# Risk Reserve Constrained Economic Dispatch Model with Wind Power Penetration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Risk Reserve Constrained Optimization Model

_{n}is the power from n th conventional generator; N is the number of conventional generators; a

_{n},b

_{n},c

_{n}are constants of fuel cost function of n th conventional generator.

_{n}is the scheduled output of the n th wind-powered generator; N

_{w}is the number of wind turbine generator; d

_{n}is the direct cost coefficient for the n th wind-powered generator. In order to simplify the problem, both the output and direct cost coefficient for each wind turbine generator are assumed to be identical in the wind farm. So the total amount of wind power generation in a wind farm is $W=n{W}_{n}$.

_{d,n}is the required down spinning reserve of the n th generator; k

_{p}is the down reserve cost coefficient.

_{u,n}is the required up spinning reserve of the n th generator; k

_{r}is the up reserve cost coefficient.

_{u}and down spinning reserve R

_{d}will be calculated in this model.

_{u}must be not more than SR

_{su}:

_{d}must be not more than SR

_{sd}:

_{L}is the forecast value of load demand; ${P}_{n}^{\mathrm{max}}$ and ${P}_{n}^{\mathrm{min}}$ are the maximum and minimum power from n th conventional generator, respectively; ${P}_{r}{N}_{w}$ is the wind power capacity; α and β are the risk thresholds; $S{R}_{su}$ is the total up spinning reserve supplied by all the conventional generators; $S{R}_{su}^{m}$ is the total up spinning reserve supplied by all the conventional generators except m th conventional generator; $S{R}_{sd}$ is the total down spinning reserve supplied by all the conventional generators; $S{R}_{sd}^{m}$ is the total down spinning reserve supplied by all the conventional generators except m th conventional generator.

## 3. Stochastic Characterizations of Wind Power and Load Demand

#### 3.1. Probability Functions of Wind Speed and Wind Power

_{i}is the cut-in wind speed; v

_{r}is the rated wind speed; v

_{0}is the cut-out wind speed; P

_{r}is the rated wind power from a wind-powered generator; $a=\frac{{P}_{r}{v}_{i}^{3}}{{v}_{i}^{3}-{v}_{r}^{3}},\text{\hspace{0.17em}}b=\frac{{P}_{r}}{{v}_{r}^{3}-{v}_{i}^{3}}$.

_{w}is the number of wind turbine generator.

_{av}are used to indicate the random variables of wind speed and available wind power output respectively. It is easy to verify that W

_{av}is a mixed random variable. For the discrete portions, the probability at minimum and maximum are given in the following two formulas.

_{av}= 0 can be calculated by the probability of $V<{v}_{i}$ or $V>{v}_{o}$ as follows:

_{av}is continuous between minimum and maximum. Since the wind speed has a given distribution, it is necessary to convert that distribution to a wind power distribution. The transformation can be accomplished according to formulas (22) and (23) as follows:

#### 3.2. Probability Function of Load Forecast Error

#### 3.3. Total Stochastic Characterization

_{av}and $\text{\Delta}{P}_{L}$:

_{av}can be derived by a transformation from wind speed to wind power generation, and the correlation between W

_{av}and $\text{\Delta}{P}_{L}$ is neglected in the following calculation.

_{av}and $\text{\Delta}{P}_{L}$:

_{Z}(z):

## 4. Method for Solving the Proposed ED Model

## 5. Numerical Simulations

_{i}= 4 m/s, v

_{r}= 12.5 m/s and v

_{o}= 20 m/s are used. It is assumed that the forecast value of wind speed is 8.5 m/s, which indicates the expectation of normally distributed wind speed. The corresponding standard deviation is set to 10% of mean value. During the calculation of minimum generation cost, the risk threshold α is set to 0.03. The risk threshold β is set to be equal to α. The direct cost of wind energy is 50 USD/MWh. The up and down spinning reserve cost coefficients are 2.4 USD/MWh and 30 USD/MWh, respectively. The generator constants of fuel cost function and the active power limits are shown in [24]. From this reference, it can be seen that the 10

^{th}conventional generator cannot be optimized because its upper and lower powers are both equal to 55 MW. So there are nine conventional generators left which will be scheduled in the optimization problem. The probability of conventional generator trip is listed in Table 1.

G | Pr | G | Pr |
---|---|---|---|

1 | 0.0008 | 6 | 0.0005 |

2 | 0.002 | 7 | 0.0008 |

3 | 0.0003333 | 8 | 0.002 |

4 | 0.002 | 9 | 0.00033 |

5 | 0.002 | 10 | 0.002 |

G | P( MW) | G | P( MW) |
---|---|---|---|

1 | 470.0 | 7 | 130.0 |

2 | 460.0 | 8 | 50.51 |

3 | 340.0 | 9 | 20.00 |

4 | 230.5 | W | 91.01 |

5 | 243.0 | R_{u} | 129.5 |

6 | 160.0 | R_{d} | 62.52 |

Method | Cost($) | Iter | Time(s) |
---|---|---|---|

IP | 59433 | 21 | 20.71 |

PCIP | 59432 | 13 | 14.68 |

#### 5.1. Computational Performance of the Predictor-Corrector IP Method

#### 5.2. Effects of Risk Threshold on the Optimal Outputs

#### 5.3. Effects of Up Reserve Cost Coefficient on the Optimal Outputs

#### 5.4. Effects of Down Reserve Cost Coefficient on the Optimal Outputs

_{p}varying from 27 USD/ MWh to 32 USD/MWh. When down spinning reserve cost coefficient is smaller than 30.5 USD/MWh, the corresponding total generation cost and wind power output depicted in Figure 8 and Figure 9 increase respectively with the rise of down spinning reserve cost coefficient. Figure 10 demonstrates the impacts of various k

_{p}on the required up and down spinning reserves. The variation trend of the required up spinning reserve in this figure is similar to that of Figure 9. However, the required down spinning reserve will be decreased due to the increasing penalty cost coefficient. If the down spinning reserve cost coefficient continues to rise, the required down spinning reserve will reach zero. That means the term of down spinning reserve cost in the objective function won’t have an effect on the solution of ED issue. Therefore, the total generation cost, scheduled wind power output and required up spinning reserve will also remain unchangeable.

## 6. Conclusions

## References

- Ummels, B.; Gibescu, C.M.; Pelgrum, E.; Kling, W.L.; Brand, A.J. Impacts of wind power on thermal generation unit commitment and dispatch. IEEE Trans. Energy Convers.
**2007**, 22, 44–51. [Google Scholar] [CrossRef] - Lee, T.Y. Optimal spinning reserve for a wind-thermal power system using EIPSO. IEEE Trans. Power Syst.
**2007**, 22, 1612–1621. [Google Scholar] [CrossRef] - Chen, C.L. Optimal wind-thermal generating unit commitment. IEEE Trans. Energy Convers.
**2008**, 23, 273–280. [Google Scholar] [CrossRef] - Chen, C.L.; Lee, T.Y.; Jan, R.M. Optimal wind-thermal coordination dispatch in isolated power systems with large integration of wind capacity. Energy Conv. Manage.
**2006**, 47, 18–19. [Google Scholar] [CrossRef] - Miranda, V.; Hang, P.S. Economic dispatch model with fuzzy wind constraints and attitudes of dispatchers. IEEE Trans. Power Syst.
**2005**, 20, 2143–2145. [Google Scholar] [CrossRef] - Wang, L.; Singh, F.C. Tradeoff between risk and cost in economic dispatch including wind power penetration using particle swarm optimization. In Proceedings of the 2006 International Conference on Power System Technology, Chongqing, China, 5 September 2006.
- Hong, Y.Y.; Li, C.T. Short-term real-power scheduling considering fuzzy factors in an autonomous system using genetic algorithms. Inst. Eng. Technol.
**2006**, 153, 684–692. [Google Scholar] [CrossRef] - Hetzer, J.; Yu, D.C. An economic dispatch model incorporating wind power. IEEE Trans. Energy Convers.
**2008**, 23, 603–611. [Google Scholar] [CrossRef] - Justus, C.G.W.; Hargraves, R.; Mikhail, A.; Graber, D. Methods for estimating wind speed frequency distributions. J. Appl. Meteorol.
**1978**, 17, 350–353. [Google Scholar] [CrossRef] - Jabr, R.A.; Pal, B.C. Intermittent wind generation in optimal power flow dispatching. IET Gener. Transm. Distrib.
**2009**, 3, 66–74. [Google Scholar] [CrossRef] - Wang, J.H.; Shahidehpour, M.Z.; Li, Y. Security-constrained unit commitment with volatile wind power generation. IEEE Trans. Power Syst.
**2008**, 23, 1319–1327. [Google Scholar] [CrossRef] - Bouffard, F.; Galiana, F.D. Stochastic security for operations planning with significant wind power generation. IEEE Trans. Power Syst.
**2008**, 23, 306–316. [Google Scholar] [CrossRef] - Soder, L. Reserve margin planning in a wind-hydro-thermal power system. IEEE Trans. Power Syst.
**1993**, 8, 564–571. [Google Scholar] [CrossRef] - Siahkali, H.; Vakilian, M. Stochastic unit commitment of wind farms integrated in power system. Electr. Power Syst. Res.
**2010**, 80, 1006–1017. [Google Scholar] [CrossRef] - Doherty, R.; Malley, M.O. A new approach to quantify reserve demand in systems with significant installed wind capacity. IEEE Trans. Power Syst.
**2005**, 20, 587–595. [Google Scholar] [CrossRef] - Lange, M. On the uncertainty of wind powr predictions-analysis of the forecast accuracy and statistical distribution of errors. Solar Energy Eng.
**2005**, 127, 177–184. [Google Scholar] [CrossRef] - Fabbri, A.; Roman, T.G.S.; Abbad, J.R.; Quezada, V.H.M. Assessment of the cost associated with wind generation prediction errors in a liberalized electricity market. IEEE Trans. Power Syst.
**2005**, 20, 1440–1446. [Google Scholar] [CrossRef] - Matevosyan, J.; Soder, L. Minimization of imbalance cost trading wind power on the short-term power market. IEEE Trans. Power Syst.
**2006**, 21, 1396–1404. [Google Scholar] [CrossRef] - Zhao, M.; Chen, Z.; Blaabjerg, F. Probabilistic capacity of a grid connected wind farm based on optimization method. Renewable Energy
**2006**, 31, 2171–2187. [Google Scholar] [CrossRef] - Valenzuela, J.; Mazumdar, M.; Kapoor, A. Influence of temperature and load forecast uncertainty on estimation of power generation production costs. IEEE Trans. Power Syst.
**2000**, 15, 668–674. [Google Scholar] [CrossRef] - Billinton, R.; Huang, D. Effects of load forecast uncertainty on bulk electric system reliability evaluation. IEEE Trans. Power Syst.
**2008**, 23, 418–425. [Google Scholar] [CrossRef] - Wu, Y.C.; Debs, A.; Marsten, R.E. A direct nonlinear predictor—Corrector primal-dual interior point algorithm for optimal power flows. IEEE Trans. Power Syst.
**1994**, 9, 876–883. [Google Scholar] - Zhou, W.; Peng, Y.; Sun, H. Probabilistic wind power penetration of power system using nonlinear predictor-corrector primal-dual interior-point method. In Proceedings of the 2008 IEEE Electric Utility Deregulation and Restructuring and Power Technologies Conference, Nanjing, China, 6–9 April 2008.
- Attaviriyanupap, P.; Kita, H.; Tanaka, E.; Hasegawa, J. A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function. IEEE Trans. Power Syst.
**2002**, 17, 411–416. [Google Scholar] [CrossRef]

© 2010 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 license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, W.; Sun, H.; Peng, Y.
Risk Reserve Constrained Economic Dispatch Model with Wind Power Penetration. *Energies* **2010**, *3*, 1880-1894.
https://doi.org/10.3390/en3121880

**AMA Style**

Zhou W, Sun H, Peng Y.
Risk Reserve Constrained Economic Dispatch Model with Wind Power Penetration. *Energies*. 2010; 3(12):1880-1894.
https://doi.org/10.3390/en3121880

**Chicago/Turabian Style**

Zhou, Wei, Hui Sun, and Yu Peng.
2010. "Risk Reserve Constrained Economic Dispatch Model with Wind Power Penetration" *Energies* 3, no. 12: 1880-1894.
https://doi.org/10.3390/en3121880