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
Emails: 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
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 C_{P}, is the aerodynamic coefficient of turbine power (it characterizes the aptitude of the aerogenerator to collect wind power), ρ is the air density, R_{p}_{ }the turbine ray and V_{W}_{ }wind speed. The power coefficient value C_{P}, 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 P_{n}_{ }from a nominal wind speed V_{n}. For wind speeds higher than V_{n}, 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 P_{w }= 0, zone 2, in which useful power depends on wind speed. V_{w}, zone 3, generally where provided power P_{w} remains appreciably equal to P_{n} and finally zone 4,P_{w} = 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) 
E_{c} is solar radiation, T_{jref} is the reference temperature of the panels of 25°C, T_{j} is the cells junction temperature (°C), P_{1} represent the characteristic dispersion of the panels and the value for one panel is included enters 0.095 to 0.105 and the parameter P_{2}=0.47%/C°; is the drift in panels temperature [14].
The addition of one parameter P_{3 }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 P_{1}, P_{2} and P_{3} 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 P_{D}. the remaining total demand_{}, will distributed between the thermal power stations. The problem is reduced for a speed wind V_{w }and solar radiation E_{c }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 V_{w}, solar radiation E_{c} and total power demand P_{D}. The neurons network is composed of three layers, the entries layer is formed by V_{w}, E_{c} et P_{D}; 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 F_{1}, the emissions of polluting gases F_{2}, 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 ith 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 SO_{2} 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 ith 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 multiobjective 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 x_{1} and x_{2} of such optimization problem, we could have one which dominates the other or none dominates the other.
In a minimization problem, a solution x_{1 }dominates other solution x_{2} 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 nondominated within the entire search space are denoted as Paretooptimal and the set of objectives vectors corresponding constitute the Paretooptimal set or Paretooptimal 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 nondominated 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 tradeoff 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 d_{c} 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 30bus 6generator 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 V_{n} 
34 m 
3 
1480 m^{2} 
45 m 
15 m/s 
Nominal characteristics of the asynchronous generator 

Interlinked voltage

Current 
Frequency 
power P_{n} 
Cos φ 
660 V 
760 A 
50 Hz 
790 Kw 
0.91 
R_{s} 
R_{r} 
L_{σ} 
X_{m} 
R_{m} 
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/m^{2 }and 1000w/m^{2} 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 

P_{w} (pu) 
P_{s } (pu) 
P_{g1 }(pu) 
P_{g2 }(pu) 
P_{g3 }(pu) 
P_{g4 }(pu) 
P_{g5 }(pu) 
P_{g6 }(pu) 
Emiss (ton/h) 
Coût ($/h) 

P_{D1} 
2.35 
0.6156 
0.0475 
0.3242 
0.4352 
0.4363 
0.2052 
0.3871 
0.3795 
0.1934 
499.5495 
P_{D2} 
3.43 
0.7059 
0.0175 
0.4095 
0.5220 
0.5516 
0.3635 
0.5071 
0.4846 
0.1866 
643.0993 
P_{D3} 
3.35 
0.8423 
0.0385 
0.3708 
0.4870 
0.5054 
0.3575 
0.4714 
0.4605 
0.1874 
598.0519 
P_{D4} 
1.16 
0.3039 
0.0252 
0.3051 
0.4284 
0.4116 
0.0939 
0.3786 
0.3598 
0.1973 
468.1725 
P_{D5} 
0.93 
0.2315 
0.0412 
0.3204 
0.4743 
0.4537 
0.1231 
0.3774 
0.3504 
0.1950 
500.7442 
P_{D6} 
1.25 
0.3392 
0.0301 
0.3107 
0.4249 
0.4091 
0.1017 
0.3630 
0.3410 
0.1977 
461.6945 
P_{D7} 
2.65 
0.4445 
0.0266 
0.3786 
0.4834 
0.5146 
0.2245 
0.4519 
0.4197 
0.1901 
569.3929 
P_{D8} 
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 

P_{w} (pu) 
P_{s } (pu) 
P_{g1 }(pu) 
P_{g2 }(pu) 
P_{g3 }(pu) 
P_{g4 }(pu) 
P_{g5 }(pu) 
P_{g6 }(pu) 
Emiss (ton/h) 
Coût ($/h) 

P_{D1} 
2.35 
0.6156 
0.0475 
0.3202 
0.4351 
0.4317 
0.2163 
0.3874 
0.3767 
0.1932 
499.6342 
P_{D2} 
3.43 
0.7059 
0.0175 
0.4105 
0.5289 
0.5448 
0.3633 
0.5111 
0.4931 
0.1864 
643.0030 
P_{D3} 
3.35 
0.8423 
0.0385 
0.3711 
0.4872 
0.5114 
0.3614 
0.4711 
0.4611 
0.1872 
598.8744 
P_{D4} 
1.16 
0.3039 
0.0252 
0.3054 
0.4126 
0.4144 
0.1013 
0.3814 
0.3437 
0.1995 
466.2881 
P_{D5} 
0.93 
0.2315 
0.0412 
0.3199 
0.4759 
0.4538 
0.1001 
0.3633 
0.3543 
0.1951 
501.1572 
P_{D6} 
1.25 
0.3392 
0.0301 
0.3106 
0.4276 
0.4097 
0.1081 
0.3665 
0.3403 
0.1972 
463.4970 
P_{D7} 
2.65 
0.4445 
0.0266 
0.3652 
0.4882 
0.5168 
0.2244 
0.4543 
0.4282 
0.1901 
569.3759 
P_{D8} 
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 P_{D}^{’ } 
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 P_{g2 }when total power demands P’_{D}is 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 P_{D}.


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.
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Appendix
Figure 1. Singleline diagram of IEEE 30bus test system with two renewable power stations
Table 1. Line data
Line N° 
Connection 
Impedance (p.u.) 
Line N° 
Connection 
Impedance (p.u.) 
1 
3029 
0.0192 + j0.0575 
22 
1312 
0.0192 + j0.0575 
2 
3024 
0.0452 + j0.1852 
23 
1211 
0.0452 + j0.1852 
3 
2923 
0.0570 + j0.1737 
24 
1911 
0.0570 + j0.1737 
4 
2423 
0.0132 + j0.0379 
25 
1914 
0.0132 + j0.0379 
5 
2928 
0.0472 + j0.1983 
26 
1910 
0.0348 + j0.0749 
6 
2922 
0.0581 + j0.1763 
27 
199 
0.0727 + j0.1499 
7 
2322 
0.0119 + j0.0414 
28 
109 
0.0116 + j0.0236 
8 
2821 
0.0460 + j0.1160 
29 
168 
0.1000 + j0.2020 
9 
2221 
0.0267 + j0.0820 
30 
97 
0.1150 + j0.1790 
10 
2227 
0.0120 + j0.0420 
31 
87 
0.1320 + j0.2700 
11 
2220 
j0.2080 
32 
76 
0.1885 + j0.3292 
12 
2219 
j0.5560 
33 
65 
0.2544 + j0.3800 
13 
2026 
j0.2080 
34 
64 
0.1093 + j0.2087 
14 
2318 
j0.2560 
35 
34 
j0.3960 
15 
1825 
j0.1400 
36 
42 
0.2198 + j0.4153 
16 
1817 
0.1231 + j0.2559 
37 
41 
0.3202 + j0.6027 
17 
1816 
0.0662 + j0.1304 
38 
21 
0.2339 + j0.4533 
18 
1815 
0.0945 + j0.1987 
39 
273 
0.0636 + j0.2000 
19 
1716 
0.2210 + j0.1997 
40 
223 
0.0169 + j0.0599 
20 
1514 
0.0824 + j0.1923 
41 
2019 
j0.1100 
21 
1613 
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 
PQ 
0.106 
0.019 
－ 
16 
PQ 
0.082 
0.025 
－ 
2 
PQ 
0.024 
0.009 
－ 
17 
PQ 
0.062 
0.016 
－ 
3 
PQ 
0.000 
0.000 
－ 
18 
PQ 
0.112 
0.075 
－ 
4 
PQ 
0.000 
0.023 
－ 
19 
PQ 
0.058 
0.020 
－ 
5 
PQ 
0.035 
0.000 
－ 
20 
PQ 
0.000 
0.000 
－ 
6 
PQ 
0.000 
0.000 
－ 
21 
PQ 
0.228 
0.109 
－ 
7 
PQ 
0.087 
0.067 
－ 
22 
PQ 
0.000 
0.000 
－ 
8 
PQ 
0.032 
0.016 
－ 
23 
PQ 
0.076 
0.000 
1.010 
9 
PQ 
0.000 
0.000 
－ 
24 
PQ 
0.024 
0.000 
1.010 
10 
PQ 
0.175 
0.112 
－ 
25 
PV 
0.000 
0.000 
1.071 
11 
PQ 
0.022 
0.007 
－ 
26 
PV 
0.000 
0.000 
1.082 
12 
PQ 
0.095 
0.034 
－ 
27 
PV 
0.300 
－ 
1.010 
13 
PQ 
0.032 
0.009 
－ 
28 
PV 
0.942 
－ 
1.010 
14 
PQ 
0.090 
0.058 
－ 
29 
PV 
0.217 
－ 
1.045 
15 
PQ 
0.035 
0.018 
－ 
30 
Bilan 
0.000 
0.000 
1.060 