842 | 731,239.0336 | 418.3461 | 409.6045 |

0.5 | 720,665.7621 | 727,088.0353 | 402.1863 | 386.9233 | |

0.1 | 717,358.1034 | 720,196.5471 | 447.1164 | 422.8967 | |

System 3 | 0.9 | 937,747.6542 | 945,013.2889 | 456.7289 | 420.0858 |

0.5 | 930,092.0212 | 938,816.0995 | 428.3801 | 416.1260 | |

0.1 | 920,580.2708 | 929,187.2603 | 470.2236 | 436.2757 |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## List of Abbreviations and Symbols

## Abbreviations

ED | Economic dispatch |

DED | Dynamic economic dispatch |

USR | Up spinning reserve |

DSR | Down spinning reserve |

URR | Up reserve requirement |

DRR | Down reserve requirement |

PSO | Particle swarm optimization |

HCSO | Hill climbing search operation |

PSO-HCSO | Particle swarm optimization with hill climbing search operation |

Probability density function | |

CDF | Cumulative density function |

FRA | Feasible region adjustment |

## Symbols

T | Number of Periods |

I | Number of thermal units |

t | Index of time period,
$t=1,2,\cdots ,T$ |

i | Index of thermal unit,
$t=1,2,\cdots ,I$ |

${f}_{\text{cos}t}$ | Total generation cost |

${p}_{i,t}$ | Power output of thermal unit i at time t |

${p}_{w,t}$ | Scheduled wind power of wind farm at time t |

${p}_{d,t}$ | Load demand at time t |

${C}_{i}({p}_{i,t})$ | Generation cost of thermal unit i at time t |

${a}_{i}$, ${b}_{i}$, ${c}_{i}$ | Cost coefficients of thermal unit i |

${E}_{i}({p}_{i,t})$ | Valve point loading effect of thermal unit i at time t |

${e}_{i}$, ${f}_{i}$ | Coefficients related to valve point effect of thermal unit i |

${p}_{i\text{min}}$, ${p}_{i\text{max}}$ | Minimum and Maximum generation limits of thermal unit i |

${w}_{t}$ | Actual wind generation, a random variable |

${w}_{\text{max}}$ | Installed capacity of wind farm |

ρ | Confidence level |

${\Delta}_{i,u}$, ${\Delta}_{i,d}$ | Upper and lower ramp rate limits of thermal unit
$i$ |

T_{10}, T_{60} | 10 min and 1 h respectively |

${r}_{w,t}^{u}$ | URFW at time
$t$ |

${r}_{w,t}^{d}$ | DRRW at time
$t$ |

α, β | Parameters of the beta function |

${\text{\mu}}_{t}$, ${\text{\sigma}}_{t}$ | Mean value and the standard deviation |

${f}_{{W}_{t}}$, ${F}_{{W}_{t}}$ | PDF and CDF of actual wind generation at time t |

${f}_{{X}_{t}}$, ${F}_{{X}_{t}}$ | PDF and CDF of normalized actual wind generation at time t |

D | Dimension of the particle |

k, K | Current number and maximum number of iteration |

${V}_{j}^{k}$, ${Y}_{j}^{k}$ | Velocity and position of the jth particle at generation k |

$\text{\omega}(k)$ | Dynamic inertia weight factor |

${\text{\phi}}_{1}(k)$, ${\text{\phi}}_{2}(k)$ | Acceleration coefficients corresponding to cognitive and social behavior |

$pbes{t}_{j}^{k}$ | Personal best position of jth particle at generation k |

$gbes{t}^{k}$ | Global best position at generation k in the whole population |

λ | Penalty factor |

$P{F}_{g,t}$ | Penalty functions |

H | Maximum number of the HCSO |

$hcount$ | The number of times that $gbest$ does not change continuously |

## Conflicts of Interest

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**MDPI and ACS Style**

Cheng, W.; Zhang, H. A Dynamic Economic Dispatch Model Incorporating Wind Power Based on Chance Constrained Programming. *Energies* **2015**, *8*, 233-256.
https://doi.org/10.3390/en8010233

**AMA Style**

Cheng W, Zhang H. A Dynamic Economic Dispatch Model Incorporating Wind Power Based on Chance Constrained Programming. *Energies*. 2015; 8(1):233-256.
https://doi.org/10.3390/en8010233

**Chicago/Turabian Style**

Cheng, Wushan, and Haifeng Zhang. 2015. "A Dynamic Economic Dispatch Model Incorporating Wind Power Based on Chance Constrained Programming" *Energies* 8, no. 1: 233-256.
https://doi.org/10.3390/en8010233