Optimal Dispatch of Power System with Stochastic Wind Generation

Shuang Wang1,a, Hongming Yang1,b, Shuang Zuo2,c and Wenjun Xu1,d 1The College of Electrical and Information Engineering, Changsha University of Science and

Technology, Changsha, Hunan Province 410114, P.R.China

2The College of Chemical Engineering, Huaqiao University, Xiamen, Fujian Province 361021, P.R.China

awangshuangzs@126.com, byhm5218@163.com, celulihualmaq@163.com, dxwj--163@163.com

Keywords: Optimal dispatch, Smart grid, Copula, Chance constrained programming, Sample average approximation, Genetic algorithm

Abstract. Large-scale wind power incorporated into power grid brings new challenges to optimal

dispatch of power system. Especially, wind power at different locations may has a significant degree

of correlation. A copula function was used to characterize the Joint Probability Distribution (JPD) of

wind power from multiple wind farms considering their correlation. An optimal dispatch model based

on Chance Constrained Programming (CCP) for power system with multiple wind farms was set up.

And Sample Average Approximation (SAA) was proposed to transform the chance constrains.

Finally Genetic Algorithm (GA) was employed to solve the optimal model. Simulation results

indicate that copula function can well express the correlation of wind power from multiple wind

farms and SAA has been a dramatic increase and improvement on the search of solution.

Introduction

In order to solve the problem of energy shortage and environmental pollution, renewable energy

generation is widely concerned. Wind becomes an important form of renewable energy generation

due to its non-pollution, renewablity and other excellent features. Especially with the construction of

Smart Grid, large-scale wind power will be connected to power grid, which brings a great challenge

to optimal dispatch of power system. Therefore, it is important to calculate the JPD of wind power

from multiple wind farms, in order to assess the maximum wind power that can be incorporated into

the system. But wind at different locations may come from the same origin, so that their wind power

has a significant degree of correlation [1]. considering the unit rate of forced shutdown, wake effects,

temperature, calculated the JPD of wind power from multiple wind farms, but in circumstance that

the wind power from multiple wind farms were independent [2]. proposed copula function to describe

the correlation between onshore and offshore wind power by Gaussion-copula function, but it only

studied their linear correlation but nonlinear. Conventional optimal dispatch incorporating wind

power was to realize the smallest generation cost, which satisfied the power balance, spinning

reserve, climbing rate and unit output constraints [3,4], but didn’t consider the correlation of wind

power from multiple wind farms. And CCP had been widely applied to solving the power system

issues incorporating wind power [3,5]. However, the chance constraints are often too complex to be

transformed into their equivalent certain problem. Thus, it’s necessary to simplify the constraints in

actual computation, and then use the existing optimization algorithms to solve it.

A copula function [6], in the following text, is employed to construct the JPD of wind power from

multiple wind farms considering their correlation. An optimal dispatch model based on CCP is set up.

And SAA [7] is employed to transform the CCP, finally the GA [8] is used to solve the model.

Probability Distribution of Wind Power from a single Wind Farm

Wind Speed Distribution. Three-parameter Weibull distribution family is extremely flexible and

can fit very well an extremely wide range of empirical observations. It exhibits a wide range of shapes

for the density and hazard functions, which are suitable to model complex failure data sets. The

Cumulative Probability Distribution (CPD) of the three-parameter Weibull function is given by:

Applied Mechanics and Materials Vols. 37-38 (2010) pp 783-786Online available since 2010/Nov/11 at www.scientific.net© (2010) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.37-38.783

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( ) 1 exp

wk

w ww w

w

vv

c

0, 1w wk c (1)

where wv ,

wc , wk ，

w are the wind speed, scale，shape and location parameters of wind farm w .

CPD of Wind Power. For the randomness of wind speed, wind power is always fluctuating, we

assume that: ( ) 0w wp v , when 0 w inv v or w outv v ; ( ) wk

w w w w wp v a v b , when in w ratev v v ; and

( )w w ratep v p , when rate w outv v v . Then we can get

1/( ) ( ) / wk

w w w w wv p p b a ， where wp is the

wind power available of farm w . inv , ratev ,

outv , ratep are the cut-in, rated, cut-out wind speeds and

rated power respectively, wk ,

wc are the shape parameter and scale parameter described above. ,w wa b

are constants given by Wind Conversion System. Here the wind speed has a given CPD in Eq. 1, so

the CPD of power available from wind farm w can be formulated as:

1 exp ( ) / exp ( ) /

( ) 1 exp / exp ( ) / 0

1

w w

w w

k k

in w w out w w w

k k

w w w w w w out w w w rate

w rate

v c v c p

F p v p c v c p p

p p

（ =0）

（ ）

（ ）

(2)

JPD of Wind Power from Multiple Wind Farms

To accurately estimate the whole outputs from multiple wind farms, it is necessary to calculate their

JPD. Copula functions can express their JPD with their univariate marginal distributions and model

their correlation. Analysis showed that the wind power from multiple farms presented upper right

fat-tail property. The Gumbel-copula function is therefore employed to construct their JPD as:

1

1 2 1 1 2 2, , , , , , expw W w w W WL p p p p InF p InF p InF p InF p

(3)

where 1 2, , , , ,w Wp p p p are wind power available from W wind farms, 1 2, , , , ,w WL p p p p is

their JPD, and 1 1 2 2, , , W WF p F p F p are their marginal distributions respectively, which can be

obtained from Eq. 2. is the parameter to be evaluated by Maximum Likelihood Estimation.

Optimal Model and Simplifying

Optimal Dispatch Model. Because the wind power is stochastic and intermittent so that it is difficult

to get a exact forecast before making a reasonable dispatch scheme. Thus, we can use CCP, which is

a better method and in probability, to describe the randomness of wind power. Based on this, we can

obtain the stochastic optimal dispatch model of power system with multiple wind farms as:

2

i i i i i i

i G

Minf P a P b P c

(4)

Subject to

1 1

G W

i w L

i w

P P P

,, ,i min i i maxP P P , , 1,2, , 1w wP P p w W (5)

where if P is cost of conventional generators; ia , ib , ic are cost coefficients of generator i ; iP is the

conventional generator power and , ,,i min i maxP P are the minimum and maximum limits; G is the number

of conventional power generators; ,w wP p are scheduled and available wind power from farm w , W is

the number of wind farms; LP is the total load of the system; is a given probability level; P is the

probability of {.} happening. The goal of Eq. 4 is to minimize the total cost of conventional generators.

The constrains are power balance, limits of conventional generator power, wind power scheduled is

not more than the available must be satisfied with a given confidence level 1 at least, respectively.

Simplifying the Constraints. The SAA method is adopted here to simplify the constraints to

convert it into certain equivalent problem，which takes maximum preserve convexity of functions to

equivalently replace a number of constrains. So the chance constraints in Eq. 5 can be written as

784 Advances in Engineering Design and Optimization

1 0 1w W w wP max P p . Meanwhile generate decision variable sample datas through Monte-Carlo

simulation to calculate the value of corresponding object function，then determine whether the

sample datas satisfy all the constraints or not. Here, indictor function I , which is 0 when meet the

constraints, or 1，is used. Therefore, chance constraints Eq. 5 can be converted into:

(0, ) 1

1

10 1

T

T w W w wm

m

I max P pT

(6)

where , , , , ,1m 2m wm Wmp p p p are samples generated from JPD described in Eq. 3； m is the

generation times and T is the maximum.

GA Optimization

Recently, GA and Particle Swarm Optimization (PSO) techniques appeared as promising algorithms

for handling the optimization problems. The stochastic optimal dispatch model proposed in this paper

is carried out using the GA optimization technique, in order to implement the search for the optimal

solution. The GA is an optimization method that employs a search process imitated from the

mechanism of biological selection and biological genetics. It searches an optimal solution to the

problems by manipulating a population of strings that represent different potential solutions, each

corresponding to a sample point from the search space.

Simulation Results

Following, we use GA to test the dispatch model and SAA in IEEE 6-Nod System. Parameters: cost

coefficients of conventional generators are 1 1 1, , 0.00145,7.69,563a b c , 1 1,, 130,620,min maxP P ,

2 2 2, , 0.00192,7.86,320a b c , 2 2,, 120,410,min maxP P , 3 3 3, , 0.00487,7.91,100a b c , 3 3,, 68,110,min maxP P ;

History wind speed datas are from Gansu Yumen wind farms. The cut-in, rated, cut-out wind speed

and rated power of two wind farms are 4m/s, 15m/s, 20m/s, 850kW, system load is 800MW;

population size 80Po ， number of populations 100Q , probability of crossover 80%Pc ,

probability of mutation 5%Pm , solution precession 0.001Pr . Simulation one: According to the MLE, we obtain the shape parameter 1 2, 2.14,2.34k k , scale

parameter 1 2, 14.96,13.55c c m/s, 1 2, 8.7,9.8 m. The scatter plot of wind power from two

wind farms is shown in Fig 1. We can see that their wind power has obvious correlation of

non-symmetrical upper tail. So the Gumbel-copula function is the best to express the JPD of them

with θ=1.3. According to the copula JPD of wind power from two wind farms, we assess the

maximum wind power that can be incorporated into the system to determine the best dispatch scheme.

Then we get the minimum total cost of system shown in Table 1, and the curve shown in Fig 2. Also

we calculated the minimum cost with independent CPD of them also shown in Table 1. By

comparison, we see that the dispatch scheme according to copula JPD of wind power from two wind

farms, can save total costs of the system, and reduce the reserve capacity and related costs.

Simulation two: We obtain the total cost 7,630.585$ without using SAA transformation, with little

difference from 7,630.610$ using SAA that can be seen from Fig. 2 (Both confidence level are at 95%

). ut the genetic generation is 13 of SAA transformation, running time of 2.93 seconds, while it gets to

23 and 4.07 seconds without SAA transformation. Therefore, the SAA transformation in the

convergence speed and computation time has been a dramatic increase and improvement.

Table 1 Total costs of the system under different probability distribution

Probability Distribution Available wind power [MW] Total cost [$]

Gumbel-Copula 177.08 7630.610

Independent CPD 150.62 7865.822

Applied Mechanics and Materials Vols. 37-38 785

100 200 300 400 500 600 700 800200

300

400

500

600

700

800

Wind Power P1

Win

d P

ow

er

P2

0 10 20 30 40 50 60 70 80 90 100

7,630

7,631

7,632

7,633

7,634

7,635

7,636

7,637

7,638

Genetic generation of GA

Tota

l cost

of

syste

m

with SAA

Fig. 1 Scatter plot of wind power Fig. 2 Curve of total cost

Conclusion

An optimal dispatch model, considering the correlation of outputs from multiple wind farms, was

formulated based on CCP. A copula function was used to express the JPD and model their correlation.

SAA was proposed to transform the CCP to non-continuous, non-differentiable optimal problem.

Simulation results show that copula function can well express the correlation of outputs from multiple

wind farms and SAA has been a dramatic increase and improvement on the search of solution.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (70601003), the

Program for New Century Excellent Talents in University of China, the National Science Fund for

Distinguished Young Scholars (70925006), and the Key Project of Science and Technology of Hunan

Province (2008FJ1006).

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786 Advances in Engineering Design and Optimization

Advances in Engineering Design and Optimization 10.4028/www.scientific.net/AMM.37-38 Optimal Dispatch of Power System with Stochastic Wind Generation 10.4028/www.scientific.net/AMM.37-38.783

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