European Journal of Scientific Research ISSN 1450-216X Vol.64 No.1 (2011), pp. 157-167 © EuroJournals Publishing, Inc. 2011 http://www.europeanjournalofscientificresearch.com
A Novel Algorithm Unit Commitment Problem by a Fuzzy tuned Particle Swarm Optimization C. Kumar Research Scholar, Anna University of Technology Coimbatore, Tamilnadu, India E-mail:
[email protected] T. Alwarsamy Liaison Officer, Directorate of Technical Education Chennai, Tamilnadu, India E-mail:
[email protected] Abstract One of the disadvantages of traditional genetic algorithms is premature convergence because the selection operator depends on the quality of the individual, with the result that the genetic information of the best individuals tends to dominate the characteristics of the population. Furthermore, when the representation of the chromosome is linear, the crossover is sensitive to the encoding or depends on the gene position. The ends of this type of chromosome have only a very low probability of changing by mutation. In this work a genetic algorithm is applied to the unit commitment problem using a deterministic selection operator, where all the individuals of the population are selected as parents according to an established strategy, and an annular crossover operator where the chromosome is in the shape of a ring. The results show that, with the application of the proposed operators to the unit commitment problem.
1. Introduction In the commercial operation of an electricity market, the correct planning of generator units is of fundamental importance [1]. The economic savings, together with efficiency in the use of energy resources, mean that new proposals to solve the unit commitment problem (UCP) continue to be sought. The UCP has been solved using deterministic methods, such as Priority List (PL) Dynamic Programming (DP) [2]. These methods are characterised principally by their speed and their capacity to handle large scale problems when the objective function is linear and when some constraints are not considered [3]. Otherwise, with PL [4,5] the quality of the final solution is not guaranteed; DP [6-9] suffers the problem of dimensionality; with LR a feasible solution is not guaranteed; and with MILP it is difficult achieve a balance between the efficiency and the accuracy of the model [10-12]. In the face of these disadvantages the search for new methods has focused on metaheuristic methods [13-15] such as genetic algorithms (GA) using these metaheuristic methods it is possible to find an optimal solution to complex problems which is their main advantage over deterministic methods. Due to their iterative nature however, metaheuristic methods require a large amount of computer time to find a solution near to the global optimum, especially in large-scale problems. Today many proposals are based on hybrid techniques which exploit the advantages of both deterministic and metaheuristic techniques, making them attractive alternatives for solving the UCP.
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GA is a global optimization method which works well and efficiently on objective functions which are complex in terms of the nonlinearities and constraints imposed. One of the disadvantages of GA is premature convergence, because when the selection is based on the quality of the individual, the genetic information of the best individuals tends to dominate the genetic characteristics of the population. Another disadvantage results from the representation of chromosomes in string form, with the result that the genetic information at the ends of the chromosome tend to remain unaltered during crossover. In this chromosome, modification of the ends can only be achieved by the mutation operator. PSO proposed by Kennedy and Eberhart in 1995 has become a candidate for optimization applications due to its flexibility and efficiency. PSO is used for solving several power system related problems [16-23]. In this paper, a PSO (IBPSO) method has been proposed to solve UCP, which is integrated an fuzzy logic. The fuzzy tunes PSO (FTPSO) is enhanced by priority list based on the unit characteristics and heuristic search strategies to repair the spinning reserve and minimum up/down time constraints. In the proposed FTPSO method is used to solve the unit-scheduling problem in support of the Nelder–Mead method [24,25]. In solving UCP, the PSO and fuzzy logic are run in parallel, adjusting their solutions in search of a better solution. Finally, the proposed FTPSO method is tested on the UCP systems with the number of units in the range of 10 to 20. Simulation results demonstrate the feasibility and effectiveness of the FTPSO method in terms of solution quality and computation time compared with those of other optimization methods reported in the literature.
2. Unit commitment formulation The objective function and constraints associated with the unit commitment problem are the following. 2.1. Objective function The mathematical model used as the objective function to obtain the unit commitment of thermal units is: H
N
OF = ∑∑ ( FC nh + SU nh + SD n ).
(1) Where OF represents the total production cost for horizon planning, H is the total number of hours and N is the total number of units This objective function includes the fuel costs of unit n in hour h as a function of the power generated, normally represented by a quadratic equation as: h =1 n =1
FC nh ( Pnh ) = a n + bn Pnh + C n ( Pnh ) 2 .
(2) The start-up cost is dependent on the number of hours during which the unit has been off (TOff n ) . Using the two-step function, the start-up cost function is given by: HS n , ifTOff n ≤ TCold ,n, SU nh = CS n , otherwise.
(3)
Where HS n is the hot start cost, CS n is the cold start cost TCold ,n = Tdn n + CSH n
(4) Where TCold , n is the number of hours that it takes for the boiler of unit n to cool down, Tdn n is minimum downtime of unit n and CSH n is the cold start hours. The shut-down cost values SDn are generally considered to be constant. 2.2. Constraints The optimization of the objective function is subject to a number of system and unit constraints as follows.
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System power balance. The total power generated by all on units must supply the load demand in the hour h. N
∑P
n
=D h .
(5)
n =1
•
Demand and spinning reserve. The maximum power generated by all on units must at least meet the demand plus the spinning reserve in hour h. N
∑ PMax
n
≥ Dh + Rh.
(6)
n =1
•
Minimum load conditions. The minimum power generated by all on units must be less than or equal to the demand in hour
h. N
∑ PMin
n
≤ Dh.
(7)
n =1
•
Minimum up and down times. The total number of hours for which unit n has been running (TOn n ) must be greater than or
equal to the minimum unit uptime (Tup n ). TOn n ≥ Tup n , n ∈ N .
(8) Similarly, the total number of hours for which unit n has been down (TOff n ) must be greater
than or equal to the minimum unit downtime (Tdn n ). TOff n ≥ Tdnn , n ∈ N
•
Generator technical limits Each unit has a generation range which is represented as: PMinn ≤ Pn ≤ PMaxn , n ∈ N .
•
(9)
(10)
Unit initial status. The initial status at the start of the scheduling period must be taken into account.
3. The proposed method 3.1. Particle Swarm Optimization PSO is a stochastic optimization algorithm; the main idea of the PSO is the mathematical modeling and simulation of the food searching activities of a flock of birds. In the multidimensional space, each particle is moved toward the optimal point by changing its position according to a velocity. The velocity of a particle is calculated by three components; inertia, cognitive, and social. The inertial component simulates the inertial performance of the bird to fly in the previous direction. The cognitive component models the memory of the bird about its previous best position. The social component models the memory of the bird about the best position among the particles. The particles move around the multidimensional search space until they find the optimal solution. Based on the above discussion, the mathematical model for PSO is as follows. (11) Vi (t +1) = ωVi (t ) + c1rand1 (⋅).(Pbesti − X i(t ) ) + c2 rand2 (⋅).(Gbest − X i(t ) ) X i(t +1) = X i( t ) + Vi (t +1) i= 1, 2, 3 ….Nswarm Where, i is the index of each particle, t is the current iteration number, rand1 (.) and rand2 (.) are random numbers between 0 and 1. Pbesti is the best previous experience of the ith particle that is recorded Gbest is the best particle among the entire population. Nswarm is the number of the swarms. Constants c1 and c2 are the weighting factors of the stochastic acceleration terms, which pull each particle towards the Pbesti and Gbest ω is the inertia weight. As indicated in (6), there are three tuning
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parameters: ω , c1 and c2 that each of them has a great impact on the algorithm performance. The inertia weight ω controls the exploration properties of the algorithm. The learning factors c1 and c2 determine the impact of the personal best Pbesti and the global best Gbest respectively. If c1 > c2 the particle has the tendency to converge to the best position found by itself Pbesti rather than the best position found by the population Gbest and vice versa. Most implementations use a setting with c1 = c2 = 2 [27–31]. To implement the PSO algorithm to solve the UC problem, the following steps should be taken: Step 1: The initial population and initial velocity for each particle should be generated randomly. Step 2: The objective function is to be evaluated for each individual. Step 3: The individual that has the minimum objective function should be selected as the global position. Step 4: The ith individual is selected Step 5: The best local position Pbest is selected for the ith individual Step 6: The modified velocity for the ith individual needs to be calculated based on the local and global positions and Eq. (6). Step 7: The modified position for the ith individual should be calculated based on Eq. (6) and then checked with its limit. Step 8: If all individuals are selected, go to the next step, otherwise i = i + 1 and go to step 4. Step 9: If the current iteration number reaches the predetermined maximum iteration number, the search procedure is stopped, otherwise go to step 2. The last Gbest is the solution of the problem. 3.2. FTPSO From experience, it is known that [29–31]: (i) When the best fitness is found at the end of the run, low inertia weight and high learning factors are often preferred; (ii) When the best fitness is stayed at one value for a long time, the number of generations for unchanged best fitness is large. The inertia weight should be increased and learning factors should be decreased. According to this knowledge, a fuzzy system is utilized to tune the inertia weight and learning factors with the best fitness (BF) and the number of generations for the best unchanged fitness (NU) as the input variables, and the inertia weight (ω) and learning factors (c1 and c2) as the output variables. The BF value determines the performance of the best candidate solution found so far. The optimization problems have different ranges of the BF values. To use a FTPSO, which is applicable to a various range of problems, the ranges of the BF and NU values are normalized into [0, 1.0]. The BF values can be normalized using the following formula: BF − BFmin (12) NBF = BFmax − BFmin
Where BFmax and BFmin are the maximum and minimum values of BF value.NU values are normalized in a similar way. Other converting methods are possible as well. The bound values for x, c1, and c2 are: 0.2 ≤ ω ≤ 1.2, 1 ≤ c1 and c2 ≤ 2. The Mamdani-type fuzzy rule is used to formulate the conditional statements that comprise fuzzy logic. For example Ri: IF ( NBF is PB ) and ( NU is PM ), THEN (ω is PB), (c1 is PM) and (c2 is PM) The fuzzy rules are used to adjust the inertia weight (ω) and learning factors (c1 and c2), respectively. Each rule represents a mapping from the input space to the output space. To obtain a deterministic control action, a defuzzification strategy is required. In this paper, the centroid method has been used. To apply the FTPSO algorithm to solve the ED problem, the following steps should be taken:
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Step 1: The initial population and initial velocity for each particle should be generated randomly. Step 2: The objective function is to be evaluated for each individual. Step 3: The individual that has the minimum objective function should be selected as the global position. Step 4: The ith individual is selected. Step 5: The best local position ( Pbest ) is selected for the ith individual. Step 6: Update the FTPSO parameters. Step 7: Calculate the next position for each individual based on the FTPSO parameters and Eq. (6) and then checked with its limit. Step 8: If all individuals are selected, go to the next step, otherwise i = i + 1 and go to step 4. Step 9: If the current iteration number reaches the predetermined maximum iteration number, the search procedure is stopped, otherwise go to step 2. The last Gbest is the solution of the problem.
4. Nelder-Mead method The Nelder-Mead method is a generally used non-linear optimization algorithm. It is a numerical method for minimizing an objective function in a multidimensional space [33–35]. The operations of the method are to rescale the simplex based on the local behavior of the function by using four basic procedures: reflection, expansion, contraction, and shrinkage [33–35]. Through these procedures, the simplex can successfully improve itself and get closer to the optimum solution. The original NM simplex procedure is outlined by the following steps: Step 1 Initialization Generate N + 1 vertex points randomly to form an initial N dimensional simplex. Evaluate the functional value at each vertex point of the simplex. N + 1 vertex points have been sorted ascending based on the objective function values as below: (N+1)*(n+1) X low . . X highs X high
J low . J highs J high
(13)
.
Where Xlow, Xhigh, Xhighs are the vertices with the lowest, the highest and, the second highest function values, respectively. Jlow, Jhigh and Jhighs represent the corresponding observed function values, respectively.N is the number of state variables. Step 2 Reflection Find Xcent the center of the simplex without Xhigh in the minimization case. Generate a new vertex Xreft by reflecting the worst point according to the following equation N +1 1 (14) X cent = ∑Xj N
j =1, X j ≠ X high
X ref 1 = (1 + α ) * X cent − α * X high
(15)
Where α is the reflection coefficient ( α >0) Nelder and Mead suggested that α = 1. If Jlow < Jhigh < Jhighs accept the reflection by replacing Xhigh with Xreft and step 2 is repeated again for a new iteration.If Jreft < Jlow go to step 3.If Jhigh >Jreft > Jhighs replace Xhigh with Xreft and go to step 4.If Jhigh < Jreft go to step 4 without the replacement of Xhigh by Xreft
Step 3 Expansion
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Should reflection produce a function value smaller than Jlow (i.e. Jreft < Jlow ) the reflection is expanded in order to extend the search space in the same direction and the expansion point is calculated by the following equation. X exp = γ * X ref 1 + (1 − γ ) * X cent (16) Where γ is the expansion coefficient ( γ >1). Nelder and Mead suggested γ =2. If Jexp < Jlow the expansion is accepted by replacing Xhigh with Xexp Otherwise, Xexp replaces Xhigh The algorithm continues with a new iteration in step 2.
Step 4 Contraction The contraction vertex is calculated by the following equation: X cont = γ * X high + (1 − β ) * X cent (17) Where β is the contraction coefficient (0 < β < 1).Nelder and Mead suggested β =0.5 If Jcont < Jlow the contraction is accepted by replacing Xhigh with Xcont and then a new iteration begins with step 2. If Jcont > Jhigh then go to step 5. Step 5 Shrinkage In this step, shrink the entire simplex except Xlow by X i = γ * X low + (1 − δ ) * X low 0〈δ 〈1 (18) Nelder and Mead suggested δ = 0.5 Exit the algorithm if the stopping criteria are satisfied; otherwise return to step 2.
5. Numerical examples All simulations have been run on MATLAB environment with Pentium-IV, 2.66GHz computer with 512 MB RAM. Base 10-unit characteristics are taken from Ref. [28] and are given in Table 1. The spinning reserve requirement is considered to be 10% of the load 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 20- unit 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.
Table 1: Pmax(MW) Pmin(MW) a($/h) b($/MWh) c($/MW2h) min up(h) min down(h) hot start cost($) cold start cost($) cold start hours(h) initial status(h) Pmax(MW) Pmin(MW) a($/h) b($/MWh) c($/MW2h)
unit data for the 10-unit system Unit1 455 150 1000 16.19 0.00048 8 8 4500 9000 5 8 Unit6 80 20 370 22.26 .00712
Unit2 455 150 970 17.26 0.00031 8 8 5000 10000 5 8 Unit7 85 25 480 27.74 0.00079
Unit3 130 20 700 16.60 0.002 5 5 550 1100 4 -5 Unit8 55 10 660 25.92 0.00413
Unit4 130 20 680 16.50 0.00211 5 5 560 1120 4 -5 Unit9 55 10 665 27.27 0.00222
Unit5 162 25 450 19.70 0.00398 6 6 900 1800 4 -6 Unit10 55 10 665 27.29 0.00173
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Table 1:
unit data for the 10-unit system - continued
min up(h) min down(h) hot start cost($) cold start cost($) cold start hours(h) initial status(h)
Table 2:
3 3 170 340 2 -3
3 3 260 520 0 -3
1 1 30 60 0 -1
1 1 30 60 0 -1
1 1 30 60 0 -1
Unit Test results of the proposed FTPSO after 10 runs No of Units Best 563,938 1,123,297
10 20
Worst 565,698 1,128,072
Total Cost($) Average 564,831 1,125,516
Variation (%) 0.34 0.42
Ten test runs were performed on 10- and 20-unit systems and the results are presented in Table 2. Population size of 20 is taken for evolving PSO parameters. Generally, it is not easy to find optimal set of parameters for PSO. This is to be done heuristically through trial and error procedure. The number of particles mis-included for evolving because it is unclear how many ants are necessary to find a very good solution in an efficient way for a problem of given size. The exploitation probability is included or evolving because its optimal value is also unclear. Table 3:
Comparison of Total cost
No of units 10 20
LR 565,825 1,130,660
GA 565,825 1,126,243
Total cost($) GAUC ELR 563,977 563,977 1,125,516 1,123,297
EP 564,551 1,125,494
ICGA 566,404 1,127,244
HPSO 563,942
FTPSO 557,952 1,127,241
Performance of the 10-unit system was considered with the best combination of PSO parameters (i.e. with fixing of PSO parameters without evolving them). The total cost obtained was $ 564,551. Table 3 shows comparison of the proposed method with other methods. Obviously FTPSO is a robust method in obtaining improved solution. Output powers of the units for FTPSO solution for 10 and 20-unit cases are given in Tables 4 and 5, respectively. It is to be observed that ramp rate limit constraints are not taken into account in this paper since we would like to compare the solutions with previous works. The fitness and total cost characteristics for 10- and 20-unit systems. FTPSO algorithm was tested with the 10-unit system given. The constraints for this system are taken same as that reference including ramp rate constraints. Table 4:
Unit output power for the 10-unit case
Hours (1-24)
units
1 2 3 4 5 6 7 8 9 10 11 12
1 455 455 455 455 455 455 455 455 455 455 455 455
2 245 295 370 455 390 360 410 455 455 455 455 455
3 0 0 0 0 0 130 130 130 130 130 130 130
4 0 0 0 0 130 130 130 130 130 130 130 130
5 0 0 25 40 25 25 25 30 85 162 162 162
6 0 0 0 0 0 0 0 0 20 33 73 80
7 0 0 0 0 0 0 0 0 25 25 25 25
8 0 0 0 0 0 0 0 0 0 10 10 43
9 0 0 0 0 0 0 0 0 0 0 10 10
10 0 0 0 0 0 0 0 0 0 0 0 10
A Novel Algorithm Unit Commitment Problem by a Fuzzy tuned Particle Swarm Optimization Table 4: 13 14 15 16 17 18 19 20 21 22 23 24
164
Unit output power for the 10-unit case - continued 455 455 455 455 455 455 455 455 455 455 455 455
455 455 455 310 260 360 455 455 455 455 425 345
130 130 130 130 130 130 130 130 130 0 0 0
130 130 130 130 130 130 130 130 130 0 0 0
162 85 30 25 25 25 30 162 85 145 0 0
33 20 0 0 0 0 0 33 20 20 20 0
25 25 0 0 0 0 0 25 25 25 0 0
10 0 0 0 0 0 0 10 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
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
The population size is taken as 20. Table 4 gives optimal solution of the proposed algorithm and comparison of the cost with the methods given in Ref. [26] is given in Table 7. The above results on the two test systems clearly indicate that the performance of FTPSO 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 FTPSO 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 FTPSO, this problem is minimized because it selects the optimal combination of the feasible states of these similar units considering spinning reserve and up time/down time constraints. Table 5:
Unit output power for the 20-unit case
Units 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Units 1 2 3 4 5 6 7 8 9
1 455 455 245 245 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 455 455 295 295 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 455 455 382.5 382.5 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0
4 455 455 455 455 0 0 0 0 40 40 0 0 0 0 0 0 0 0 0 0
5 455 455 455 455 0 0 0 130 25 25 0 0 0 0 0 0 0 0 0 0
13 455 455 455 455 130 130 130 130 162
14 455 455 455 455 130 130 130 130 97.5
15 455 455 455 455 130 130 130 130 30
16 455 455 310 310 130 130 130 130 25
17 455 455 260 260 130 130 130 130 25
Hours(1-24) 6 7 455 455 455 455 455 425 455 425 130 130 0 0 130 130 130 130 25 45 25 45 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Hours(1-24) 18 19 455 455 455 455 360 455 360 455 130 130 130 130 130 130 130 130 25 30
8 455 455 455 455 130 130 130 130 30 30 0 0 0 0 0 0 0 0 0 0
9 455 455 455 455 130 130 130 130 97.5 97.5 20 20 25 0 0 0 0 0 0 0
10 455 455 455 455 130 130 130 130 162 162 33 33 25 25 10 10 10 10 0 0
11 455 455 455 455 130 130 130 130 162 162 73 73 25 25 10 10 10 10 0 0
12 455 455 455 455 130 130 130 130 162 162 80 80 25 25 43 43 10 10 10 10
20 455 455 455 455 130 130 130 130 162
21 455 455 455 455 130 130 130 130 105
22 455 455 455 455 0 0 0 130 105
23 455 455 432.5 432.5 0 0 0 0 0
24 455 455 345 345 0 0 0 0 0
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Table 5:
Unit output power for the 20-unit case - continued 162 33 33 25 25 10 10 0 0 0 0
10 11 12 13 14 15 16 17 18 19 20
97.5 20 20 25 0 0 0 0 0 0 0
30 0 0 0 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0 0 0 0
30 0 0 0 0 0 0 0 0 0 0
162 43 43 0 0 10 10 10 10 10 0
105 20 20 0 0 10 0 0 0 0 0
105 20 20 0 0 0 0 0 0 0 0
25 0 0 0 0 0 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
Table 6:
FTPSO solution for the 10-unit system Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Table 7:
For 10-unit system Unit status 1111111101 1111111101 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111011111 1111011111 1101011111 1101011111 1101011111
Comparison of total fuel cost for the 10-unit system
No. of units 10
Total cost ($) DP [26] 83652.40
BB[26] 83475.25
ACS[26] 83491.42
FTPSO 83347.72
6. Conclusion This paper presents a new approach the FTPSO, for the UC as well as scheduling problem. The proposed method is successfully applied to well known test systems, 10-unit-based system and its multiples. The significant results are compared with the other methods from both total operating costs and computational time aspects. Simulation results confirm that the proposed algorithm may achieve better results. In the 10-unit-based system the proposed method gives the best results for both total costs and execution time among different methods. Comparing FTPSO with the other heuristic method, it leads to the lower costs for the 10- and 20-unit systems, while from computational time point of view FTPSO seems much faster than ACO and BP. Results for the 10 and 20-unit test system show that FTPSO is a cost-effectiveness technique that may also improve the reliability of power systems. Also
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results show the usefulness of the proposed method which is capable of solving both small-scale and large-scale power systems UC as well as scheduling problems.
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