Applied Soft Computing 11 (2011) 2863–2870
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Evolving ant colony optimization based unit commitment K. Vaisakh a,∗ , L.R. Srinivas b a b
Department of Electrical Engineering, AU College of Engineering, Andhra University, Visakhapatnam 530003, AP, India Department of Electrical and Electronics Engineering, S.R.K.R. Engineering College, Bhimavaram 534204, AP, India
a r t i c l e
i n f o
Article history: Received 6 June 2009 Received in revised form 2 August 2010 Accepted 28 November 2010 Available online 4 December 2010 Keywords: Evolving ant colony optimization Unit commitment problem Pheromone matrix Genetic algorithm
a b s t r a c t Ant colony optimization (ACO) was inspired by the observation of natural behavior of real ants’ pheromone trail formation and foraging. Ant colony optimization is more suitable for combinatorial optimization problems. ACO is successfully applied to the traveling salesman problem. Multistage decision making of ACO gives an edge over other conventional methods. This paper proposes evolving ant colony optimization (EACO) method for solving unit commitment (UC) problem. The EACO employs genetic algorithm (GA) for finding optimal set of ACO parameters, while ACO solves the UC problem. Problem formulation takes into consideration the minimum up and down time constraints, startup cost, spinning reserve, and generation limit constraints. The feasibility of the proposed approach is demonstrated on two different systems. The test results are encouraging and compared with those obtained by other methods. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Unit commitment (UC) is used to schedule the generating units for minimizing the overall cost of the power generation over the scheduled time horizon while satisfying a set of system constraints. UC problem is a nonlinear, combinatorial optimization problem. The global optimal solution can be obtained by complete enumeration, which is not applicable to large power systems due to its excessive computational time requirements [1]. Up to now, many methods have been developed for solving the UC problem such as priority list methods [2,3], integer programming [4,5], dynamic programming (DP) [6–8], branch-and-bound methods [9], mixed-integer programming [10] and Lagrangian relaxation (LR) [11–13]. These methods have only been applied to small UC problems and have required major assumptions which limit the solution space [14,15]. Lagrange relaxation for UC problem was superior to dynamic programming due to its faster computational time. However, it suffers from numerical convergence and solution quality problems in the presence of identical units. Furthermore, solution quality of LR depends on the method to initialize and update Lagrange multipliers [16]. Ant colony optimization (ACO) was proposed by Dorigo et al. to solve difficult combinatorial optimization problems. ACO is a random stochastic population based algorithm that simulates the
∗ Corresponding author. Tel.: +91 891 2844840; fax: +91 891 2747969. E-mail address: vaisakh
[email protected] (K. Vaisakh). 1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2010.11.019
behavior of ants for cooperation and learning in finding shortest paths between food sources and their nest [17–20]. In ACO, the ants’ behavior is simulated to solve the combinatorial problems such as traveling salesman problem and quadratic assignment problem [19,20]. Artificial ant colony search algorithm is applied to solve large-scale economic dispatch problem in Ref. [21]. In Ref. [22], economic dispatch of power systems was solved by generalized ant colony optimization. Ant colony search algorithm is applied to distribution network reconfiguration for loss reduction in Ref. [23]. Ant colony search algorithm for Optimal Reactive Power Optimization is given in Ref. [24]. The ACO is applied to solve the UC problem by Refs. [25,26]. This paper proposes a new method, evolving ant colony optimization (EACO) for solving UC problem for a period of 24 h. In this approach, the ACO is used to obtain the unit commitment schedule and genetic algorithm technique is used to find optimal set of parameters required for ACO. The Lagrangian multiplier method is applied to obtain the economic dispatch for the 24-h schedule. To illustrate the effectiveness of the proposed method, it is tested on two different systems one with 10 and 20 units and the other with 10 units. Simulation results are presented and compared with other methods.
2. Problem formulation The objective of unit commitment problem is to minimize the production cost over the scheduled time horizon (24 h) under the generator operational and spinning reserve constraints. The objec-
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minimum up and down time constraints Nomenclature CSCi Fit
cold startup cost of unit i generator fuel cost in quadratic form
Ui,t =
2
Fit = ai + bi Pit + ci (Pit ) fuel cost in $/h HSCi hot startup cost of unit i N total number of generator units Pi,min minimum real power generation of unit i (in MW) maximum real power generation of unit i (in MW) Pi,max Pit real power generation of unit i at hour t (in MW) PDt load demand at hour t (in MW) spinning reserve at hour t (in MW) Rt STit startup cost of unit i at hour t T total number of hours cold start hours of unit i (in h) Ti,cold Ti,down minimum down time of unit i (in h) Ti,off continuously off time of unit i (in h) Ti,on continuously on time of unit i (in h) Ti,up minimum up time of unit i (in h) Ui,t status of unit i at hour t (on = 1, off = 0) m number of ants pheromone intensity Cij production cost occurred during a stage ij heuristic function for visibility n number of eligible states (size of pheromone matrix)
tive function to be minimized is F(Pit , Ui,t ) =
N T
[Fi (Pit ) + STi,t (1 − Ui,t−1 )]Ui,t
(1)
t=1 i=1
subject to the following constraints: power balance constraint N
Pit Ui,t = PDt
(2)
i=1
spinning reserve constraint N
Pi,max Ui,t ≥ PDt + Rt
(3)
i=1
generator limit constraints Pi,min Ui,t ≤ Pit ≤ Pi,max Ui,t ,
i = 1, . . . , N
(4)
S0 Initial Condition
Transition Cost
⎧ ⎪ ⎨ 1, ⎪ ⎩
0, 0 or 1,
if Ti,on < Ti,up if Ti,off < Ti,down
(5)
otherwise
startup cost
STi,t =
HSC
if Ti,down ≤ Ti,off ≤ Ti,cold + Ti,down
CSCi
if Ti,off > Ti,cold + Ti,down
(6)
3. Implementation of the proposed method The implementation of EACO algorithm for solving UC problem involves two phases. In the first phase, all possible states of the tth hour (using exhaustive enumeration) that satisfy the load demand with spinning reserve constraints are found. For 10-unit system, a maximum of 256 eligible states are found in any hour by taking first two generators as base units. i.e. first two generators are in ‘on’ condition for 24 h and only 256 feasible states are available for remaining eight generators. In 20-unit case exhaustive enumeration is not possible. First four units are base units and in the remaining 16 units, last six peak units are not considered for light load conditions to try the combinations. In that way, a maximum of 1024 feasible states are found in any hour. Economic dispatch using Lagrangian multiplier method is carried out for all feasible states to calculate the optimal generator output and production cost for each hour and startup cost is added to production cost to get transition cost for each hour. This process is continued for the complete scheduling period of 24 h to get total cost for each state of all feasible states which constitutes the ant search space (ASS). The ASS which involves multi decision states is shown in Fig. 1. St is the eligible state satisfying load demand and spinning reserve at tth hour. Once the search space is identified, the second phase involves the artificial ants allowed to pass continuously through the ASS. Each ant starts its journey from the starting node (initial condition, i.e. 1st hour), reaches the final node (24th hour) to complete its tour. Whenever an ant reaches the final node, overall generation cost for 24 h including startup cost is calculated. For each transit stage (t to t + 1 h), the ant selects a state satisfying minimum uptime, minimum down time, ramp rate constraints, etc. The generation cost together with startup cost is calculated for all units which becomes transition cost. This process is continued till the time period becomes T (24 h) and a tour is completed for that particular ant. Whenever a tour is completed by an individual ant and if the total generation cost is found is lesser than the minimum cost paths taken by the previous
S1
S1
S2
S2
S3
S3
St t=1
St t=2 Transition Cost
Fig. 1. Multi decision space.
K. Vaisakh, L.R. Srinivas / Applied Soft Computing 11 (2011) 2863–2870
ants, the present cost path is captured. The procedure is continued for all the remaining ants available at the starting nodes, which enables to trace the optimal path.
The evolving ant colony search mechanism can be mainly divided into initialization, pseudo-random probabilistic transition rule, fitness function and genetic algorithm, pheromone update rule. 4.1. Initialization During initialization, the parameters such as number of ants m, the relative importance of the pheromone trail ˛, relative importance of the visibility ˇ, amplifying or decaying factor for updating of pheromone , evaporation factor , the exploitation probability q0 and scaling factor for the modification of the trail Q are randomly generated as binary strings to be subjected to GA search and are converted into the values within the limits shown in Table 1 as mentioned in Ref. [27] and pheromone trail 0 is initialized heuristically to a small value. 4.2. Transition strategy The transition strategy used as given in Ref. [17] .The transition probability for the kth ant from one state i to next state j is given by
argmaxu ∈ J k
i
˛
[iu (t)] [iu ]ˇ
J
if q ≤ q0 if q > q0
(7)
where ij trail intensity on edge (i,j); ij = (1/Cij ) is called heuristic function; Cij is the production cost occurred for that particular stage; q is a random variable in the interval [0, 1]; q0 is a tunable parameter. J ∈ Jik is a state that is randomly selected according to the probability, PiJk (t) =
[iu (t)]
u ∈ J k [iu i
˛
ˇ iJ
(t)]
is the weighted sum of the components given below F=
3
ci Fi
(9)
i=1
4. Working of EACO
j=
2865
ˇ ˛
where ci is the weight of component i. (1) F1 = 1/(L + 1 − L+ ), where L is the best cost found by the colony and L+ is the best cost found by all of the colonies thus far. This component is relative to the performance of other colonies. c1 = 2.0–3.0. (2) F2 = e−v/5n , where v is the iteration in which best tour was found. F2 reflects fact that it is important for the best tour to be found quickly. c2 = 0.5–0.8. (3) F3 = e−m/10n encourages m to be as small as possible. It is important to try to minimize m as the CPU time per iteration scales linearly with m. c3 = 0.5. F3 does not directly depend on how well the algorithm performs but rather on parameters that influence its speed of convergence toward a good solution. Maximization of the function F in Eq. (9) is considered in this algorithm. Standard GA implementation is used here which includes reproduction by Roulette-wheel selection, single-point crossover and mutation procedures to find the optimal set of parameters for ACO. The parameters taken for GA search are: the length of the strings 16, the crossover probability 0.95 and the mutation probability 0.01. 4.4. Pheromone update rule The ant that performed the best tour since the beginning of the trail is allowed to globally update the concentrations of pheromone on corresponding edges. In ACO the global trail updating rule is ij (t + 1) ← (1 − )ij (t) + ij (t) where is a parameter governing trail decay and
(8)
iJ
when q ≤ q0 which corresponds to an exploitation of the knowledge available about the problem, that is the heuristic knowledge about cost between states and the learned knowledge memorized in the form of pheromone trails, whereas q > q0 favors more exploration. Cutting exploration by tuning q0 allows the activity of the system to concentrate on the best solutions instead of letting it explore constantly. Here ‘u’ is the allowable states [17,18]. 4.3. Fitness function and genetic algorithm The fitness function of genetic algorithm used in this method is given in Ref. [27]. In the GA, each colony, characterized by a set of parameters, is an individual. Colonies are in competition to make it to the next generation. The fitness function F used in this study
(10)
ij (t) =
⎧ ⎨ Q ⎩
(L+ )
0
if (i, j) ∈ T+
(11)
otherwise
where Q is the scaling factor for the modification of the trail and T+ is the best tour since the beginning of the trail and L+ is best cost. Only the best tour is reinforced through the global update. Local updates are also performed, so that other solutions also emerge. It is performed as follows: when, while performing a tour, ant k is in state i and selects a state j ∈ Jik , the pheromone concentration of (i,j) is updated by the following equation: ij (t) ← (1 − )ij (t) + 0
(12)
where 0 initial value of the pheromone trail. The algorithm for the EACO approach to UC problem is given below. The flowchart of EACO is shown in Fig. 2. 4.5. Algorithm
Table 1 Parameters evolved and their ranges. Range 1 ≤ m ≤ 2n 0.0 ≤ ˛ ≤ 5.0 0.0 ≤ ˇ ≤ 10.0 0.1 ≤ ≤ 3.0 0.0 ≤ ≤ 1.0 0.0 ≤ q0 ≤ 1.0 0.0 ≤ Q ≤ 100.0
• Step 1: Initialize ACO parameters, pheromone matrix • Step 2: For each hour, generate ant search space • Step 3: Ants start search from starting node. Ants are placed at different nodes randomly • Step 4: Select the next state of ants by applying pseudo-random probability transition rule • Step 5: Check for satisfying constraints. If any constraint is violated, large penalty value is added to the total cost
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Table 2 Unit data for the 10-unit system.
Pmax (MW) Pmin (MW) a ($/h) b ($/MWh) c ($/MW2 h) Min up (h) Min down (h) Hot start cost ($) Cold start cost ($) Cold start hours (h) Initial status (h)
Pmax (MW) Pmin (MW) a ($/h) b ($/MWh) c ($/MW2 h) Min up (h) Min down (h) Hot start cost ($) Cold start cost ($) Cold start hours (h) Initial status (h)
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
455 150 1000 16.19 0.00048 8 8 4500 9000 5 8
455 150 970 17.26 0.00031 8 8 5000 10,000 5 8
130 20 700 16.60 0.002 5 5 550 1100 4 −5
130 20 680 16.50 0.00211 5 5 560 1120 4 −5
162 25 450 19.70 0.00398 6 6 900 1800 4 −6
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
80 20 370 22.26 0.00712 3 3 170 340 2 −3
85 25 480 27.74 0.00079 3 3 260 520 0 −3
55 10 660 25.92 0.00413 1 1 30 60 0 −1
55 10 665 27.27 0.00222 1 1 30 60 0 −1
55 10 670 27.79 0.00173 1 1 30 60 0 −1
• Step 6: Evaluate the fitness function and obtain the local and global parameters for each generation • Step 7: Terminating Criteria Satisfied? If yes, print the UC schedule, if no, update pheromone trails and evolving parameters.
5. Simulation results All simulations have been run on MATLAB environment with Pentium-IV, 2.66 GHz computer with 512 MB RAM. Base 10-unit characteristics are taken from Ref. [28] and are given in Table 2. The spinning reserve requirement is considered to be 10% of the load
Start Initialize ACO parameters, pheromone matrix
For each hour, generate ant search space
Ants start search from starting node. Ants are placed at different nodes randomly.
Select the next state of ants by applying pseudo- random probability transition rule.
Check for satisfying constraints. If any constraint is violated, large penalty value is added to the total cost.
Evaluate the fitness function and obtain the local and global parameters for each generation.
Table 3 Control parameter settings for the 10-unit system. c1
c2
2.1 2.1 2.1 2.1 2.5 2.5 2.5 2.5 3.0 3.0 3.0 3.0
0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8
No
Terminating Criteria Satisfied?
Yes Print the UC schedule
Exit Fig. 2. Flowchart of the proposed method.
565,352 564,950 564,094 563,938 565,804 565,825 565,827 565,508 567,367 567,367 566,404 566,107
Table 4 Units test results of the proposed EACO after 10 runs. No. of units
Total cost ($) Best
10 20
Update pheromone trails and evolving parameters
Total cost ($)
563, 938 1, 123, 297
Worst 565,869 1,128,072
Average 564,831 1,125,516
Variation (%) 0.34 0.42
Table 5 Set of ACO parameters evolved. ACO parameter
10-Unit system
20-Unit system
m ˛ ˇ q0 Q
428 2.6127 7.1177 0.5408 0.1663 0.9342 54.8196
1305 1.9054 3.8413 0.3016 0.5747 0.6721 97.6349
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Table 6 Comparison of total cost. Total cost ($) LR [30]
GA [30]
EP [31]
GAUC [32]
ELR [28]
ICGA [33]
HPSO [34]
EACO
565, 825 1, 130, 660
565,825 1,126,243
564,551 1,125,494
563,977 1,125,516
563,977 1,123,297
566,404 1,127,244
563,942 –
563,938 1,123,297
demand; cold startup cost is double that of hot startup cost and total scheduling period is 24 h. The simulations include test runs for 10 and 20 generator systems. For the 20-unit system, the base 10 generators are duplicated and load demand is multiplied by two. The number of eligible states (size of pheromone matrix) for 10 and 20unit systems are 256 and 1024, respectively. Maximum generations are 100 and 50 for 10 and 20-unit systems, respectively. The solution is tested for different weights of the fitness function within the ranges mentioned previously. Based on the results, c1 = 2.1, c2 = 0.8 and c3 = 0.5 are chosen, as EACO performed best with these settings. The solutions with different weights of the fitness function for 10unit system are given in Table 3. Ten test runs were performed on 10- and 20-unit systems and the results are presented in Table 4. Population size of 20 is taken for evolving ACO parameters. In normal ACO procedure the ant colony parameters m, ˛, ˇ, , , q0 , Q are fixed. Generally, it is not easy to find optimal set of parameters for ACO. This is to be done heuristically through trail and error procedure. In this paper, these parameters are not fixed, but are evolved using genetic algorithm [27] so that optimal parameters for better solution can be found. The evolved set of ACO parameters obtained by GA is given in Table 5. The number of ants m is included for evolving because it is unclear how many ants are necessary to find a very good solution in an efficient way for a problem of given size. The exploitation probability q0 is included for evolving because its optimal value is also unclear. Performance of the 10-unit system was considered with the best combination of ACO parameters given in Ref. [26] (i.e. with fixing of ACO parameters without evolving them). The total cost
5.66
Graph between cost and generation
x 105
5.655
Cost for 10 units
10 20
5.65
5.645
5.64
5.635
0
10
20
30
40
50
60
70
80
90
100
45
50
45
50
Generations Fig. 4. Total cost for 10-unit system.
Graph between fitness & generation 3.34
3.335
Fitness
No. of units
3.33
3.325
Graph between fitness & generation 3.34 3.32 3.335 3.33
3.315
0
5
10
15
Fitness
20
25
30
35
40
Generations
3.325 3.32
Fig. 5. Fitness for 20-unit system.
3.315 3.31 1.128
3.305
Graph between cost and generation
x 106
1.1275
3.3
1.127 0
10
20
30
40
50
60
70
Generations Fig. 3. Fitness for 10-unit system.
Table 7 Set of ACO parameters evolved for the system in Ref. [26]. ACO parameters
10-Unit system
m ˛ ˇ q0 Qsf
234 1.7108 7.0986 1.6114 0.6684 0.5767 80.5814
80
90
100
Cost for 20 units
3.295 1.1265 1.126 1.1255 1.125 1.1245 1.124 1.1235 1.123
0
5
10
15
20
25
30
35
Generations Fig. 6. Total cost for 20-unit system.
40
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Table 8 EACO solution for the 10-unit system in Ref. [26].
8.36
Hour
For 10-unit system Unit status
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1111111101 1111111101 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111011111 1111011111 1101011111 110101111 1 1101011111
8.355
Cost for 10 units
8.35
8.34 8.335
8.325 8.32
0
10
20
30
40
50
60
70
80
90
100
Generations Fig. 7. Convergence of total cost for 10-unit system for the system in Ref. [26].
Table 9 Comparison of total cost for the system in Ref. [26].
10
8.345
8.33
obtained was $ 564,551. Table 6 shows comparison of the proposed method with other methods. Obviously EACO is a robust method in obtaining improved solution. Output powers of the units for EACO solution for 10 and 20-unit cases are given in Tables 10 and 11, respectively. It is to be observed that ramp rate limit constraints [29] are not taken into account in this paper since we would like
No. of units
x 104
Total cost ($) DP [26]
BB [26]
ACS [26]
EACO
83652.40
83475.25
83491.42
83240.17
to compare the solutions with previous works [28,30–34]. The fitness and total cost characteristics for 10- and 20-unit systems are shown in Figs. 3–6. EACO algorithm was tested with the 10-unit system given in Ref. [26]. The constraints for this system are taken same as that reference including ramp rate constraints. The population size is taken as 20. The evolved ant colony parameters are given in Table 7. Table 8 gives optimal solution of the proposed algorithm and comparison of the cost with the methods given in Ref. [26] is given in Table 9. The total cost characteristic for this system is shown in Fig. 7. The above results on the two test systems clearly indicate that the performance of EACO is better than ACO run with fixed parameters and other methods. One advantage of this method is the simplicity of the algorithm and another advantage is that EACO effectively handles the problem of same/similar units. In methods like Lagrangian relaxation (LR), all the similar units will be switched on simultaneously because decision criterion would be same for all similar units. This leads to increase of production cost due to excessive spinning reserve. In EACO, this problem is minimized
Table 10 Units output power for the 10-unit case. Hours (1–24)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Units 1
2
3
4
5
6
7
8
9
10
455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455
245 295 370 455 390 360 410 455 455 455 455 455 455 455 455 310 260 360 455 455 455 455 425 345
0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 25 40 25 25 25 30 85 162 162 162 162 85 30 25 25 25 30 162 85 145 0 0
0 0 0 0 0 0 0 0 20 33 73 80 33 20 0 0 0 0 0 33 20 20 20 0
0 0 0 0 0 0 0 0 25 25 25 25 25 25 0 0 0 0 0 25 25 25 0 0
0 0 0 0 0 0 0 0 0 10 10 43 10 0 0 0 0 0 0 10 0 0 0 0
0 0 0 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0
Total fuel cost for the above system = 559,848, startup cost = 4090 and total cost = 563,938.
K. Vaisakh, L.R. Srinivas / Applied Soft Computing 11 (2011) 2863–2870
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Table 11 Units output power for the 20-unit case. Units Hours (1–24)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
455 455 245 245 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
455 455 295 295 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
455 455 382.5 382.5 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0
455 455 455 455 0 0 0 0 40 40 0 0 0 0 0 0 0 0 0 0
455 455 455 455 0 0 0 130 25 25 0 0 0 0 0 0 0 0 0 0
455 455 425 425 130 0 130 130 25 25 0 0 0 0 0 0 0 0 0 0
455 455 455 455 130 0 130 130 45 45 0 0 0 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 30 30 0 0 0 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 97.5 97.5 20 20 25 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 162 162 33 33 25 25 10 10 0 0 0 0
455 455 455 455 130 130 130 130 162 162 73 73 25 25 10 10 10 10 0 0
455 455 455 455 130 130 130 130 162 162 80 80 25 25 43 43 10 10 10 10
455 455 455 455 130 130 130 130 162 162 33 33 25 25 10 10 0 0 0 0
455 455 455 455 130 130 130 130 97.5 97.5 20 20 25 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 30 30 0 0 0 0 0 0 0 0 0 0
455 455 310 310 130 130 130 130 25 25 0 0 0 0 0 0 0 0 0 0
455 455 260 260 130 130 130 130 25 25 0 0 0 0 0 0 0 0 0 0
455 455 360 360 130 130 130 130 25 25 0 0 0 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 30 30 0 0 0 0 0 0 0 0 0 0
455 455 455 455 130 130 130 130 162 162 43 43 0 0 10 10 10 10 10 0
455 455 455 455 130 130 130 130 105 105 20 20 0 0 10 0 0 0 0 0
455 455 455 455 0 0 0 130 105 105 20 20 0 0 0 0 0 0 0 0
455 455 432.5 432.5 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0
455 455 345 345 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Total fuel cost for the above system = 1,114,897, startup cost = 8400, and total cost = 1,123,297.
because it selects the optimal combination of the feasible states of these similar units considering spinning reserve and up time/down time constraints. Similar units are switched on only when spinning reserve or up time or down time constraints are not satisfied (Tables 10 and 11). 6. Conclusion ACO is more suitable for combinatorial optimization problems. In this paper, evolving ACO is proposed and effectively implemented to solve the UC problem. The effectiveness of this proposed method is tested on two different systems. Results demonstrate that EACO is a very competent method to solve the UC problem. EACO generates better solutions because the ACO parameters are evolved instead of fixing them. EACO handles similar units effectively so that excessive spinning reserve is reduced. The results obtained from simulation are most encouraging in comparison with other methods and EACO production costs are found to be less expensive. Accordingly, EACO is very suitable for UC problem due to the production cost savings. References [1] A.J. Wood, B.F. Wollenberg, Power Generation, Operation and Control, 2nd ed., Wiley, New York, 1996. [2] R.M. Burns, C.A. Gibson, Optimization of priority lists for a unit commitment program, in: Proceedings of IEEE/Power Engineering Society Summer Meeting, Paper A 75-453-1, 1975. [3] G.B. Sheble, Solution of the unit commitment problem by the method of unit periods, IEEE Trans. Power Syst. 5 (1) (1990) 257–260. [4] T.S. Dillon, K.W. Edwin, H.D. Kochs, R.J. Taud, Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination, IEEE Trans. Power App. Syst. 97 (6) (1978) 2154–2166. [5] L.L. Garver, Power generation scheduling by integer programming development of theory, IEEE Trans. Power App. Syst. 18 (6) (1963) 730–735. [6] W.L. Snyder Jr., H.D. Powel Jr., J.C. Rayburn, Dynamic programming approach to unit commitment, IEEE Trans. Power App. Syst. 2 (1987) 339–350. [7] P.G. Lowery, Generation unit commitment by dynamic programming, IEEE Trans. Power App. Syst. 102 (1983) 1218–1225. [8] Z. Ouyang, S.M. Shahidepour, An intelligent dynamic programming for unit commitment application, IEEE Trans. Power Syst. 6 (3) (1991) 1203–1209. [9] A.I. Cohen, M. Yoshimura, A branch-and-bound algorithm for unit commitment, IEEE Trans. Power App. Syst. 102 (2) (1983) 444–451. [10] J.A. Muckstadt, R.C. Wilson, An application of mixed-integer programming duality to scheduling thermal generating systems, IEEE Trans. Power App. Syst. (1968) 1968–1978.
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[34] T.O. Ting, M.V.C. Rao, C.K. Loo, A novel approach for unit commitment problem via an effective hybrid particle swarm optimization, IEEE Trans. Power Syst. 21 (1) (2006) 411–418. K. Vaisakh received the B.E. degree in Electrical Engineering from Osmania University, Hyderabad, India in 1994, M.Tech. degree from JNT University, Hyderabad, India in 1999, and Ph.D. degree in Electrical Engineering from the Indian Institute of Science, Bangalore, India in 2005. Currently, he is working as a Professor in the Department of Electrical Engineering, AU College of Engineering, Andhra University, Visakhapatnam, AP, India. His research interests include optimal operation of power system, voltage stability, FACTS, power electronic drives, and power system dynamics.
L.R. Srinivas received the B.Tech. degree in Electrical and Electronics Engineering from JNT University, Hyderabad, India in 1992, M.Tech. degree from JNT University, Hyderabad, India in 2005. He is pursuing Ph.D. in the Department of Electrical Engineering, AU College of Engineering, Andhra University, Visakhapatnam, AP, India. He is currently working as an Associate Professor in the Department of Electrical and Electronics Engineering, SRKR Engineering College, Bhimavaram, W.G. Dist., AP, India. His research interest includes power system operation and control, power system analysis, power system optimization, and soft computing applications.
