Scenario-based Multi-objective Optimal Dispatch Considering Forecast Uncertainties

ZHANG Xiao-hui1, a, YAN Ke-ke2, b, ZHONG Jia-qing3 1,2,3Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province,

Yanshan University, 066004, China

aemail: xhzhang@ysu.edu.cn, bemail:yankeke1987@163.com

Keywords: component; Stochastic Dynamic Economic Emission Dispatch; Multi-objective improved Particle Swarm Optimization; Wind Power.

Abstract. Renewable energy recourses and Wind Power Generators are playing an ever-increasing

role in power generation. In this paper, a two stage scenario-based approach considering load/wind

forecast uncertainty is implemented for multi-objective economic emission dispatch problem which

concludes minimize total fuel cost and emission simultaneously. Besides, an improved optimization

algorithm of multi-objective based on Particle Swarm Optimization (MOIPSO) is implemented to

extract the best solution for the stochastic problem. The proposed method is tested on a power

system having 6-unit in order to measure its efficiency and feasibility.

Introduction

The advantage of wind power is that it costs low and almost does not produce greenhouse gas

emissions. The main form of wind energy around the world is large wind farm integration. Wind

power generation has taken great effect to overcome carbon emission of electricity generation and

save the total fuel cost for power generation. On the other hand, Wind power generator has random

nature, the concern for the prediction of load/wind is more increased over time. Two methods have

been developed to model uncertainty in the stochastic programming model[1]

, fuzzy-based

approaches[2]

and scenario-based approaches[3]

. A scenario-based approach is developed to model

the uncertain wind power and load demand. The multi-objective problem is concerned to determine

the optimal outputs of generation units over a period of time while satisfying the constraints. By the

way, we developed MOIPSO approach to solve the multi-objective optimization problem in power

system.

Uncertainty characterization

A scenario based method is used to consider the load/wind uncertainty based on load/wind

forecast error. Fig. 1 shows a typical continuous PDF of the load/wind forecast error along with its

discretization[4]

. On the basis of different load/wind forecast levels and their probabilities obtained

from the probability distribution function, roulette wheel mechanism[5]

is implemented to generate

load/wind scenarios for each hour. For this purpose, at first, the probabilities of different load/wind

forecast levels are normalized such that their summation becomes equal to unity. Then the range of

[0,1] is accumulated by the normalized probabilities as shown in Fig. 2. Random numbers are

generated between [0-1]. Each random number falls in the normalized probability range of a

load/wind forecast level in the RWM. That load/wind forecast level is selected by the RWM for

each hour of a scenario.

Applied Mechanics and Materials Vols. 427-429 (2013) pp 337-340Online available since 2013/Sep/27 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.427-429.337

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σ2 σσ σ2σ3 0 σ31α2α

3α

4α

Fig.1: Typical discretization of the PDF of the load/wind forecast error

Fig. 2: The roulette wheel mechanism for the normalized probabilities of the load/wind forecast error levels

In the stochastic programming, the probability distribution of a random variable is represented

using a finite set of scenarios. Each scenario has an associated probability of occurrence. Thus, the

levels for each scenario can be expressed as follows:

( ) ( )∏ ∑ ∑∈ = =

⋅⋅=Tt

Nl

l

Mw

w

tw

W

stwtl

L

stl WWs1 1

,,,,,, )( βαπ (1)

where Nl and Mw is the number of wind and load forecast levels in each hour, tl ,αand tw,β

represents the probability of the lth load interval, wth wind power interval, respectively. L

stlW ,, andW

stwW ,, are binary parameters indicating whether the intervals are selected in or not. In other

word, for each time interval and each random variable, a random number is produced between 0 and

1.

Finally, scenario reduction techniques can be employed to reduce the number of scenarios, to

eliminate a scenario with very low probability and scenarios that are very similar [4]

. The most

probable and dissimilar scenarios can be extracted.

( ) ( )( )∑

=

=NS

s

s

ss

1

Pr

π

π (2)

where NS is the number of all the scenarios.

Stochastic dynamic economic emission dispatch (EED)

The objective functions[6]

:

)()Pr()()Pr(min2

,,,,

1 1 1

,1

1

1 stiistii

NS

s

T

t

NG

i

isG

NS

s

PcPbasPFsF ++== ∑ ∑∑∑= = −=

(3)

)()Pr()()Pr(min 2

,,,,

1 1 1

,2

1

2 stiistii

NS

s

T

t

NG

i

isG

NS

s

PPsPFsF γβα ++== ∑ ∑∑∑= = −=

(4)

Where ia , ib , ic are the cost coefficients, iα, iβ , iγ

are the emission coefficients; stiP ,, .is the generating

power of unit i; NG is the number of power plant.

Constraints:

stD

NW

j

stw

NG

i

sti PPP ,,

1

,,

1

,, =+∑∑==

(5)

TrPPTr up

ististi

down

i ∆≤−≤∆− − ,1,,, (6)

max

,,,,

min

,, stististi PPP ≤≤ (7)

Where NW is the number of wind power plant; stwP ,, .is the generating power of wind power;

stDP ,, is the load demand;up

ir ,down

ir are the Up/down ramp rates; max

,, stiP ,min

,, stiP are the generation limits.

338 Mechanical Engineering, Industrial Electronics and Information TechnologyApplications in Industry

Multi-objective improved particle swarm optimization (MOIPSO)

The improved strategies of optimal and worst particle, is that the particle are not easily falling

into local optimum and is easier to find the global optimal solution.

( ) ( )( ) ( )

1

,

,

1 1 2 2

3 3 4 4

k k k k k k

best i i gbest ii i

k k k k

i worst i i gworst

V w V c r P X c r P X

c r X P c r X P

+= ∗ + ∗ ∗ − + ∗ ∗ −

+ ∗ ∗ − + ∗ ∗ −

�� �� �� ��� �� ���

��� �� ��� ��

(8)

where c1,c2, c3,c4 is the acceleration coefficient; r1, r2, r3, r4 is the random number.

For multiple objectives, there are usually a heap of solutions which are not simply comparable,

which are often referred as Pareto optimal solution [7]

.Optimal solution is the objective function

point cut, it always falls in the search area boundary.

Simulation and numerical results

In this paper we use 6-unit system as an example to verify the rationality of the proposed

optimization model. Thermal power unit parameters are shown in[6], the wind and load forecast

data is shown in figure 3 and figure 4 respectively.

0 3 6 9 12 15 18 21 2460

80

100

120

140

160

t/h

Win

d F

ore

cast/M

W

0 3 6 9 12 15 18 21 24200

225

250

275

300

325

350

t/h

Load F

ore

cast/M

W

Fig.3 Wind power forecast data Fig.4 Load forecast data

In order to verify the effectiveness and superiority of the proposed MOIPSO algorithm for the

EED problem, it is applied on the 6-unit system. Figure 5 shows the comparison of different

algorithms for the Pareto front. We can see that the algorithm of MOIPSO is more efficient and

superior than the algorithm of MOPSO.

For a better illustration of the performance of the proposed method in the stochastic

optimization, three cases are demonstrated as follows: case 1.EED model without error; 2.The

scenario-based EED model with stochastic error; 3.EED model with determined error, for the wind

forecast error is 30% and the load forecast error is 1%.

All the cases are computed by MOIPSO and the Pareto results are shown in figure 6.From the

results, we get that case 1has a lower costs than case 2 and 3, for the error is not concerned; case 2

has a lower costs than case 3, compared with the determined model, the scenario based stochastic

model has reduced the total costs.

Applied Mechanics and Materials Vols. 427-429 339

1400 1450 1500 1550 1600 16504400

4600

4800

5000

5200

5400

5600

5800

6000

Fuel Cost(thousand $)

Em

issio

n(t

)

EED Front with the algorithm

of MOIPSO

EED Front with the algorithm

of MOPSO

1400 1450 1500 1550 1600 16504400

4600

4800

5000

5200

5400

5600

5800

6000

Fuel Cost(thousand $)

Em

issio

n(t

)

EED Front without Error

EED Front with Wind Stochastic Error

EED Front with Determind Wind/Load

Forecast Error

Figure.5 Results of different algorithms Figure.6 Results of different cases

Conclusion

This Paper provides a scenario-based approach to characterize the load/wind forecast error in the

6-unit test system can lead to a more efficient utilization of energy resources and the total fuel cost

of thermal units and emission can be reduced significantly using wind power. Results were showed

accuracy of presented multi-objective EED model.

Acknowledgement

In this paper, the research was sponsored by the Natural Science Foundation of Hebei Province

(Project No. E2013203113).

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340 Mechanical Engineering, Industrial Electronics and Information TechnologyApplications in Industry

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