Economic Dispatch for Power System included Wind and Solar Thermal energy

 

 

Saoussen BRINI, Hsan Hadj ABDALLAH, and Abderrazak OUALI

 

ENIS, Dép. Génie Electrique, 3038 Sfax, Tel.:74 274 088

E-mails: ingenieurbrini@yahoo.fr, hsan.haj@enis.rnu.tn,  abderrazak.ouali@enis.rnu.tn

 

 

Abstract

With the fast development of technologies of alternative energy, the electric power network can be composed of several renewable energy resources. The energy resources have various characteristics in terms of operational costs and reliability. In this study, the problem is the Economic Environmental Dispatching (EED) of hybrid power system including wind and solar thermal energies. Renewable energy resources depend on the data of the climate such as the wind speed for wind energy, solar radiation and the temperature for solar thermal energy. In this article it proposes a methodology to solve this problem. The resolution takes account of the fuel costs and reducing of the emissions of the polluting gases. The resolution is done by the Strength Pareto Evolutionary Algorithm (SPEA) method and the simulations have been made on an IEEE network test (30 nodes, 8 machines and 41 lines).

Keywords

economic dispatch, total cost, active losses, multi objectives optimization, evolutionary algorithms, SPEA, renewable energy

 

 

Introduction

 

The economic and environmental problems in the power generation have received considerable attention. The apparition of the energy crisis and the excessive increase of the consumption have obliged production companies to implant renewable sources. However, this production poses many technical problems for their integration in the electric system.

The economic dispatch [8, 16] is a significant function in the modern energy system. It consists in programming correctly the electric production in order to reduce the operational cost [4, 7, 10, 15]. Recently, the wind power and solar thermal power attracted much attention like promising renewable energy resources [1, 6, 11, 18, 19, 22].

The problem is formulated as a multiobjective optimization problem [3, 5, 9, 24, 25]. It consists in distributing the active and renewable productions between the power stations of the most economic way, to reduce the emissions of the polluting gases and to maintain the stability of the network after penetration of renewable energy. The number of decision variables of the problem is related to all the nodes of the network. 

 

Renewable energy

In this study, it is interested in two types of energies; wind power and thermal solar energy.

 

Wind energy

The mechanical power recovered by a wind turbine can be written in the form [6, 17, 23]:

                       

(1)

where CP, is the aerodynamic coefficient of turbine power (it characterizes the aptitude of the aerogenerator to collect wind power), ρ is the air density, Rp the turbine ray and  VW wind speed. The power coefficient value CP, depends on the rotation speed of turbine and wind speed.

 

Mechanical adjustment of the wind power

Wind turbine is dimensioned to develop a nominal power Pn from a nominal wind speed Vn. For wind speeds higher than Vn, the wind mill must modify these aerodynamic parameters in order to avoid the mechanical overloads, so that the power recovered by the turbine does not exceed the nominal power for which the wind mill was designed.

Figure 1. Diagram of the useful power according to the wind speed.

 

According to the figure1, the characteristic of power according to the wind speed comprises four zones. Zone 1, where Pw = 0, zone 2, in which useful power depends on wind speed. Vw, zone 3, generally where provided power Pw remains appreciably equal to Pn and finally zone 4,Pw = 0

 

Solar energy

Solar energy is energy produced by the solar radiation, directly or in a diffuse way through the atmosphere. Thanks to various processes, it can be transformed into another form of useful energy for the human activity, in particular in electricity or heat [14, 17, 23].

The maximum power provided by a solar panel is given by the following characteristic [14, 17]:

                       

(2)

Ec is solar radiation, Tjref is the reference temperature of the panels of 25°C, Tj is the cells junction temperature (°C), P1 represent the characteristic dispersion of the panels and the value for one panel is included enters 0.095 to 0.105 and the parameter P2=-0.47%/C°; is the drift in panels temperature [14].

The addition of one parameter P3 to the characteristic, gives more satisfactory results:

                       

(3)

This simplified model makes it possible to determine the maximum power provided by a group of panels for solar radiation and panel temperature given, with only three constant parameters P1, P2 and P3 and simple equation to apply.

A thermal solar power station consists of a production of solar system of heat which feeds from the turbines in a thermal cycle of electricity production.

 

 

Formulation of problem

The control system problem can be treated as follows:

Of absence of the auxiliary elements, the problem consists in extracting the maximum of power from the renewable sources. Then, we slice this power of the total demand PD. the remaining total demand, will distributed between the thermal power stations. The problem is reduced for a speed wind Vw and solar radiation Ec given to minimize the thermal cost functions and the emissions of polluting gases.

To approach to the reality, it is obvious that to must take account of the variation of the wind and solar radiation that can be done by using the techniques of the neurons networks which consists in forming a data base for various wind speed Vw, solar radiation Ec and total power demand PD. The neurons network is composed of three layers, the entries layer is formed by Vw, Ec et PD; the hidden internal layer which the number of neurons is variable and the exit layer which consists of 10 neurons which represent the minimal cost F1, the emissions of polluting gases F2, the generating nodes powers . The structure of this network is given by the figure (2).

Figure 2. Structure du réseau de neurones utilisé

 

Objective functions

Fuel cost function

 

The fuel cost function  in $/h is represented by a quadratic function as follow [2, 5, 19]:

                       

(4)

The coefficients,  and are appropriate to every production unit,  is the real power output of i-th generator and  is the number of thermal generators.

 

                        Emission fonction

The atmospheric emission can be represented by a function that links emissions with the power generated by every unit. The emission of SO2 depends on fuel consumption and has the same form as the fuel cost [8, 13].

The emission of NOx is difficult to predict and his production is associated to many factors as the temperature of the boiler and content of the air [12].

The emission function in ton/h which represents SO2 and NOx emission is a function of generator output and is expressed as follow [20]:

                       

(5)

Where, and  are the coefficients of emission function corresponding to the i-th generator. These three parameters are determined by adjustment techniques of curves based on reel tests [13].

 

Problem constraints

The problem constraints are five types:

·        Production capacity constraints

The generated real power of each generator at the bus i is restricted by lower limit and upper limit:

                       

(6)

 

·        Power balance constraint

The total power generation and the wind power must cover the total demand and the power loss p in transmission lines, so we have:

 

                       

(7)

·        Active power loss constraint

Active power loss of the transmission and transport lines, are positives:

                       

(8)

·        Renewable power constraint:

The renewable power used for dispatch should not exceed the 30% of total power demand:

                       

(9)

Thus, the problem to be solved is formulated as follow:

·        Minimize:()

Under:

 

 

Multi objectives Optimization

 

Principle

The multi-objective optimization problem is formulated in general as follow:

                       

(10)

with:

    : number of objectives functions

   : number of equality and inequality respectively constraints

         : decision vector.

            Two solutions x1 and x2 of such optimization problem, we could have one which dominates the other or none dominates the other.

In a minimization problem, a solution x1 dominates other solution x2 if the following two conditions are satisfied:

                       

(11)

Define by  the satisfiesability set, that is to say:

where

 and

A decision vector is none dominated compared to a set, if:

                       

(12)

The optimize solutions set that are non-dominated within the entire search space are denoted as Pareto-optimal and the set of objectives vectors corresponding constitute the Pareto-optimal set or Pareto-optimal front.

 

SPEA approach (Strength Pareto Evolutionary Algorithm)

In [18], Zitzler and Thiele propose an elitist evolutionary approach to solve a multi objective problem which is called Strength Pareto Evolutionary Algorithm (SPEA). The Elitism is introduced by an external Pareto set. This set stores the non-dominated solutions funded during the resolution of the problem. In order to reduce the size of the external set, an average linkage based on hierarchical clustering algorithm is used without destroying the characteristics of the trade-off front.

Noting by:

 P : the current population.

: the external population.

: the size of current population.

 the fitness of an individual i.

the strength of an individual i.

The assignment procedure to calculate the fitness values is the following:

·        Step 1: For each individual  is assigned a reel value called strength. is proportional to the number of individuals in the current population dominated by the individual i in the external Pareto set. It can be calculated as follows:

For an individual

                       

(13)

The strength of a Pareto solution is also its fitness:

·        Step 2: The fitness of an individual is the sum of the strengths of all external Pareto individuals  dominated by. We add one in odder to guarantee that Pareto solutions are most likely to be produced.

                       

(14)

where

The clustering algorithm is described by the following steps:

·        Step 1: To initialise clustering set C; each individual constitutes a distinct cluster:

                       

(15)

·        Step 2: if the number of cluster is lower or equal to maximum size of external set (), go to step 5. Else, go to step 3.

·        Step 3: Calculate the distance between each pair of clusters. The distance dc between two clusters and  is defined as the average distance between two pairs of individuals from each cluster:

                       

(16)

 and  are respectively the numbers of individuals in clusters and.

·        Step 4: Find the pair of clusters corresponding to the minimal distance  between them. Combine into a large one and return to step 2.

·        Step 5: Find the centroid of each cluster. Select the nearest individual in this cluster to the centroid as a representative individual and remove all other individuals from the cluster.

·        Step 6: Thus, the reduced Pareto set  is computed by uniting these representatives:

 

 

Numeric Simulations and Comments

 

Presentation of the test network

The structure of the test system is shown in fig.1.Appendix A1. It was derived from the standard IEEE 30-bus 6-generator test system while adding to him two renewable generators. The characteristics of the wind mill are presented in table 1. The values of the fuel and emission coefficients are given in table2. The lines data and bus data are given respectively in tables 1 and 2 in Appendix A1.

 

Table 1. Wind mill Data

Characteristics of the wind mill

propeller

Diameter

 

blades

number

Surface

 swept

chechmate

Height

Nominal wind speed Vn

34 m

3

1480 m2

45 m

15 m/s

Nominal characteristics of the asynchronous generator

Interlinked voltage

 

Current

Frequency

power Pn

Cos φ

660 V

760 A

50 Hz

790 Kw

0.91

Rs

Rr

Lσ

Xm

Rm

0.00374 Ω

0.00324 Ω

0.23 mH

5.8 mH

83.85 Ω

 

Table 2: Generator cost and emission coefficients

 

 

G1

G2

G3

G4

G5

G6

 

Cost

a

10

10

20

10

20

10

b

200

150

180

100

180

150

c

100

120

40

60

40

100

 

 

Emission

4.091

2.543

4.258

5.326

4.258

6.131

-5.554

-6.047

-5.094

-3.550

-5.094

-5.555

6.490

5.638

4.586

3.380

4.586

5.151

2.0 10-4

5.0 10-4

1.0 10-6

2.0 10-3

1.0 10-6

1.0 10-5

2.857

3.333

8

2

8

6.667

 

Lower limit and upper limit of the generated real power of each generator at the bus it is shown by (17):

                       

(17)

 

 

 

Results and Comments

 

Implementation and test of the neurons network

The neurons network is used to calculate in real time the active production in the thermal generating nodes and of the renewable origins. The structure of this network is given by the figure (2).

To ensure a good training of the neurons network, the base data is formed by 600 random solutions calculated by method SPEA and corresponding at a wind speed included enters 8 and 12 ms-1, solar radiation vary between 0w/m2 and 1000w/m2 and the total power demand vary between 0.8 and 4 pu.

Training curves of wind speed, solar radiation and the total power demand are given by figures (3, 4 and 5).

 

Figure 3. Training curve of wind speed

Figure 4. Training curve of solar radiation

Figure 5. Training curve of total power demand

 

During this phase, some of new examples are presented to the neurons network. The same examples were already simulated by SPEA method and we studied the quality of these answers given to table 3.


Table 3. Results of test neurons network

 

Test exemple

Pw

(pu)

Ps

(pu)

Pg1 (pu)

Pg2 (pu)

Pg3 (pu)

Pg4 (pu)

Pg5 (pu)

Pg6 (pu)

Emiss

(ton/h)

Coût

($/h)

PD1

2.35

0.6156

0.0475

0.3242

0.4352

0.4363

0.2052

0.3871

0.3795

0.1934

499.5495

PD2

3.43

0.7059

0.0175

0.4095

0.5220

0.5516

0.3635

0.5071

0.4846

0.1866

643.0993

PD3

3.35

0.8423

0.0385

0.3708

0.4870

0.5054

0.3575

0.4714

0.4605

0.1874

598.0519

PD4

1.16

0.3039

0.0252

0.3051

0.4284

0.4116

0.0939

0.3786

0.3598

0.1973

468.1725

PD5

0.93

0.2315

0.0412

0.3204

0.4743

0.4537

0.1231

0.3774

0.3504

0.1950

500.7442

PD6

1.25

0.3392

0.0301

0.3107

0.4249

0.4091

0.1017

0.3630

0.3410

0.1977

461.6945

PD7

2.65

0.4445

0.0266

0.3786

0.4834

0.5146

0.2245

0.4519

0.4197

0.1901

569.3929

PD8

3.10

0.5576

0.0388

0.4071

0.5251

0.5541

0.3173

0.5067

0.4879

0.1871

638.1037

 

Response of neurons network

Pw

(pu)

Ps

(pu)

Pg1 (pu)

Pg2 (pu)

Pg3 (pu)

Pg4 (pu)

Pg5 (pu)

Pg6 (pu)

Emiss

(ton/h)

Coût

    ($/h)

PD1

2.35

0.6156

0.0475

0.3202

0.4351

0.4317

0.2163

0.3874

0.3767

0.1932

499.6342

PD2

3.43

0.7059

0.0175

0.4105

0.5289

0.5448

0.3633

0.5111

0.4931

0.1864

643.0030

PD3

3.35

0.8423

0.0385

0.3711

0.4872

0.5114

0.3614

0.4711

0.4611

0.1872

598.8744

PD4

1.16

0.3039

0.0252

0.3054

0.4126

0.4144

0.1013

0.3814

0.3437

0.1995

466.2881

PD5

0.93

0.2315

0.0412

0.3199

0.4759

0.4538

0.1001

0.3633

0.3543

0.1951

501.1572

PD6

1.25

0.3392

0.0301

0.3106

0.4276

0.4097

0.1081

0.3665

0.3403

0.1972

463.4970

PD7

2.65

0.4445

0.0266

0.3652

0.4882

0.5168

0.2244

0.4543

0.4282

0.1901

569.3759

PD8

3.10

0.5576

0.0388

0.4019

0.5132

0.5537

0.3083

0.4982

0.4800

0.1875

637.9193

 

Figures (6, 7 and 8) present respectively the forecasts of wind speed, solar radiation and total power demand.

 

Figure 6. Forecast of wind speed

Figure 7. Forecast of total power demand

 

 

 

Figure 8.Forecast of solar radiation

 

After the training phase of the neurons network, the simulation results are presented in the figures 9, 10, 11 and 12.

 

Figure 9. Power of the thermal generated nodes and PD’

 

According to figure 9, we notice that the bus thermal generators powers remain variable within their limits.

Bus generator 2 has a remarkable participate by its active power Pg2 when total power demands P’Dis significant because it is the machine which has the more high cost.

 

Figure 10.Total power demand and renewable power

 

            Figure 10; show the variation of solar thermal power, wind power and their resultant which remains lower than 30%of the total power demand PD.

 

Figure 11. Total cost function

Figure 12. Emissions function

 

                     Figures 11 and 12 who represent respectively the variations of the total cost function and emissions of polluting gases function show that if the emissions of polluting gases decrease in the course of time then the total cost increases and conversely.

 

 

Conclusion

 

In this study we presented a method allowing the resolution of the problem of the Environmental Economic Dispatching of an electrical network including renewable energy sources. We made an optimization without auxiliary elements and the problem consists to extract the maximum of power from the renewable sources and to distribute the remainder of the power on the power stations. To have the solutions of the problem in real time we established them on a neurons network.

 

 

References

 

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Appendix

 

Figure 1. Single-line diagram of IEEE 30-bus test system with two renewable power stations

 

Table 1. Line data

Line N°

Connection

Impedance (p.u.)

Line N°

Connection

Impedance (p.u.)

1

30-29

0.0192 + j0.0575

22

13-12

0.0192 + j0.0575

2

30-24

0.0452 + j0.1852

23

12-11

0.0452 + j0.1852

3

29-23

0.0570 + j0.1737

24

19-11

0.0570 + j0.1737

4

24-23

0.0132 + j0.0379

25

19-14

0.0132 + j0.0379

5

29-28

0.0472 + j0.1983

26

19-10

0.0348 + j0.0749

6

29-22

0.0581 + j0.1763

27

19-9

0.0727 + j0.1499

7

23-22

0.0119 + j0.0414

28

10-9

0.0116 + j0.0236

8

28-21

0.0460 + j0.1160

29

16-8

0.1000 + j0.2020

9

22-21

0.0267 + j0.0820

30

9-7

0.1150 + j0.1790

10

22-27

0.0120 + j0.0420

31

8-7

0.1320 + j0.2700

11

22-20

j0.2080

32

7-6

0.1885 + j0.3292

12

22-19

j0.5560

33

6-5

0.2544 + j0.3800

13

20-26

j0.2080

34

6-4

0.1093 + j0.2087

14

23-18

j0.2560

35

3-4

j0.3960

15

18-25

j0.1400

36

4-2

0.2198 + j0.4153

16

18-17

0.1231 + j0.2559

37

4-1

0.3202 + j0.6027

17

18-16

0.0662 + j0.1304

38

2-1

0.2339 + j0.4533

18

18-15

0.0945 + j0.1987

39

27-3

0.0636 + j0.2000

19

17-16

0.2210 + j0.1997

40

22-3

0.0169 + j0.0599

20

15-14

0.0824 + j0.1923

41

20-19

j0.1100

21

16-13

0.1070 + j0.2185

 

Table 2. Bus data

Line N°

Type

Active power (p.u)

Reactive power

(p.u)

Bus voltage (p.u)

Line

N°

Type

Active power

(p.u)

Reactive power

(p.u)

Bus

voltage (p.u)

1

P-Q

0.106

0.019

－

16

P-Q

0.082

0.025

－

2

P-Q

0.024

0.009

－

17

P-Q

0.062

0.016

－

3

P-Q

0.000

0.000

－

18

P-Q

0.112

0.075

－

4

P-Q

0.000

0.023

－

19

P-Q

0.058

0.020

－

5

P-Q

0.035

0.000

－

20

P-Q

0.000

0.000

－

6

P-Q

0.000

0.000

－

21

P-Q

0.228

0.109

－

7

P-Q

0.087

0.067

－

22

P-Q

0.000

0.000

－

8

P-Q

0.032

0.016

－

23

P-Q

0.076

0.000

1.010

9

P-Q

0.000

0.000

－

24

P-Q

0.024

0.000

1.010

10

P-Q

0.175

0.112

－

25

P-V

0.000

0.000

1.071

11

P-Q

0.022

0.007

－

26

P-V

0.000

0.000

1.082

12

P-Q

0.095

0.034

－

27

P-V

0.300

－

1.010

13

P-Q

0.032

0.009

－

28

P-V

0.942

－

1.010

14

P-Q

0.090

0.058

－

29

P-V

0.217

－

1.045

15

P-Q

0.035

0.018

－

30

Bilan

0.000

0.000

1.060