GABased HTSMay 2011
GABased HTSMay 2011
GABased HTSMay 2011
a r t i c l e i n f o a b s t r a c t
Article history: This paper presents an algorithm for solving the hydrothermal scheduling through the application of
Received 3 August 2007 genetic algorithm (GA). The hydro subproblem is solved using GA and the thermal subproblem is solved
Received in revised form 9 July 2008 using lambda iteration technique. Hydro and thermal subproblems are solved alternatively. GA based
Accepted 23 November 2010
optimal power flow (OPF) including line losses and line flow constraints are applied for the best hydro-
thermal schedule obtained from GA. A 9-bus system with four thermal plants and three hydro plants and
a 66-bus system with 12 thermal plants and 11 hydro plants are taken for investigation. This proposed GA
Keywords:
reduces the complexity, computation time and also gives near global optimum solution.
Hydrothermal scheduling
Genetic algorithm
Ó 2011 Elsevier Ltd. All rights reserved.
Optimal power flow
Lambda iteration technique
0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijepes.2010.11.008
828 V.S. Kumar, M.R. Mohan / Electrical Power and Energy Systems 33 (2011) 827–835
List of symbols
am, bm, cm cost coefficients of the mth thermal plant PHij active power generation of the ith hydro plant during
ALij inflow into the ith hydro reservoir during the jth time the jth time interval
interval in CMS PLj transmission loss during the jth time interval
as, bs, cs cost coefficients of the slack bus plant PTmj active power generation of the mth thermal plant dur-
Ci correction factor for head variation of ith hydro plant ing the jth time interval
Dij discharge of the ith hydro reservoir during the jth time PTsj active power generation of the slack bus plant during
interval CMS the jth time interval
Di min minimum discharge limit of the ith hydro reservoir in PTm min minimum generation limit of mth thermal plant
CMS PTm max maximum generation limit of mth thermal plant
Di max maximum discharge limit of the ith hydro reservoir in TFCj total fuel cost for the jth interval
CMS TPC total production cost
Fmj fuel cost function of the mth thermal plant in the jth Yij water storage level in the ith hydro reservoir at the
time interval beginning of the jth time interval in Cubic Meter per
G constant used in the determination of hydro power gen- Second-hour (CMS-hour)
eration Yi min minimum water storage level limit in the ith hydro res-
Hoi basic head of the ith hydro reservoir in meter ervoir in CMS-hour
N number of time intervals Yi max maximum water storage level limit in the ith hydro res-
NB number of buses in the system ervoir in CMS-hour
NG number of generating plants in the system NH + NT /k phase angle of the kth transmission line
NH number of hydro plants in the system / k, (max) maximum phase angle rating of the kth transmission
NL number of Lines in the system line
NT number of thermal plants in the system / k, (min) minimum phase angle rating of the kth transmission
PDj system demand during the jth time interval line
In the present work, genetic algorithm is applied to solve the subject to: the power balance constraints
hydrothermal scheduling problem with optimal power flow. The
X
NT X
NH
hydro subproblem is solved using genetic algorithm and thermal PT mj þ PHij ¼ PDj þ PLj ; j ¼ 1; 2; . . . ; N ð2Þ
subproblem is solved using lambda iteration technique without m¼1 i¼1
line losses. Both the hydro and thermal subproblems are solved
the water balance equation
alternatively. Total fuel cost is calculated including line losses for
the best hydrothermal schedule obtained using proposed GA. Line Y i;j þ 1 ¼ Y ij þ ALij Dij þ; i ¼ 1; 2; . . . ; NH; j ¼ 1; 2; . . . ; N ð3Þ
flow constraints are checked for its limits at each interval. GA
the active power generation of hydro plants
based Optimal Power Flow (OPF) is implemented only for the con-
straint violated intervals [1][17]. Fast Decoupled Load Flow (FDLF) PHij ¼ ðHoi =GÞ½1 þ C i ðY ij þ Y i;j þ 1Þ=2Dij ; i ¼ 1; 2; . . . ; NH;
method is used to calculate the line losses. Computation of line j ¼ 1; 2; . . . ; N ð4Þ
losses in each generation of genetic algorithm increases the com-
putational time and increases the complexity of the problem. This the limits on water storage level in reservoirs
proposed GA reduces the complexity, computation time and also Y i min 6 Y ij 6 Y i max ; i ¼ 1; 2; . . . ; NH; j ¼ 1; 2; . . . ; N ð5Þ
gives near global optimum solution. The proposed algorithm has
been tested on two sample systems, one with an Indian utility sys- with Yi1 and Yi,N+1 fixed for i = 1, 2, . . . , NH the limits on water
tem comprising 66 buses, 93 transmission lines, 12 thermal plants discharge
and 11 hydro plants and other with an nine buses, 11 transmission Di min 6 Dij 6 Di max ; i ¼ 1; 2; . . . ; NH; j ¼ 1; 2; . . . ; N ð6Þ
lines, four thermal plants and three hydro plants.
the limits on active power generation of hydro units
2. Problem formulation PHi min 6 PHij 6 PHi max ; i ¼ 1; 2; . . . ; NH; j ¼ 1; 2; . . . ; N ð7Þ
The optimal power flow problem for jth interval with transmis-
The time span of the day is subdivided into 24 equal hourly
sion security constraint is formulated as:
intervals and the load is assumed to remain constant over each
interval. The reservoir inflows, correction factors for head varia- X
NT
tions and generating units available for scheduling each interval TFC j ¼ ðam PT 2mj þ bm PT mj þ cm Þ þ as PT 2sj þ bs PT sj þ cs ;
m¼1
are assumed as deterministically known. In all hydro plants, evap- m–s
oration, water travel time and spill over of water in the hydro res- j ¼ 1; 2; . . . ; N ð8Þ
ervoirs are neglected.
The one-day hydrothermal scheduling problem is stated as [7]: the power balance constraints
Determine the water discharge Dij for the ith reservoir, i = 1, 2, X
NT
. . ., NH during the jth discrete time interval, j = 1, 2, . . ., N and the cor- PT mj ¼ PDth;j þ PLj ; j ¼ 1; 2; . . . ; N ð9Þ
responding generation schedule of the hydro plants, PHij and the m¼1
the limits on active power generation of thermal plants intervals be N, then each parent population is represented as
shown in Fig. 1.
PT m min 6 PT mj 6 PT m max ; m ¼ 1; 2; . . . ; NT; m–s; j ¼ 1; 2; . . . ; N
ð10Þ 3.1.1 A number of initial binary-coded solutions (genotypes) are
generated at random to form the initial parent of population
the slack bus constraint
size Np. Each string (S) represents the discharge for that par-
Ps min 6 Psj 6 P s max ; j ¼ 1; 2; . . . ; N ð11Þ ticular interval of that unit.
3.1.2 Binary strings are decoded to real values Dij (discharges) for
the transmission line flows the ith reservoir, i = 1, 2, . . ., NH during the jth discrete time
/k;ðminÞ 6 /k 6 /k; ðmaxÞ ; k ¼ 1; 2; . . . ; NL ð12Þ interval, j = 1, 2, . . ., N.
3.1.3 Each discharge Dij is checked for minimum and maximum
the power flow equations limits. If discharge Dij is less than the minimum discharge
level it is made equal to minimum discharge and if the dis-
F i ðX; U; CÞ ¼ 0; i ¼ 1; 2; . . . ; NB ð13Þ
charge Dij is greater than the maximum discharge level it is
The state vector X comprises of the bus voltage phase angles made equal to maximum discharge.
and magnitudes. The control vector U comprises of all the control- 3.1.4 Corresponding generation schedule of the hydro plants, PHij;
lable system variables like real power generations. The parameter i = 1, 2, . . ., NH is calculated.
vector C includes all the uncontrollable system parameters such 3.1.5 In each time interval, j = 1, 2, . . ., N compute the balance
as line parameters, and loads. demand to be met from thermal units PDth, j by taking the
difference between the system demand and the total hydro-
3. Proposed approach for the solution of hydrothermal power generation.
scheduling through GA 3.1.6 The scheduling of thermal units were done for the demand
of PD th, j. This economic dispatch problem is solved using
The hydro subproblem is solved using GA by creating the initial lambda iteration technique. The fuel cost is calculated for
populations randomly. The strings generated in the population each interval excluding transmission losses.
represent the discharge of each interval for all the plants. For the 3.1.7 The fitness function for each parent population Fpi is computed as:
discharge, equivalent hydropower generations are calculated. The
sum of hydropower generations of all the plants for each interval
X
NH X
N
Fpi ¼ FC Tpi þ k1 DHpi;lim
i þ k2 PT pi;lim
j ; pi ¼ 1; 2; . . . ; Np
gives the demand for thermal subproblem. The thermal subprob- i¼1 j¼1
lem is solved using lambda iteration technique without consider-
ð14Þ
ing losses. The cost obtained from economic dispatch and the
penalty functions for the constraint violations are considered as where K1, and K2 are penalty factors for the constraint violations,
objective function. Line losses and line flow constraints are consid- FCTpi is the total fuel cost for pith parent and the constraint viola-
ered only for the best solution obtained using GA. tions are given by
8 ! !
>
> P Dij ðY i;min Y i;max Þ þ P ALij ; if P Dij > P
N N N N
>
>
> ðY i;min Y i;max Þ þ ALij
< j¼1 j¼1 j¼1 j¼1
DHpi;lim
i ¼ ! ! ð15Þ
>
> P
N P
N PN P
N
> ðY
>
>
: i;min Y i;max Þ þ ALij Dij ; if Dij < ðY i;min Y i;max Þ þ ALij
j¼1 j¼1 j¼1 j¼1
To get the new patterns of genetic strings during the evolution The outputs of the NT thermal plants are determined so as to
process, two levels of crossover operation, i.e. string level crossover minimize the total operating cost (8) subject to power balance con-
and population level crossover are introduced. Both type of cross- straint (9), generator limit constraint (10) and slack bus constraint
over is done with fixed probability of 0.7. (11), line flow constraint (12) and power flow Eq. (13). Initially, the
outputs of the n = NT 1 ‘‘free generators’’ can be chosen arbi-
3.3.1. String level crossover trarily within the limits while the output of the ‘‘slack generator’’
A good scheduling could be expected by exchanging the strings is constrained by the power balance Eq. (9). For arbitrary outputs
of the units within the genotype. Since the partial string of geno- PTij, i = 1, 2, . . ., n, the output of the slack generator is:
type has no fitness function value, the selection processes are per-
X
NT1
formed randomly with certain probability. PT sj ¼ PDth;j þ PLj PT ij ; j ¼ 1; 2; . . . ; N ð17Þ
i¼1
3.3.2. Population level crossover
The various steps of the algorithm for solving the OPF problem
This operator is applied with certain probability. When applied,
with line flow constraint are given below.
the parent genotypes are combined to form two new genotypes
that inherent solution characteristics from both parents. In the
4.1. Initialisation of parent population
opposite case the offspring are identical replications of their par-
ents. Crossover is done between the parent genotypes obtained
The output of each one of the free generators is encoded in a 15-
from roulette wheel parent selection. The crossover scheme used
bit string, which gives a resolution of 215 = 32,768 discrete power
is single-point crossover.
values in the range (Pi (min), Pi (max)). These n strings are concatenated
to form a concatenated solution bit string of 15 n bits called
3.4. Mutation
genotype. Each genotype is decoded uniquely to an n-dimensional
generator power output vector called the phenotype, which is a
Mutation introduces new genetic material into the gene at some
real solution to the problem. The individuals in a parent are the
low rate. With a small probability, randomly chosen bits of the off-
real power outputs of the units excluding the slack bus unit. The
spring genotypes change from ‘0’ to ‘1’ and vice versa.
initial parent population is generated randomly as follows.
Consider the jth parent, Ij = [P1h, P2h, . . . , Pih, . . . , Pn h] of the popu-
3.5. Selection lation size Np. The components of Ij are generated as Pi h
U(Pi (min), Pi (max)), where U(Pi (min), Pi (max)) denotes a uniform random
The entire population, including parent and offspring are ar- variable ranging over U(Pi (min), Pi (max)). The remaining parents are
ranged in descending order. The best Np solutions, which survive generated in the same way.
are transcribed along with their elements to form the basis of the The system transmission loss, slack bus generation and line
next generation. The above procedure is repeated until the given phase angles are evaluated, by running decoupled load flow with
maximum generation count is reached. unit generations of each parent. The fitness value for each parent
For the best hydrothermal schedule obtained from the proposed of the population is computed as
GA, fuel cost is calculated for each interval including line losses.
Line flow constrains are checked at each interval and GA based X
NL
OPF is implemented only for the violated intervals to reduce the fj ¼ ðF i ðPih ÞÞj þ k1 Plim
sh þ k2 /lim
k ; j ¼ 1; 2; . . . ; Np ð18Þ
computational time. k¼1
90000000
80000000 ind 1
ind 2
70000000
ind 3
objective function (Rs)
60000000 ind 4
ind 5
50000000
40000000
30000000
20000000
10000000
0
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
No. of iterations
Fig. 2. Convergence characteristics of 9-bus system.
V.S. Kumar, M.R. Mohan / Electrical Power and Energy Systems 33 (2011) 827–835 831
where k1 and k2 are penalty factors for the constraint violations, new offspring genotype is produced by means of the two basic ge-
(Fi(Pi h))j is the total fuel cost for the jth parent. netic operators namely crossover and mutation.
Psh;min Psh ; if P sh < Psh;min
Plim
sh ¼ ð19Þ 4.3. Crossover
Psh Psh;min ; if P sh > Psh;min
Two genotypes are selected using Roulette wheel parent selec- With a small probability, random bits of the offspring genotype
tion algorithm that selects a genotype with a probability propor- flip from 0 to 1 and vice versa to give characteristics that don not
tional to genotypes relative fitness within the population. Then, a exist in the parent population.
25000000
ind 1
ind 2
20000000
ind 3
objective function (Rs)
ind 4
ind 5
15000000
10000000
5000000
0
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
No. of Iterations
Fig. 3. Convergence characteristics of 66-bus system.
2500
2000
Load (MW)
1500
1000
500
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Time Interval (hours)
Fig. 4. Load curve – 9-bus system.
832 V.S. Kumar, M.R. Mohan / Electrical Power and Energy Systems 33 (2011) 827–835
400
hydro plant 1
hydro plant 2
hydro plant 3
300
Discharge (cms)
200
100
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Time interval (hour)
Fig. 5. Discharge trajectories of hydro plants – 9-bus system.
Table 1 are transcribed along with their elements to form the basis of the
Comparison of cost and time of a 66-bus utility system. next generation. The above procedure is repeated until the given
DA and LP method [7] Proposed GA
maximum generation count is reached.
1800
1600
1400
1200
Load (MW)
1000
800
600
400
200
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Time Interval (hour)
Fig. 6. Load curve of 66-bus system.
V.S. Kumar, M.R. Mohan / Electrical Power and Energy Systems 33 (2011) 827–835 833
Interval Violated line Rating u Before GA based After GA based A.1. Thermal plant data
no. (°) OPF OPF
1 7 2.44 2.8077 2.4282
2 2.5776 2.4086 Unit Bus VSCH Pmax Pmin Cost coefficients
24 2.6252 2.4292
no. no. (pua) (MW) (MW)
6 6
Cost (Rs) 4.310906 10 4.341229 10 a (Rs/ b (Rs/ c (Rs/
MW2/h) MW/h) h)
1 1 1.00 600 60 1.0 50.0 0.0
starting with different random populations. The population size
2 2 1.02 300 30 1.2 52.0 0.0
was 50 genotypes in all the runs. The simulation was carried out
3 3 1.02 300 30 1.6 56.0 0.0
on a Pentium(R) D, 2.80 GHz processor.
4 4 1.02 200 20 1.6 40.0 0.0
The hydrothermal scheduling convergence characteristic of fit-
ness function for the best five individuals of a 9-bus and 66-bus
systems using proposed GA is presented in Figs. 2 and 3 respec-
tively. The fitness function convergence characteristic is drawn
A.2. Hydro plant data
by taking the parent with minimum fitness value at the end of iter-
ations. It is seen from Figs. 2 and 3 that fitness function converges
smoothly to the optimum value without any abrupt oscillations.
Description P1 Bus P2 Bus P3 Bus
This shows the convergence reliability of the proposed algorithm.
no. 5 no. 6 no.7
Fig. 4 shows the daily load curve of the 9-bus system. Fig. 5
shows the optimal discharge trajectories of the hydro plants of a Ymax CMS-hour 3000 2750 3000
9-bus system. It is seen from the Fig. 5 that the hydro discharge Ymin CMS-hour 1320 720 1320
trajectories obtained by the proposed GA closely matches with Y0 CMS-hour 2330 2030 2330
the daily load curve. YF CMS-hour 2330 2030 2330
To prove the effectiveness of the proposed GA, it is compared Dmax CMS 350 390 370
with Decomposition Approach (DA) and Linear Programming (LP) Dmin CMS 0.0 0.0 0.0
method as reported in the literature [7]. The data of a 66-bus In- PHmax MW 600 500 550
dian utility system from the literature is considered for compari- PHmin MW 0.0 0.0 0.0
son. Table 1 gives the comparison of execution time and cost of a Water equivalent Ho/G 1.78 1.28 1.58
9-bus and 66-bus Indian utility system. The cost of the proposed MW/CMS
GA is comparable with that of the DA and LP method of the litera- AL CMS 200 200 200
ture and the execution time of the proposed GA are very less as
compared with that of the DA and LP method.
Fig. 6 shows the daily load curve of an adapted 66-bus Indian
utility system and the data for this system is given in appendix. A.3. Factors of bus load to system load
A set of 12 limiting lines is chosen for observing line flow con-
straints. Table 2 gives the line flows of the violated line number
7. Hydrothermal scheduling obtained by the proposed GA gives Bus no. Factor Bus no. Factor Bus no. Factor
the line flow violations at intervals 1, 2 and 24. Using GA based
1 0.14 4 0.13 7 0.03
OPF the line flow violations are removed by adjusting the real
2 0.13 5 0.04 8 0.13
power generations and the cost is increased from Rs.
3 0.13 6 0.12 9 0.14
4.310906 106 to Rs. 4.341229 106.
6. Conclusion
A.4. Line data
Hydrothermal scheduling is a complex mathematical optimiza-
tion problem with a highly nonlinear and computational expensive
environment. This paper presents an algorithm for solving the Branch From To R (pu) X (pu) Y/2 Line phase
hydrothermal scheduling using GA in which all the problem vari- no. (pu) angle (°)
ables are taken into account without making the usual simplifying 1 1 2 0.0338 0.1743 0.1614 19.5
assumptions required by conventional techniques. This paper 2 2 3 0.0228 0.1171 0.1084 19.5
demonstrates with clarity, chronological development and suc- 3 3 4 0.0670 0.3489 0.3230 19.5
cessful application of GA to the solution of hydrothermal schedul- 4 5 6 0.0270 0.1359 0.1258 19.5
ing. Network flow constraints have been effectively enforced 5 6 7 0.0371 0.1906 0.1765 19.5
through GA based OPF. The proposed algorithm has been tested 6 7 4 0.0670 0.3489 0.3230 19.5
on two sample systems, the first one consisting of nine buses, 11 7 5 9 0.0143 0.0735 0.0680 19.5
transmission lines, four thermal plants and three hydro plants 8 1 9 0.0338 0.1743 0.1614 19.5
and the second with an Indian utility system comprising 66 buses, 9 9 3 0.0199 0.1024 0.0941 19.5
93 transmission lines, 12 thermal plants and 11 hydro plants. This 10 3 8 0.0109 0.0561 0.0520 19.5
proposed GA reduces the complexity, computation time and also 11 8 7 0.0187 0.0963 0.0892 19.5
gives near global optimum solution.
834 V.S. Kumar, M.R. Mohan / Electrical Power and Energy Systems 33 (2011) 827–835
Unit no. Bus no. VSCH (pua) Pmax (MW) Pmin (MW) Cost coefficients
a (Rs/MW2/h) b (Rs/MW/h) c (Rs/h)
1 1 1.0 210 20 1.6 40.0 0.0
2 210 20 1.6 40.0 0.0
3 210 20 1.6 40.0 0.0
4 210 20 1.6 40.0 0.0
5 2 1.02 210 20 1.0 50.0 0.0
6 210 20 1.0 50.0 0.0
7 210 20 1.0 50.0 0.0
8 3 1.02 60 10 1.2 52.0 0.0
9 60 10 1.2 52.0 0.0
10 4 1.02 110 10 1.6 56.0 0.0
11 110 10 1.6 56.0 0.0
12 110 10 1.6 56.0 0.0
Branch From To R (pu) X (pu) Y/2 (pu) Line Branch From To R (pu) X (pu) Y/2 (pu) Line
no. phase no. phase
angle (°) angle (°)
13 15 14 0.00670 0.03443 0.03369 – 72 56 55 0.01370 0.02040 0.00215 –
14 14 16 0.01889 0.09705 0.09400 – 73 56 51 0.10540 0.15700 0.01625 9.84
15 14 18 0.02276 0.11696 0.10830 – 74 54 55 0.01640 0.02700 0.00284 –
16 17 18 0.01308 0.07020 0.06500 – 75 62 53 0.12024 0.20596 0.02168 –
17 18 16 0.03327 0.17100 0.15830 – 76 53 52 0.01376 0.02356 0.00444 –
18 18 25 0.01911 0.09822 0.09090 – 77 52 51 0.04735 0.08111 0.00854 –
19 15 25 0.01416 0.07278 0.06724 – 78 65 30 0.01406 0.07222 0.07592 –
20 18 19 0.00540 0.02500 0.02370 – 79 36 27 0.02034 0.10448 0.09674 –
21 19 20 0.00540 0.02560 0.04492 – 80 24 27 0.01752 0.07000 0.06481 –
22 18 20 0.00994 0.05106 0.04720 – 81 27 40 0.00000 0.03601 0.00000 5.63
23 2 20 0.00792 0.04070 0.03768 7.54 82 36 65 0.01905 0.09783 0.09060 –
24 20 21 0.00710 0.03647 0.03369 – 83 13 51 0.00000 0.08667 0.00000 –
25 2 21 0.00710 0.03647 0.00334 – 84 14 49 0.00000 0.06059 0.00000 –
26 2 23 0.02778 0.14275 0.13215 11.63 85 34 64 0.00000 0.06053 0.00000 5.33
27 23 24 0.02172 0.11160 0.10331 – 86 16 5 0.01728 0.08875 0.08170 –
28 32 14 0.00000 0.06059 0.00000 – 87 38 63 0.03324 0.08540 0.00898 –
29 25 26 0.01005 0.05165 0.04780 – 88 63 64 0.03324 0.08540 0.00898 –
30 25 27 0.02109 0.10831 0.13688 – 89 57 58 0.03070 0.03760 0.00158 –
31 28 29 0.02056 0.10565 0.09761 – 90 57 60 0.09220 0.11280 0.00474 8.10
32 29 25 0.01440 0.07396 0.06750 – 91 26 27 0.01195 0.06136 0.05680 –
33 29 30 0.00300 0.15650 0.01450 – 92 1 8 0.00244 0.01252 0.01159 –
34 30 31 0.00411 0.02113 0.01950 – 93 1 36 0.01570 0.08061 0.07462 7.29
35 31 66 0.00130 0.00665 0.02464 –
36 34 31 0.00442 0.02270 0.02101 –
37 34 33 0.00244 0.01232 0.01140 –
38 33 4 0.00122 0.00626 0.00580 – References
39 4 34 0.00366 0.01878 0.01739 –
40 35 36 0.01493 0.07670 0.07100 – [1] Wood AJ, Wollenberg BF. Power generation, operation & control. 2nd ed. John
Wiley & Sons, Inc.; 2003.
41 36 5 0.01234 0.06339 0.05869 – [2] Yang J-S, Chen N. Short-term hydrothermal coordination using multi-pass
42 4 3 0.00000 0.09524 0.00000 8.40 dynamic programming. IEEE Trans Power Syst 1989;4:1050–6.
43 30 38 0.00000 0.04275 0.00450 – [3] Shang SC, Chen CH, Fong IK, Luh PB. Hydroelectric generation scheduling with
an effective differential dynamic programming algorithm. IEEE Trans Power
44 38 39 0.13553 0.34822 0.03660 – Syst 1990;5:737–43.
45 28 39 0.00000 0.16066 0.00000 – [4] Li C-A, Jap PJ, Streiffert DL. Implementation of network flow programming to
46 39 40 0.02263 0.05815 0.00612 – the hydrothermal coordination in energy management systems. IEEE Trans
Power Syst 1993;8:1045–53.
47 7 27 0.00000 0.02500 0.00000 –
[5] Dillon TS, Anderson S, Sjelvgren D. Optimal operations planning in a large
48 41 47 0.27000 0.53000 0.05580 – hydro-thermal power system. IEEE Trans Power Apparatus Syst 1983;PAS-
49 42 43 0.54442 0.56756 0.05974 – 102:3644–51.
50 43 45 0.00000 0.09333 0.00000 – [6] Sjelvgren D, Brannlund H, Dillon TS. Large-scale non-linear programming
applied to operations planning. Int J Elec Power Energy Syst 1989;11:213–7.
51 45 44 0.06602 0.13082 0.01377 8.10 [7] Mohan MR, Kuppusamy K, Khan MA. Optimal short-term hydro-thermal
52 20 45 0.00000 0.08000 0.00000 7.76 scheduling using decomposition approach and linear programming method.
53 18 46 0.00000 0.08027 0.00000 – Int J Elect Power Energy Syst 1992;14:39–44.
[8] Soares S, Lyra C, Tavares H. Optimal generation scheduling of hydro-thermal
54 46 47 0.05366 0.12127 0.01276 – power systems. IEEE Trans Power Apparatus Syst 1980;PAS-99:107–1115.
55 17 47 0.00000 0.01050 0.00000 – [9] Pereira MVF, Pinto LMVG. Application of decomposition techniques to the mid-
56 35 31 0.01361 0.06093 0.08315 – and short-term scheduling of hydrothermal systems. IEEE Trans Power
Apparatus Syst 1983;PAS-102:3611–8.
57 25 41 0.00000 0.06484 0.00000 – [10] Bai Xaiomin, Shahidehpour SM. Hydro-thermal scheduling by tabu search and
58 66 63 0.00000 0.06063 0.00000 – decomposition method. IEEE Trans Power Syst 1996;11:968–74.
59 22 31 0.00000 0.03968 0.00000 – [11] Werner TG, Verstege JF. An evolution strategy for short-term operating
planning of hydrothermal power systems. IEEE Trans Power Syst
60 45 48 0.02109 0.41668 0.04380 – 1999;14:1362–8.
61 47 48 0.04311 0.09759 0.02244 – [12] Huang Shyh-Jier. Enhancement of hydroelectric generation scheduling using
62 49 50 0.03477 0.07029 0.01365 – ant colony system based optimization approaches. IEEE Trans Power Syst
2001;16:296–301.
63 49 51 0.13620 0.23354 0.02458 –
[13] Goldberg DE. Genetic algorithms in search, optimization and machine
64 49 62 0.24700 0.40500 0.04263 – learning. Addison Wesley; 1989.
65 62 59 0.09220 0.11280 0.01180 – [14] Wu yong-Gang, Ho Chun-Ying, Wang Ding-Yi. A diploid genetic approach to
66 62 61 0.03070 0.03760 0.00237 – short-term scheduling of hydro-thermal system. IEEE Trans Power Syst
2000;15:1268–74.
67 6 62 0.00000 0.11312 0.00000 5.23 [15] Kazarlis SA, Bakirtzis AG, Petridis V. A genetic algorithm solution to the unit
68 60 61 0.01401 0.01725 0.00158 – commitment problem. IEEE Trans Power Syst 1996;11:83–92.
69 58 59 0.01401 0.01725 0.00158 – [16] Gil E, Bustos J, Rudnick H. Short-term hydrothermal generation scheduling
model using a genetic algorithm. IEEE Trans Power Syst 2003;18:1256–64.
70 57 8 0.01401 0.01725 0.00158 – [17] Somasundaram P, Kuppusamy K, Kumudini Devi RP. Evolutionary
71 56 10 0.00000 0.05250 0.00000 – programming based security constraint optimal power flow. Elec Power Syst
Res 2004;72:137–45.