Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013
ThBD.2
A PSO Algorithm for the Economic Load Dispatch Including a Renewable Wind Energy F. Benhamida, Y. Salhi(1), I. Ziane
S. Souag, R. Belhachem, A. Bendaoud(2)
Irecom laboratory, dept. of electrotechnics UDL university of Sidi Bel Abbes Sidi Bel Abbes, Algeria (1)
[email protected]
Apelec laboratory, dept. of electrotechnics UDL university of Sidi Bel Abbes Sidi Bel Abbes, Algeria (2)
[email protected]
Abstract— Economic load dispatch among thermal units is one of the most important problems in power systems operation. Usually so called equal marginal cost criterion is adopted to this calculation. Recently global trend of utilizing more and more renewable energy such as wind power makes this problem more important than ever. With the continuing search for alternatives to conventional energy sources, it is necessary to include wind energy generators (WEG) in the ELD problem. This paper presents a solution of economic load dispatch incorporation wind energy using a particle swarm optimization algorithm (PSO). The effect of wind energy generators system inclusion on ELD problem is investigated, with the source wind susceptible to short duration variations, which is the uncertainty of wind speed around a short-duration- stable mean value. A 6 and 20 unit test system is resolved using PSO to illustrate the variation in the optimal cost, losses, and systemλ with the variation of short-duration-stable mean wind speed. Keywords—Economic load dispatch (ELD), wind energy, particle swarm optimization (PSO) algorithm.
I. INTRODUCTION As one of the most promising non pollution renewable energy resources wind power has given more consideration [1]. Comparing with the conventional generators, wind generator has advantage of reducing the dependences on fossil fuels and transmission losses, enhancing the independence and flexibility of large power grids [2]. The classic problem of economic load dispatch (ELD) has inducing new interest with debate on how wind energy generators (WEG) are to be taken into consideration within dispatch schedules [3]-[4] taking into consideration the variability of wind speed. In the past, this problem has been studied for some time as an advance of ELD [5]; while recently works focus on WEG units independently [6], with proper cost components. Availability of wind power is used to formulate ELD problem constraints in [7] and [8]. Most of these works [6]–[8] used a valid statistics distribution [9], [10] to represent variability of wind known as Weibull distribution. The optimal solution of an ELD is defined for a short time duration as the validity interval of ELD for many applications, where the Weibull distribution is not the best statistical model for wind speed variations [10], [11]. Short time duration wind speed variations include turbulence and gusts. The turbulence is the random variations on a stable mean wind speed value (u), while the gusts are surges within
turbulent wind fields [12]. The Gaussian distribution has been used to model the turbulence and gusts modeled around the stable mean wind speed. In ELD problem model, the stochastic models of power outputs are not taken into account. Practically the conventional units are under influence of small variations over the set point of power, while the load demand varies according to consumer behavior. The generation allocation levels of ELD do not attempts to meet instantaneous values of power demand but a total equivalent power demand in a valid interval. The B-coefficients method has been used in the classical ELD to simulate the total transmission loss [3], [4]. The Bcoefficients representation is more compatible with WEG units generation output only if an acceptable total WEG output power generation is used [12], [13]. In this paper we propose a PSO method to solve the ELD problems by including WEG units in the power system to show the effect on optimal generation cost. To study these aspects separately, some of the conventional constraints of the ELD problem have been ignored. II.
ELD INCLUDING TRANSMISSION LOSSES
The B-coefficients loss formula used for conventional ELD problems [3] is PL ({ Pn }) =
N
N
∑∑P
b
n1 n1, n 2
n1
n2
N
Pn 2 + ∑ bn ,0 . Pn + b0 ,0 (1) n
where the parameters {bn1,n2}, {bn,0}, and b0,0 are Bcoefficients known for a specific unit. The augmented loss function due to the including of WEG units within a power system would add three additional summations N ⎫ Pn 2 + ∑ bn ,0 .Pn + b0 ,0 ⎪ n1 n 2 n ⎪ N W W ⎪ + ∑ ∑ Pn bn ,ω Pω + ∑ bω ,0 .Pω ⎬ ω n1 ω ⎪ W W ⎪ + ∑ ∑ Pω 1bω 1,ω 2 Pω 2 ⎪ ω1 ω 2 ⎭
PL ({Pn } , { Pω }) =
N
N
∑∑P
978-1-4799-0275-0/13/$31.00 ©2013 IEEE
b
n 1 n1, n 2
(2)
Proceedings of the 3rd International Conference on
ThBD.2 (12) dC n 2 ( Pn 2 ) / dPn 2 ≤ λ (1 − (∂PLeq / ∂Pn 2 )) for Pn2 = Pnmax 2
where the parameters {bω1, ω2}, {bω,0}, and bn,ω are Bcoefficients known for a specific WEG unit ω; and the WEG generation output {Pω} are treated as extern-variables, So (2) can be reformulated as
(13) dCn3 ( Pn3 ) / dPn3 ≥ λ (1 − (∂PLeq / ∂Pn3 )) for Pn3 = Pnmin 3 N
N
N
N
n1
n2
n
eq PLeq ({ Pn } , { Pω } ) = ∑ ∑ Pn1bn1, n 2 Pn 2 + ∑ bneq,0 . Pn + b0,0
n1
bneq,0 = bn ,0 + ∑ bn ,ω Pω W
W
W
ω
ω1 ω 2
= b0,0 + ∑ bω ,0 Pω + ∑ ∑ Pω 1bω 1,ω 2 Pω 2
(5)
Considering N conventional generating units and W WEG units, all of which are utility owned, have to supply a total demand of PD for a given interval. The cost of generation C(Pn) of the n-th conventional unit, is given by [3], [4] C ( Pn ) = c 0 , n + c1, n . Pn + c 2, n . Pn2
Cω ( Pω ) = c1,ω . Pω
(7)
N
W
n
ω
min C ({ Pn } , { Pω }) = min( ∑ C n ( Pn ) + ∑ C ω ( Pω )) (8)
Pnmin ≤ Pn ≤ Pnmax ; n = 1, ..., N
ω
n2
n3
∑ Pn + ∑ Pω = PD + PLeq ({Pn } , {Pω })
(9) (10)
Conventional units has been divided between units with inactive power limits (subset n1), active maximum power limit (subset n2), and active minimum power limit (subset n3). The KT conditions for (8) (for a specific power {Pω} generated by the WEG units), can be obtained as max dCn1 ( Pn1 ) / dPn1 = λ (1 − (∂PLeq / ∂Pn1 )) for Pnmin 1 ≤ Pn1 ≤ Pn1
(11)
N
∑b
eq n1, n 2
.Pn*2 ) (15)
Due to nonlinearity of (15), it is practically impossible to have a direct relation between the averages of λ* and {Pn1*}. The convergence to the optimal solution may be expected for (8)-(10) if the derived loss coefficients{bn,0eq}and b0,0eq are approximately constant within the validity interval of ELD, the set of relations that requires a specified deterministic value of λ* can be hold and the demand constraint in (14) becomes
(6)
Pω is subject to variations of wind speed at the ω-th hub (the hub is made of cast iron and connects the turbine's blades to the main shaft). Within the validity interval it is assumed that Pω can be absorbed by the system without any overgeneration or reliability problems. The new ELD can be defined as the following optimization problem:
n
ω
n 2 ≠ n1
N
∑P n1
where Pn is the active power output and c0,n, c1,n, c2,n are the cost coefficient of the n-th conventional unit. While for the ω-th WEG units, the cost expression is
W
N
where the set of conventional units has been divided using subsets n1,n2 and n3 . The KT relation for the n1-th unit, becomes
n1
N
N
(4)
ω
subject to
W
(14) = PD + PLeq − ∑ Pω − ∑ Pnmax − ∑ Pnmin 2 3
2c2, n1 Pn*1 + c1, n1 = λ * (1 − 2bn1, n 2 Pn*1 − bneq1,0 −
W
b
n1
(3)
where the derived B-coefficients of augmented loss formula in (3) have variable in terms of {Pω}, and the original Bcoefficients {bn,0} and b0,0 of (1) and are given by
eq 0 ,0
∑P
N
N
n1
n2
N
eq = PD + ∑ ∑ Pn1bn1, n 2 Pn 2 + ∑ bneq,0 . Pn + b0,0 n
W
N
N
ω
n2
n3
(16)
− ∑ Pw − ∑ Pnmax − ∑ Pnmin 2 3
The optimal generation levels {Pn*} evaluation may now be simplified as follow: •
Neglect the impact of long duration variation of wind speed at WEG installation sites, across the validity interval of an ELD. For such situation, assume a superposed short-duration-stable mean wind speed (uωm at the WBG hub, ω-h site) [9]- [12].
•
The no proportionality of the mean wind speed uωm (at a WEG hub across the validity interval of ELD) with WEG power output. For pitch angle controlled WEG [9], [14], the output can be obtained in terms of the rated power Pωrat and a no-turbulence output coefficient as μω(uωm) using an analytical description of the standard output curve as Pω ( uωm ) = μ ω ( uωm ). Pωrat
(17)
where μω(uωm) ∈{0,…,1}, according to the following description: ⎧ 0; if uωm ≤ uωin ⎪ m in out in in m rat ⎪ ( uω − uω ) / ( uω − uω ) ; if uω < uω ≤ uω μ ω (uωm ) = ⎨ if uωrat < uωm ≤ uωout ⎪1; ⎪ if uωm ≤ uωin ⎩ 0;
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(18)
Proceedings of the 3rd International Conference on where uωrat, uωin, uωout are the rated, cut-in and cut-out speeds respectively, at the ω-th WEG. So, the possibility of a specified dispatch values from the conventional units in the ELD in the validity interval depends on the approximate the total wind power generation, for the lossless case; and approximate constancy of derived loss coefficients{bn,0eq} and b0,0 eq, for the lossy case. III. PARTICLE SWARM OPTIMIZATION METHOD In 1995, Kennedy and Eberhart first introduced the PSO method, motivated by social behavior of organisms such as fish schooling and bird flocking. PSO, as an optimization tool, provides a population-based search procedure in which individuals called particles change their positions (states) with time. In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors. The swarm direction of a particle is defined by the set of particles neighboring the particle and its history experience [15]. The particles are manipulated according to the following equations. (t ) (t ) vi(t +1) = Ivi( t ) + c1r1 ( xgbest − xi(t ) ) + c2 r2 ( xipbest − xi(t ) ) (19)
xi( t +1) = xi( t ) + vi( t +1) vi(t)
•
Initialize the PSO parameters like population size, acceleration constants, maximum and minimum value of inertia weight factor, etc. • Read the input data concerning the fuel cost functions, MW limits, the B-coefficient and the total demand for conventional and WEG unit. • At the first iteration, allocate a random population of individuals of active power satisfying the MW limits. • At iteration t, for each individual, calculate the fitness function using (19) and obtain pbest by comparison which is itself compared to obtain the gbest. • At each step the value of gbest is compared to a prespecified ε. • gbest final value represent the minimum generation cost and the corresponding individual vector represent the ELD solution. The PSO solution mapped to the ELD problem can be easily modified to account for transmission losses. The only modification required is in the computation of the power output of the reference generator Pk from (10):
The inertia constant controls the exploration of the search space. To balance between local and global explorations, we must select a suitable value of inertia weight I. The inertia weight factor I is set as follow, I = I max − iter × ( I max − I min ) / itermax .
(21)
where Imax and Imin are the maximum and minimum value of inertia weight factor, respectively and itermax is the maximum iteration number. APPLICATION OF PSO METHOD TO ECONOMIC SCHEDULING To map the PSO for solving ELD, we have to follow the following steps: • Using (8) for fitness function initialization. Calculate the total cost function including generation cost of WEG units if present.
W
i≠k
ω
(22)
where the losses PL is computed using the B-coefficients. The loss formula (6), separating the terms containing the power output of the reference generator k, PGk, gives: N
N
N
PL = Bkk Pk2 + 2∑ Pi Bik Pk + ∑∑ PB i ij Pj i≠k
xi(t)
The acceleration constants c1 and c2 serve to attract each particle to pbest and gbest positions, respectively. According to past experiences [16], c1 and c2 are often set to be 2.0.
N
Pk = PD + PLeq ({ Pn } , { Pω } ) − ∑ Pi − ∑ Pω
(20)
where is the velocity of particle i at iteration t; is the current position of particle i at iteration t; t is iterations pointer (generations); c1, c2 are the acceleration constants; r1, r2 are uniform random values in the range (0,1) ; I is the inertia weight;, xipbest(t) is the previous best position of particle i at iteration t; xgbest(t) is the best position among in the population at iteration t; vi(t+1) and xi(t+1) is the velocity and position of particle i at iteration t+1, respectively .
IV.
ThBD.2
(23)
i≠k j≠k
It is evident that for the computation of the reference generation the solution of a quadratic equation in Pk is required. V.
WIND-BASED GENERATION OUTPUT WITHIN THE VALIDITY INTERVAL OF AN ELD According to [17], the short time duration wind speed variations are represented by the following modified relation: (24)
〈 Pω 〉 = 〈 Pω ( uωm ) 〉 = μ ωeq ( uωm , τ ). Pωrat
where µωeq(uωm,τ ) is the per unit output under turbulence of the ω-th WEG unit; uωm is the mean speed and τ is the turbulence intensity. The appropriate estimation of the per unit output under turbulence µωeq(uωm,τ ) from experimental data is important to use (24) in ELD problems (8). According to [17], the per unit output under turbulence can be defined as
μ ωeq (u ωm , τ ) = 1 − exp( − ((( u ωm / uωm
rat
) / v (τ ))
k (τ )
)) (25)
where v(τ) is a scaling coefficient and k(τ) is an index coefficient which are both positive and in term of τ. In (25), the expression approaches the values of (18) at limits values of mean speed (0 and ∞), while it increase smoothly between uωin and uωrat according to v(τ) and k(τ) :
978-1-4799-0275-0/13/$31.00 ©2013 IEEE
Proceedings of the 3rd International Conference on k (τ ) ≈ 3.49 − 6.01.τ ;
⎫⎪ 0.1 ≤ τ ≤ 0.4 2⎬ v (τ ) ≈ 0.71 + 0.21.τ − 1.26.τ ⎪⎭
ThBD.2
(26)
W
(27)
ω
b
W
1658.5596 1356.6592
P i (MW)
Unit
1 32.586 2 4 136.044 5 Demand =800 MW System λ = 50.661 $/MW
36.32782 38.27041
0.02111 0.01799
P i (MW)
Unit
P i (MW)
14.484 3 141.548 257.662 6 243.007 Total cost = 41899.705 $/hr Losses= 25.331 MW
The convergence of fitness of 6-unit test system for load demand 800 MW was converged to the optimal solution in 150 iterations as shown in Fig. 2. 41899.7 = psoeld( [ 6 inputs ] )
W
≈ b0,0 + ∑ bw ,0 Pw + ∑ ∑ Pω bω 1,ω 2 Pω ω
130 125
THE BASE CASE OPTIMUM OF THE 6-UNIT TEST CASE (WITHOUT WEG INCLUSION) USING PSO
Unit
4.627
10
ω1 ω 2
Approximations made in (27) make the optimization processes of the ELD problem (8) simpler. Examples to validate these approximations are given in simulation results section.
4.626
10
gbest val.
eq 0 ,0
W
325 315
TABLE II.
(24) and (25) are applied to calculate the output under turbulence µωeq(uωm,τ) as depicted in Fig. 1. If the rated output Pωrat of WEG unit is known, Fig. 1 can be used to calculate the power output 〈Pω〉. We must mention that if the validity interval of ELD is carefully determinate, the range of variations with short time duration of wind speed can be decreased, in order to minimize the variations in Pω. In these cases, (4)-(5) can be reformulated as bneq,0 ≈ bn ,0 + ∑ bn ,ω Pω
5 6
4.625
10
Power output by WEG (pu)
4.624
10
1,0 0,8 0,6
4.623
0,0
0,5
1,0
1,5
0,4 0,2 0,0
Wind speed of WEG (pu)
2,0
τ = 5% τ = 10% τ = 15% τ = 20% τ = 25% τ = 30% τ = 35% τ = 40%
Figure. 1. Power output curve (pu) according to (25) and (26) for a typical WBG at different levels of turbulence (τ).
VI. SIMULATION RESULTS Case A: a 6 conventional unit ELD problem is used for simulation. The ELD problem, defined by the conventional unit parameters presented in Table I and loss parameters (bn1,n2) taken from [18], is to be modified to include WEG up to a maximum of 120 MW in the unit (1) (modification 1) and 325 MW in the unit (5) (modification 2) in each modified ELD. The power demand to meet is 800 MW. The base case optimum of the 6 unit test case (without WEG inclusion) obtained by PSO is shown in Table III. The processes of convergence is shown in Fig. 2. The PSO algorithm parameters are as follow: the population size is 100; the maximum number of iteration is 1000; the inertia weight factors (I) are Imax = 0.9 and Imin = 0.4; the acceleration constant are c1=2 and c2=2 and the maximum number of iterations is chosen as the stopping criterion. TABLE I. Unit 1 2 3 4
GENERATOR PARAMETERS FOR A 6-UNIT ELD PROBLEM
Gen.limits (MW) Max Min 125 10 150 10 225 35 210 35
Generation cost parameters c0,n($/hr) c1,n($/hr-MW) c2,n($/hr-MW2) 756.79886 38.53973 0.15240 451.32513 46.15916 0.10587 1049.9977 40.39655 0.02803 1243.5311 38.30553 0.03546
10
0
100
200
300
400 500 600 Number of iteration
700
800
900
1000
Figure. 2. Convergence of fitness of case A for load demand 800 MW.
A. Modification 1 The base case is modified to realize the 125 MW of wind power. So Unit 1 (Table I) is replaced by a WEG unit of 125 MW capacities where the output varies from 0 to 125MW. The generation cost function by the WEG unit is approximated by a linear model and evaluated from the conventional unit in the base case by setting c1,ω equal to 17.79764 $/MW-h (46.17% of conventional unit according to [17]). The cost function parameters of WEG unit 1 c0,n($/hr) and c2,n($/hr-MW2) are set to be zeros . Cω ( Pω ) = c1,ω . Pω , ω = 1, c1,ω = 17.79764 $ / MW-h (28)
For different values of wind generation levels of Unit 1 (0 to 125 MW), Fig. 3 displays the ELD optimal total generation cost C solutions showed in the 1st Y-axis. The 2nd and 3rd Yaxis, respectively present the evolution of transmission line losses, and system-λ all in term of WEG unit power output (MW). As power output by WEG unit increase for important values of uωm as depicted by Fig. 1, the total cost decrease from 41315.371 $/h which is less that the base case (41899.705) to 37574.222 $/h. In the same way, the losses drop from significant levels (26.061 MW for zero wind generation) to a minimum of about 24.780 MW and finally keep on 24.869 MW (for a maximum wind generation) witch close to the base case, because this correspond to the optimal output of the base case. The system-λ is higher than the base case at low levels of WEG (0 to 30 MW) and start from 51.120$/MW for zero wind generation, eventually dropping to values lower than the base case as generation by Unit 1
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increase (35 to 125 MW) and finally reach 49.372$/MW for 125 MW of WEG of Unit 1. Total cost Losses System lambda
41500 41000 40500
51,0
25,8
50,8
25,6
40000
25,4
39500
51,2
26,0
50,6
51,0
50,2
50,8
25,2
50,0
38500
25,0
49,8
38000
24,8
37500
24,6 20
40
60
80
100
0%(negligible turbulence ) 5% 40% Base case optimal values
51,2
50,4
39000
0
turbulence index is similar between the two cases (Modification 1 and 2 of the 6-unit). Thus the influence of τ is considerably unlike between ELD including unit of lowest capacity and ELD including of highest capacity (WEG units)
Systemλ ($/MW)
26,2
42000
ThBD.2
49,6 49,4
50,6 50,4 50,2 50,0 49,8
49,2
49,6
120
49,4
P1 (MW)
49,2 0,0
0,5
1,0
1,5
2,0
Wind speed of WEG (Unit 1) (pu)
Optima for the modified 6-unit problem at different values of rated output power of WEG (Unit 1).
Figure. 3.
B. Modification 2 In this case, the unit of highest capacity (Unit 5) is replaced by a WEG generator, whose rated output power is identical to the maximum value of 325 MW indicated in Table I. The maximum possible share of wind power at the above demand level is thereby fixed at 40.62 %.
56
0%(negligible turbulence ) 5% 40% Base case optimal values
55
System λ ($/MW)
(a)
54 53 52 51 50
43000
49 0,0
Total Cost change
42000
0% ( negligeable turbulance) 5% 40% base case optimal value
41000
1,0
1,5
40000
39000
0,0
0,5
1,0
1,5
2,0
Wind speed of WEG (Unit 1) (pu)
(a)
44000 42000 40000
0%(negligible turbulence ) 5% 40% Base case optimal values
38000 36000 34000 32000 30000 0,0
0,5
1,0
1,5
Wind speed of WEG (Unit 5) (pu)
2,0
(b)
Figure. 5. Change in system -λ for the 6-unit problem (with and without a turbulence) at different values wind speed for both cases modification 1 (a) and 2 (b)
38000
Total Cost change
0,5
Wind speed of WEG (Unit 5) (pu)
2,0
(b)
Figure. 4. Change in total cost for the 6-unit problem (with and without a turbulence) at different values wind speed for both cases modification 1 (a) and 2 (b)
(uωm/
The short time duration average wind speed in pu uωm rat) and turbulence intensity τ are assumed to be the same for WEG units 2, 3 and 4, involving the operation at the same point in Fig. 1. Fig. 4 and Fig. 6 shows the change in total cost, and systemλ with respect to base case with negligible turbulence for both case modification 1 and 2, respectively, due to significant levels of turbulence, in terms of short duration mean wind speed (uωm/ uωm rat) ∈{0,…,2 pu}. A comparison shows that other than an important reduction of total cost at high turbulence in case modification 2, the sensitivity to the
Case B: a 20 conventional unit ELD problem is also used for simulation. The unit and loss parameters (bn1,n2) taken from [19], is to be modified to include WBG up to a maximum of 600 MW. Together all units are expected to meet a demand of 2500 MW (maximum 24% possible share of wind power in each modified ELD), for which the base case optimum obtained by PSO in terms of the total cost C (P1,..., P20), the total transmission losses, and the marginal cost of generation (system-λ) shown is given 62456.6330$/hr, 91.966 MW, 20.958 $/MW, respectively as shown Table V. The processes of convergence is shown in Fig. 6. For the short time duration average wind speed in pu (uωm/uωm rat) and turbulence intensity τ (40% ) values of wind generation levels of Unit 1 in Table VI we summarize and compare results of the total cost in the test case obtained by the PSO and general algebraic modeling system (GAMS [20]). TABLE III.
THE BASE CASE OPTIMUM OF THE 20-UNIT TEST CASE
Unit 1 2 3 4 5
P i (MW) Unit P i (MW) Unit P i (MW) Unit P i (MW) 512.783 6 073.575 11 150.234 16 036.254 169.101 7 115.293 12 292.763 17 066.860 126.889 8 116.392 13 119.111 18 087.962 102.874 9 100.405 14 030.832 19 100.803 113.685 10 106.021 15 115.806 20 054.313 Demand =2500 MW Total cost = 62456.6330 $/hr System λ = 20.958 $/MW Losses= 91.966 MW
The convergence of fitness of 20 unit test system for load demand 2500 MW was converged in 3 second to the optimal solution in only 110 iterations for the entire run algorithm as shown in Fig. 6.
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Proceedings of the 3rd International Conference on
ThBD.2 [2]
62456.6 = psoeld( [ 20 inputs ] ) 4.7966
10
4.7964
[3]
gbest val.
10
4.7962
10
10
4.796
[4]
4.7958
[5]
10
[6]
4.7956
10
0
100
200
300
400 500 600 Number of iteration
700
800
900
1000
Figure. 6. Convergence of fitness of case B for load demand 2500 MW
TABLE IV.
uωm /uωm,rat 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2
RESULT OF 20UNIT (PD=2500 MW)(WITH WEG INCLUSION) µωeq(uωm, 40%) * Total cost ($/hr) with τ=40% Prat = P1 PSO GAMS 0 62170,279 62170,215 1,117 61036,041 61035,848 10,121 59971,030 59970,915 36,072 59049,504 59049,456 86,258 58267,167 58267,154 162,667 57609,462 57609,203 259,210 57059,203 57058,937 362,052 56600,053 56599,944 454,384 56218,438 56218,284 523,649 55901,500 55901,441 566,487 55639,167 55639,039 587,974 55421,871 55421,797 596,556 55242,390 55242,362 599,232 55094,291 55094,230 599,870 54972,145 54972,035 599,983 54871,539 54871,486 599,998 54788,998 54788,854 599,999 54720,995 54720,827 599,999 54665,176 54665,081 600 54619,461 54619,320 600 54415,797 54415,760
From the results obtained in the table VI we see that the result obtained from the proposed PSO algorithm and GAMS is better than the results obtained in the base case optimum. VII. CONCLUSION The economic load dispatch including WEG units is examined in this paper. The impact of short time duration variations on the ELD optimal solution is considered without supposing any probability distribution. The solution of such problem is done using PSO. The effect of short time duration wind variations has been considered as a constant mean wind speed at hub. The features of the WEG units inclusion in ELD problem are studied through a modified 6 and 20unit ELD test case. From comparison to the conventional base case, it is observed that for high values of WEG power the optimal generation cost, transmission losses, and systemλ, decrease, all essentially because the WEG units run costs is low compared to conventional units.
[7]
[8]
[9]
[10] [11] [12] [13]
[14]
[15]
[16]
[17]
[18]
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