International Journal of Printing, Packaging & Allied Sciences, Vol. 5, No. 1, February 2017
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An Efficient ITA Algorithm to Solve Economic Load Dispatch with Valve Point Loading Effects G. Sivagnanam, K.S. Chandragupta Mauryan, R. Maheswar and R. Subramanian Abstract--- The economic load dispatch (ELD) is the online process for allocating the generation among the available generating unit to fulfil the load demand in such a way to minimize the total generation cost and to satisfy necessary constraints. This paper helps in solving ELD problem with an Improved TANAN’s Algorithm (ITA). The main objective of this paper is to solve ELD valve point loading effects. ITA is a simple numerical search technique based on an improved parabolic TANAN function. Standard IEEE test systems have been taken for the validation of the proposed method and is compared various popular optimization methods. Keywords--- Economic Load Dispatch, Valve Point-effect, Improved TANAN Function, Numerical Method.
I. INTRODUCTION Most of power system optimization problems including economic load dispatch (ELD) have complex and nonlinear characteristics with heavy equality and inequality constraints. The practical ELD problems with valvepoint and multi-fuel effects are represented as a non-smooth optimization problem with equality and inequality constraints, and this makes the problem of finding the global optimum difficult. For simplicity, the cost function for each unit in the ELD problems has been approximately represented by a single quadratic function and is solved using mathematical programming techniques. Generally, these mathematical methods require the derivative information of the cost function. This research work proposes a novel ITA algorithm which solves the variety of ELD problems. The power demand of the system and the generator power generation limit is taken as taken as equality and inequality constraints; the individual objective function is cumulated to multi objective function. The ITA algorithm solves the ELD problem by considering both the equality and inequality constraints. A constraint handling methodology is proposed in ITA by reducing the complexity of the constraints. The computational time to reach global optimum solution in ELD problem by ITA is addressed herewith.
II. ECONOMIC LOAD DISPATCH WITH VALVE POINT LOADING The objective function of the ELD problem is given in equation (1)
n MinimizeFt = ∑ Fi ( Pi ) i =1
(1)
Where Ft is total fuel cost in $/h and Fi (Pi )is the fuel cost equation of the ‘i’th plant expressed as follows.
G. Sivagnanam, AP, EEE, Sri Krishna College of Technology, Coimbatore. E-mail:
[email protected] K.S. Chandragupta Mauryan, Professor, EEE, Sri Krishna College of Technology. E-mail:
[email protected] R. Maheswar, Professor, ECE, Sri Krishna College of Technology. E-mail:
[email protected] R. Subramanian, Professor, EEE, Sri Krishna College of Technology. E-mila:
[email protected]
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International Journal of Printing, Packaging & Allied Sciences, Vol. 5, No. 1, February 2017
n Fi ( P i ) = ∑ a i P i2 + b i P i + c i i =1
122
(2)
whereai, bi and ci are the fuel cost coefficients of ith Generator in $/MW2 h, $/MW, and $/h respectively. The total fuel cost to be minimized is subject to the following constraints.
n ∑ Pi =Pd + Pl i =1
(3)
where Pi is the output power of ith Generator in MW, Pd and Pl are the system power demand and power loss in MW respectively. The system power loss is given by the relation n
PL =
n
∑∑
n
Pi Bij P j +
i =1 j =1
∑ B0i Pi + B00
(4)
i =1
where B and Bo are the loss coefficient matrices and Boo is the loss coefficient constant. The inequality constraint is given by
Pimin ≤ P i ≤ Pimax
(5)
where Piis the power output of ith Generator in MW, Pimin and Pimax are the minimum and maximum generation limit of ith Generator in MW respectively. The valve-point loading effect has been modelled as a recurring rectified sinusoidal function, such as the one shown in Fig. 1and equation (6) represents fuel cost including valve point effects.
Figure 1: Fuel Cost Characteristics with Valve Point effects where Pi is the power output of ith Generator in MW, Pimin and Pimax are the minimum and maximum generation limit of ith Generator in MW respectively. The valve-point loading effect has been modelled as a recurring rectified sinusoidal function, such as the one shown in Fig.1 and equation (6) represents fuel cost including valve point effects. Fi (Pi ) = ai Pi2 + bi Pi + ci + ei sin( f i ( Pimin − Pi ))
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(6)
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III. IMPROVED TANAN’S ALGORITHM Fig.2 illustrates the proposed ITA system model, the objective function is subjected to equality and inequality constraints. The total fuel cost is given as the output to the system.
Figure 2: Proposed ITA System Model where a, b and c are the fuel cost coefficients, e and f are the valve point loading effect coefficients, PD is the power demand and Pimin and Pimax are the lower limit and upper limit of ith generator respectively. The numerical random search methodology employed in ITA, makes it more suitable for ELD problems. The main objective of the proposed work is to minimize the fuel cost. The major constraints are taken into account as given below. a)
Equality constraint for serving the power demand of the system
b) Inequality constraint to meet the generator limit and safe generator operation in the system c)
Valve point loading effect coefficients.
d) Emission constraint to make the proposed system eco-friendlier in nature. The ITA algorithm is the modification of NTA (Subramanian et al. 2013) algorithm and the TANAN function variable x is varied from 0 to 1 instead of 0 to 2 for achieving quick computational time. The pseudo code for the proposed ITA is given as follows.
IV. ITA ALGORITHM The necessary steps of ITA algorithm is stated as follows. •
Initialize the input parameters and TANAN function variable.
•
Assign improved TANAN function as Ti=ri-six -tix2
•
Assign ri,si and ti as Pimax
•
Calculate Ti and assign Pi = Ti
•
If Pi ≤ Pimin then fix Pi = Pimin and if Pi ≥ Pimax then fix Pi = Pimax N
•
•
Redefine Ti
= PD + PL − ∑ Ti j =1 j ≠i
Modify ‘x’ value and determine the minimum fuel cost and minimum emission
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International Journal of Printing, Packaging & Allied Sciences, Vol. 5, No. 1, February 2017
•
Repeat the procedure for other values of iupto N
•
Notify PD, PL, cost and emission
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V. RESULTS AND DISCUSSION The simulation for the proposed ITA algorithm is carried out in a computer of windows 2010 operating system with 4GB Ram memory and Intel(R) core i3 Processor with a processing speed of 2.00 GHz CPU. For the validation of the proposed ITA algorithm, the following three different IEEE standard test system cases have been taken for the simulation analysis in MATLAB software. The ELD problem with VPL effect is analyzed by (Subramanian 2013) by NTA method and introducing a parabolic function with a numerical search technique of 2000×N solutions for each system where N represents the number of generators. The NTA algorithm limits the search technique when the optimum solution is reached immediately, but the optimum solution lies within the entire search space. IEEE standard test systems of 3, 5 and 6 machines systems with the power demands of 850MW, 259 MW and 283.4 MW are simulated in MATLAB and are compared with various optimization methods. They are given in the following three test cases as follows. Case 1: IEEE 3 machine standard test system with a power demand of 850MW with and without loss. The simulation result for the system without loss is illustrated in Table 1 with various optimization algorithms and Table 2 illustrate the comparison of simulation results of IEEE 3 machine system with PD=850 MW with loss for TM [11],GA [1,2],EP [15],MPSO [12,16,17],ABC [6],NTA [19] and ITA algorithms. Case 2: IEEE 5 machine standard test system with a power demand of 259MW with valve point loading coefficients. The simulation result for this system is illustrated in Table 3and is compared with NTA[19] algorithm. Case 3: IEEE 6 machine standard test system with a power demand of 283.4MW and 1263 MW with valve point loading coefficients. The simulation result for this system is illustrated in Table 4 and Table 5 and is compared with PSO [8,10,12], GA [1,2], ABC [6,13], EP [15], ACO and NTA [19] algorithms. From the comparison tables 1 to 5, it is clear that the total fuel cost, power loss and computational time are very less when compared to other optimization methods mentioned in this paper. The graphical representation for the simulation results given in table 1 to table 5 are illustrated in Fig.3 to Fig.7 as given below. Table 1: Comparison for Total Fuel Cost for IEEE 3 Machine System with PD=850 MW Without Power Loss Coefficients S.No
Algorithms
P1(MW)
P2(MW)
P3(MW)
PD(MW)
FT ($/h)
1
TM
300.27
400
149.73
850
8234.07
2
GA
300
400
150
850
8237.6
3
EP
300.26
400
149.74
850
8234.07
4 5
MPSO ABC
300.27 300.26
400 400
149.73 149.74
850 850
8234.07 8234.07
6
NTA
300.30
399.55
150.15
850
8231.91
7
ITA
300.30
399.55
150.15
850
8231.91
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Fuel cost in $/h
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8239 8238 8237 8236 8235 8234 8233 8232 8231 8230 8229 TM
GA
EP
MPSO
ABC
NTA
ITA
Algorithms Figure 3: Comparison of Fuel Cost for IEEE 3Machine System with PD=850 MW Without Considering the Power Loss Coefficients Table 2: Comparison for Total Fuel Cost for ELD with Valve Point Loading Effects for IEEE 3 Machine System with PD=850 MW and EP, ACO, DEIANT, NTA and ITA Algorithms Generator P1(MW) P2(MW) P3(MW) Total Power (PG) Power Loss(PL) Total Loss(MW) TANAN function variable (x) Average simulation time (sec)
NTA 399.360 399.360 67.925 866.645 16.645 8417.115 1.301 0.20
ITA 402.527 324.403 134.176 861.106 11.106 8369.928 0.574 0.05
8430 8420 Fuel cost in $/h
8410 8400 8390 8380 8370 8360 8350 8340 NTA
ITA Algorithm
Figure 4: Comparison of Fuel Cost for IEEE 3 Machine System with PD=850 MW With Power Loss Coefficients
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Table 3: Comparison for total fuel cost for ELD with Valve Point Loading Effects for IEEE 5 Machine System with PD=259 MW with NTA and ITA algorithms
Fuel cost in $/h
Generator P1(MW) P2(MW) P3(MW) P4(MW) P5(MW) Total Power (PG) Power Loss(PL) Total Loss(MW) TANAN function variable (x) Average simulation time (sec)
NTA 217.87 20.00 15.00 10.00 10.00 272.87 13.87 881.244 0.0 0.24
ITA 199.648 22.200 16.650 11.100 11.100 278.608 19.608 805.958 0.100 0.020
900 880 860 840 820 800 780 760 NTA
ITA Algorithm
Figure 5: Comparison of Fuel Cost for IEEE 5 Machine System with PD=259 MW With Power Loss Coefficients Table 4: Comparison for Total Fuel Cost for ELD with Valve Point Loading Effects for IEEE 6 Machine System ith PD=283.4 MW with NTA and ITA Algorithms Unit P1(MW) P2(MW) P3(MW) P4(MW) P5(MW) P6(MW) Loss, MW Total Cost, $/hr
PSO 198.00 52.00 15.00 10.00 10.00 12.00 13.6 816.11
GA 186.43 61.36 15.00 10.33 10.00 12.00 11.72 829.14
ABC 197.54 50.56 15.00 10.00 10.00 12.00 11.70 815.33
NTA 205.430 24.330 18.247 12.165 12.165 14.598 3.534 791.193
GA
ABC
NTA
ITA 189.663 28.122 21.092 14.061 14.061 16.873 9.234 778.381
Fuelcost in $/h
840 820 800 780 760 740 PSO
ITA
Algorithms
Figure 6: Comparison of Fuel Cost for IEEE 6 Machine System with PD=283.4 MW With Power Loss Coefficients
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Table 5: Comparison for Total Fuel Cost for ELD with Valve Point Loading Effects for IEEE 6 Machine System with EP, ACO, DEIANT, NTA and ITA Algorithms for the PD=1263 MW NTA Without With Valve-Point ValvePoint 468.424 465.991 166.278 166.795 249.417 250.193 124.709 125.096 166.278 166.795 99.767 100.077 15441.009 15468.088 11.873 11.947 0.04 0.04
EP Without ValvePoint 448.06 173.26 263.63 138.85 166.26 87.026 15463.17 12.83 13
With ValvePoint 449.63 173.87 264.55 139.34 166.85 87.33 15471.33 18.56 17
ACO Without With ValveValvePoint Point 447.01 448.57 172.70 173.30 261.56 262.47 138.37 138.85 168.97 169.56 87.11 87.42 15446.28 15454.68 12.72 17.18 11 15
With ValvePoint
Without ValvePoint
With ValvePoint
Without ValvePoint
With ValvePoint
Without ValvePoint
15480 15470 15460 15450 15440 15430 15420 15410 With ValvePoint
P1 P2 P3 P4 P5 P6 Ctot ($/hr) Ploss (MW) Time (s)
Fuel cost in $/h
Output Power (MW)
ITA Without With Valve-Point ValvePoint 450.009 450.009 153.629 153.629 245.806 245.806 150.000 150.000 153.629 153.629 120.000 120.000 15433.677 15450.807 10.073 10.073 0.05 0.06
Without ValvePoint
Technique
Algorithms Figure 7: Comparison of Fuel Cost for IEEE 6 Machine System with PD=1263 MW With and Without Valve Point Loading Effects
VI. CONCLUSION The proposed ITA to solve ELD problem with valve point effect has been presented in this paper. It is clear that the ITA is a simple numerical search technique for solving ELD problems. From the simulations, it can be seen that ITA gave the best result with very less computational time. In future, the proposed ITA can be used to solve ELD considering ramp rate limits and prohibited operating zones and also for finding the optimal valve points by developing standard search techniques.
REFERENCES [1] [2] [3] [4]
D.C. Walters and G.B. Sheble, “Genetic Algorithm Solution of Economic Dispatch with Valve Point Loading”, IEEE Transactions on Power Systems, Vol. 8, No.3, Pp.1325–1332, 1993. P. H. Chen and H.C. Chang, “Large-Scale Economic Dispatch by Genetic Algorithm”, IEEE Transactions on Power Systems, Vol. 10, No.4, Pp. 1919–1926, 1995. D. Simon, “Biogeography-based optimization”, IEEE Trans. Evol. Comput., Vol. 12, No. 6, Pp.702–713, 2008. G.B. Sheble and K. Brittig, “Refined genetic algorithm- economic dispatch example”, IEEE Trans. Power Systems, Vol. 10, Pp.117-124, 1995.
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International Journal of Printing, Packaging & Allied Sciences, Vol. 5, No. 1, February 2017
[5] [6] [7] [8] [9]
[10] [11] [12] [13]
[14] [15] [16]
[17]
[18] [19]
128
B.K. Panigrahi and V.R. Pandi. “Bacterial foraging optimization: Nelder-Mead hybrid algorithm for economic load dispatch”, IET Gener. Transm,Distrib. Vol. 2, No. 4. Pp. 556-565, 2008. D. Karaboga and B. Basturk, “Artificial Bee Colony (ABC) Optimization Algorithm for Solving Constrained Optimization Problems”, Springer-Verlag, IFSA LNAI 4529, Pp. 789–798, 2007. R. Stornand and K. Price, “Differential Evolution—A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces”, International Computer Science Institute, 1995. I.A. Selvakumar and K. Thanushkodi, “A new particle swarm optimization solution to non convex economic load dispatch problems”, IEEE Transactions on Power Systems, Vol. 22, No. 1, 2007. M.D. Mohammad, M.I. Behnam, A. Najafi and A.Rabiee Erratum to “Continuous quick group search optimizer for solving non-convex economic dispatch problems”, Electr. Power Syst. Res., Vol. 93, Pp. 93–105, 2012. J.B. Park, K.S. Lee, J.R. Shin and K.Y. Lee, “A Particle Swarm Optimization for Economic Dispatch with Non smooth Cost Functions”, IEEE transactions on power systems, Vol. 20, No. 1, 2005. D. Liu and Y. Cai, “Taguchi Method for Solving the Economic Dispatch Problem with Non smooth Cost Functions”, IEEE transactions on power systems, Vol. 20, No. 4, 2005. G. Zwe-Lee, “Particle swarm optimization to solving the economic dispatch considering the generator constraints”, IEEE Transactions onPower Systems, Vol. 18, Pp. 1187-1195, 2003. S.K. Nayak, K.R. Krishnanand, B.K. Panigrahi and P.K. Rout, “Application of Artificial Bee Colony to economic load dispatch problem with ramp rate limit and prohibited operating zones”, IEEE word congress on Nature and Biologically inspired computing (NaBIC), Pp. 1237–1242, 2009. A. Bakirtzis, V. Petridis and S. Kazarlis, “Genetic Algorithm Solution to the Economic Dispatch Problem”, Proceedings. Inst. Elect. Eng. Generation, Transmission Distribution, Vol. 141, No. 4, Pp. 377–382, 1994. E. Zitzler, M. Laumanns and S. Bleuler, “A Tutorial on Evolutionary Multi-objective Optimization”, Swiss Federal Institute of Technology, Computer Engineering and Networks Laboratory, 2004. J. Sun, W. Fang, D. Wang and W. Xu, “Solving the economic dispatch problem with a modified quantumbehaved particle swarm optimization method”, Energy Conversion and Management, Vol. 50, Pp. 2967-2975, 2009. K.T. Chaturvedi, M. Pandit and L. Srivastava, “Self-Organizing Hierarchical Particle Swarm Optimization for Non-Convex Economic Dispatch”, IEEE Transactions on Power Systems, Vol. 23, Pp. 1079-1087, 2008. A. J. Wood and B. F. Wollenberg, “Power Generation Operation and Control”, 2nd Edition, Wiley, New York, 1996. R. Subramanian, K. Thanushkodi and P.N. Neelakantan, “Solution of Economic Load Dispatch problems with Valve-Point Effect using Novel TANAN’s Algorithm”, International conference on Current Trends In Engineering and Technology(ICCTET’13), Pp.141-145, 2013.
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