Price Penalty factors Based Approach for Emission constrained economic dispatch Problem Solution using Whale Optimization Algorithm Mahesh H. Pandya
Department of Electrical Engineering Govt. Engineering College Gandhinagar (Gujarat) India
[email protected]
Department of Electrical Engineering Lukhdhirji Engineering College Morbi-Gujarat, India
[email protected]
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Indrajit N. Trivedi
R.H.Bhesdadiya Department of Electrical Engineering Lukhdhirji Engineering College Morbi-Gujarat, India
[email protected]
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Narottam Jangir
Department of Electrical Engineering Lukhdhirji Engineering College Morbi-Gujarat, India
[email protected]
Arvind Kumar Department of Electrical Engineering S.S. Engineering College Bhavnagar- Gujarat, India
[email protected]
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Pradeep Jangir
Department of Electrical Engineering Lukhdhirji Engineering College Morbi-Gujarat, India
[email protected]
paper term used emission constrained economic dispatch (ECED) problem is similar to term combined economic emission dispatch (CEED) problem.
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Abstract— In The main ambition of utility is to provide continuous reliable supply to customers, satisfying power balance, transmission loss while generators are allowed to be operated within rated limits. Meanwhile, achieve this purpose emission value and fuel cost should be as less as possible. An allowable deviation in fuel cost and feasible tolerance in fuel cost has been called emission constrained economic dispatch (ECED) problem. A new nature– inspired Whale optimization algorithm (WOA) is based on concept of bubble-net hunting strategy is applied to solve ECED problem. ECED is a multi-criteria problem can transformed to single criteria using price penalty factor method. In this paper quadratic function together with emission value and fuel cost are considered as individual objective makes it multi-criteria problem. The effect of six penalty factors like “Min-Max”, “Max-Max”, “Min-Min”, “MaxMin”, “Average”, “Common” price penalty factors and emission value of various pollutants gases exhalation are included. The emission constrained economic dispatch (ECED) problem is analysed for an IEEE-30 Bus system with six operational generator units. Results prove capability of WOA in solving ECED problem with different penalty factors.
WOA algorithm [1] in exploration phase uses spiral path covers a broader area so it guarantees to obtain global solution and algorithm has a capability to avoid local stagnation or local optima.
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The multi-objective power system dispatch problem can be transformed into single objective by using two techniques:
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Price penalty factor technique and
Weighted sum method (WSM)
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In this paper price penalty factor based technique is used to analysis ECED problem.
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Every utilities desire that generation cost and emission value should be as least as possible, but both objectives are contradictory so cannot be achievable at a single time. In this
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I. INTRODUCTION
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Keywords—Whale Optimization Algorithm; Bubble-net hunting strategy; Emission constrained economic dispatch; Fuel cost; Emission value.
In past there is only objective to minimize cost while generation of power, but now a big concern about saving environment from pollution to rectify problem of global warming so some rules are imposed on private and government utilities to reduce emission of toxic gases exhalation with possible least fuel cost.
The ECED problem consists of either single objective or multi-objective is solved using various algorithms such as: Neural network, Fuzzy system and Lagrange’s algorithm (LA) [2], Emission Standards [3], Dispatch problem on different power system using Stochastic algorithm [4-5], Security
constrained economic scheduling of generation considering generator constraints [6], Penalty factor based approach [9, 10], ECED with valve point effect [11, 12], with WSM technique [13], various novel evolutionary computational algorithm [14], Based on AI technique [15] and on multiobjective with different algorithms [16,17]. II. WHALE OPTIMIZATION ALGORITHM
distance between i-th whale to the prey mean best solution so far. Note: We assume that there is 50-50% probability that whale either follow the shrinking encircling or logarithmic path during optimization. Mathematically we modelled as follows:
X *(t ) A.D X (t 1) bl D '.e .cos(2 l ) X *(t )
if if
p 0.5 p 0.5
(6)
Where: p expresses random number between [0, 1].
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In meta-heuristic algorithm, a newly purposed optimization algorithm called Whale optimization algorithm (WOA) [1], which inspired from the bubble-net hunting strategy. Algorithm describes the special hunting behavior of humpback whales, the whales follows the typical bubbles causes the creation of circular or ‘9-shaped path’ while encircling prey during hunting. Simply bubble-net feeding/hunting behavior could be understand such that humpback whale went down in water approximate 10-15 meter and then after start to produce bubbles in a spiral shape encircles prey and then follows the bubbles and moves upward the surface. Mathematic model for Whale Optimization algorithm (WOA) is given as follows:
Where: l is a random number [-1, 1], b is constant defines logarithmic shape, D ' X * (t ) X (t ) expresses the
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2.2.3
Search for prey
A Vector can be used for exploration to search for prey;
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vector A also takes the values greater than one or less than -1. Exploration follows two conditions (7) D C . X rand X X (t 1) X rand A.D
2.1 Encircling prey equation
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Humpback whale encircles the prey (small fishes) then updates its position towards the optimum solution over the course of increasing number of iteration from start to maximum number of iteration. (1) D C . X * (t ) X (t )
(8)
Finally follows these conditions: A 1 enforces exploration to WOA algorithm to find out
global optimum avoid local optima A 1 For updating the position of current search agent/best solution is selected.
III. MATHEMATICAL FORMATION OF EMISSION CONSTRAINED ECONOMIC DISPATCH (ECED) PROBLEM
X (t ) is position vector.
Mathematic equation for ECED problem [2-5] is given as follows:
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(2) X (t 1) X * (t ) A.D Where: A , D are coefficient vectors, t is current iteration, X * (t ) is position vector of optimum solution so far and
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Coefficient vectors A , D are calculated as follows: A 2a * r a
ECED=Minimum (Generation cost) + penalty factor [2] * minimum (Emission value) [3]
2.2 Bubble-net attacking method In order to mathematical equation for bubble-net behavior of humpback whales, two methods are modelled as: 2.2.1 Shrinking encircling mechanism: This technique is employed by decreasing linearly the value
ui =Cost coefficient of ith generator in [$/MW2h]
(9)
i 1
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vi =Cost coefficient of ith generator in [$/MWh]
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wi =Cost coefficient of ith generator in [$/h]
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Fc = Generation cost of the ith generator NG = Number of generators n
ET = Total Emission Value
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Where,
(10)
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i 1
A
ET xi Pi 2 yi Pi zi
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X (t 1) D '* e *cos(2 l ) X *(t ) bt
NG
Min( FC ) ui Pi 2 vi Pi wi
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of a from 2 to 0. Random value for vector in rang between [-1, 1]. 2.2.2 Spiral updating position: Mathematical spiral equation for position update between humpback whale and prey that was helix-shaped movement given as follows:
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(3) (4) Where: is a variable linearly decrease from 2 to 0 over the course of iteration and r is a random number [0, 1]. C 2*r
xi =Emission coefficient of ith generator in [kg/MW2h] yi = Emission coefficient of ith generator in [kg/MWh]
(5)
zi = Emission coefficient of ith generator in [kg/h] Price penalty factor hi is used to transform multi-objective
ECED problem into a single objective [11-15] problem:
FT ui Pi 2 vi Pi wi hi xi Pi 2 yi Pi zi
Where,
n
(11)
i 1
Where,
FT = Total ECED Cost PRICE PENALTY FACTORS (PPF)
Min-Max price penalty factor is formulated as:
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u P x P i
i
2
i min 2
i max
vi Pi min wi
yi Pi max 2 zi
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hi
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The PPF [7-10] for multi-objective ECED problem is formulated taking the ratio fuel cost to emission value of the corresponding generators as follows:
[$/kg]
2
i
i min
i
i min
vi Pi min wi
yi Pi min 2 zi
2
coefficients. Generator limits [2]
[$/kg]
Pi Min Pi Pi Max
(20)
VI. SIMULATION RESULTS In this paper data like bus number, generator operating limits and the fuel cost and emission coefficients [2] and loss coefficients [2] of standard IEEE-30 bus system [4]. Various power demands from 125-250 MW are considered with the interval of 25 MW. ECED problem is solved in MATLAB R2014b software. Table II, III, IV, V, VI, and VII shows solution of ECED problem with different power demand using “Min-Max”, “Max-Max”, “Min-Min”, “Max-Min”, “Average” and “Common” price penalty factors respectively.
(14)
Max-Min price penalty factor is formulated as:
u P x P i
i
2
i max 2
i min
vi Pi max wi
yi Pi min 2 zi
[$/kg]
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(16)
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(17)
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Average price penalty factor is formulated as: hi ( Min / Max ) hi ( Min / Min ) hi ( Max / Max ) hi ( Max / Min ) hAVGi 4 Common price penalty factor is formulated as: h hCOMi AVGi n Where: n is operational generating unit
(15)
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hi
u P x P
B0i and B00 is the transmission loss
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(13)
Min-Min price penalty factor is formulated as:
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hi
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(12)
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Max-Max price penalty factor is formulated as: ui Pi max 2 vi Pi max wi [$/kg] hi xi Pi max 2 yi Pi max 2 zi
respectively. Bij ,
The power output of a generating unit must lie within lower and upper power limits is given as:
hi = Price penalty factor (PPF) IV.
Pi and Pj is the active power of unit i th and j th
Fig. 1. Comparision of ECED/CEED fuel cost w.r.t. power demand
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V. VARIOUS CONSTRAINTS USED Power Balance constraint [6]: (18)
i 1
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n
PL PiBijPj BoiPi Boo i 1 j 1
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n
P
n
A
load demand and transmission line loss of the system respectively. Transmission line loss constraint can be given as, (Dhillon et al, 1994).
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PG , PDemand and PLoss is the total generated power,
Where,
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n
PG Pi PDemand PLoss
(19)
i 1
Fig. 2. Comparision of fuel cost w.r.t. power demand
The optimal overall cost for emission constrained economic dispatch (ECED) or CEED fuel cost and fuel cost is less when using “Min-Max” Price penalty factor compared to other penalty factors. The “Max-Max” price penalty factor is good to yield a minimum emission compared to other penalty factors. The “Max-Min” price penalty factor is good to get lowest transmission loss compared to other penalty factors “Min-Max”, “Min-Min”, “Max-Max”.
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Table I. concludes that for only ECED/CEED fuel cost prospective plant operates with “Min-Max” price penalty factor. If generators are scheduled to operate according to “Common” price penalty factor with increase almost 17% in CEED fuel cost, we can reduce emission upto 10% compare to “Min-Max” price penalty factor. But now a day’s thermal power plant operates with “Min-Max” price penalty factor having least CEED fuel cost without caring emission that causes environment pollution results in premature deaths of human being living near thermal power plant. In this research work future is multi-objective ECED formulation [16-18] and analysis generator scheduling so that optimal solution gained and all objectives fulfilled.
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Fig. 3. Comparision of emission value w.r.t. power demand
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References
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[1] Seyedali Mirjalili, Andrew Lewis, “The Whale Optimization Algorithm”, Elsevier Science Direct Advances in Engineering Software 95 (2016) 51–67. http://www.alimirjalili.com/WOA.html.
Fig. 4. Comparision of transmission loss w.r.t. power demand
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Table I compares the results obtained with all six penalty factors assuming PD=250 MW is reference. For each penalty factor with all PD search agents and maximum iterations are 30 and 2000 respectively. PD=250 MW is assumed reference because on higher PD the results shows significant variation or clearance in obtained results. Further “Min-Max” price penalty factor is taken as base and results for other penalty factor is given in percentages w.r.t. “Min-Max” penalty factor in Table I.
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[2] Senthil Krishnamurthy and Raynitchka Tzoneva, "Investigation of the Methods for Single area and Multi area Optimization of a Power System Dispatch Problem", International review of Electrical Engineering (IREE), Praise worthy prize, Feb 2012. [3] Sarath K. Guttikunda, Puja Jawahar,"Atmospheric emissions and pollution from the coal-fired thermal power plants in India", Elsevier, Atmospheric Environment 92 (2014) 449-460. [4] D.P. Kothari and J.S. Dhillon, "Power System Optimization, Text Book, Prentice - Hall of India Private Limited, New Delhi, 2nd Edition 2006. [5] J.S. Dhillon , S.C. Parti and D.P. Kothari, "Stochastic economic emission load dispatch", Electric Power Systems Research, Vo1.26, 1993, pp.179-186. [6] Zwe-Lee Gaing and Rung-Fang Chang, "SecurityConstrained economic scheduling of generation considering generator constraints", International Conference on Power System Technology, 2006, pp.I-6. [7] M.T. Tsai, and C.W. Yen “An Improved Particle Swarm Optimization for Economic Dispatch with Carbon Tax Considerations”, International Conference on Power System Technology, 2010, pp 1-6. [8] T. Thakur, K. Sem, S. Saini, and S. Sharma “A Particle Swarm Optimization Solution to NO2 and SO2 Emissions for Environmentally Constrained Economic Dispatch
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The ECED problem is solved using Whale optimization algorithm using six different penalty factors and their effect is analyzed on IEEE-30 bus 6 generating unit system. On the basis of results obtained some conclusion is made:
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Conclusion
[9]
[10]
[14]
[15]
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[11]
Problem”, IEEE/PES, Transmission & Distribution Conference and Exposition: Latin America, 2006. R. Gnanadass, Narayana Prasad Padhy, K. Manivannan,"Assessment of available transfer capability for practical power systems with combined economic emission dispatch", Elsevier science direct Electric Power Systems Research 69 (2004) 267–276. Hadi Hamedi, "Solving the combined economic load and emission dispatch problems using new heuristic algorithm", Elsevier Electrical Power and Energy Systems 46 (2013) 10–16. S. Hemamalini and S.P. Simon, “Maclaurin series-based Lagrangian method for economic dispatch with valve-point effect”, Generation, Transmission & Distribution, IET,Volume 3,Issue 9,September 2009, pp 859-871. Binod Shaw, V. Mukherjee, S.P. Ghoshal,"A novel opposition-based gravitational search algorithm for combined economic and emission dispatch problems of power systems", Elsevier Electrical Power and Energy Systems 35 (2012) 21–33. A. Chatterjee, S.P. Ghoshal, V. Mukherjee,"Solution of combined economic and emission dispatch problems of power systems by an opposition-based harmony search
[16]
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[17]
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[13]
[18]
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algorithm",Elsevier Electrical Power and Energy Systems 39 (2012) 9–20. P.Venkatesh, R.Ganadass and Narayana Prasad Padhy, "Comparsion and Application of evolutionary programming techniques to combined economic emission dispatch with line flow constraints", IEEE Transactions on Power Systems, vol.18, No.2, May 2003, pp.688-697. I. Jacob Raglend, Sowjanya Veeravalli, Kasanur Sailaja, B. Sudheera, D.P. Kothari,"Comparison of AI techniques to solve combined economic emission dispatch problem with line flow constraints", Elsevier Electrical Power and Energy Systems 32 (2010) 592–598. J .S. Dhillon , S.C. Parti and D P Kothari, “ Multi-objective optimal thermal power dispatch”, Electrical Power & Energy Systems, Volume 16, Number 6, 1994, pp 383-389. S. Dhanalakshmi, S. Kannan, K. Mahadevan, S. Baskar,"Application of modified NSGA-II algorithm to Combined Economic and Emission Dispatch problem", Elsevier Electrical Power and Energy Systems 33 (2011) 992– 1002. M. Basu, "Economic environmental dispatch using multiobjective differential evolution", Elsevier Applied Soft Computing-11(2011)2845–2853.
Criterion
Min-Max price penalty factor 100 %
Max-Max price penalty factor
Min-Min price penalty factor
Max-Min price penalty factor
Average penalty factor
Common penalty factor
100.38
116.33%
118.05%
117.82%
109.08%
100 %
157.05%
162.31%
413.37%
220.09%
116.85%
100 %
89.36%
101.42%
100.71%
100.91%
90.76%
100 %
74.41%
51.27%
46.75%
47.65%
58.26%
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Fuel cost Fc [$/hr] ECED fuel cost FT [$/hr] Emission value ET [kg/hr] Power loss PL [MW]
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TABLE I. COMPARISION OF SIMULATION RESULTS OBTAINED FOR “MIN-MAX”, “MAX-MAX”, “MIN-MIN”, “MAX-MIN”, “AVERAGE” AND “COMMON” PRICE PENALTY FACTOR FOR POWER DEMAND 250 MW.
TABLE II. SOLUTION OF ECED PROBLEM USING “ MIN-MAX ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND P1 [MW]
P2 [MW]
P3 [MW]
P4 [MW]
P5 [MW]
P6 [MW]
PL [MW]
FC [$/hr]
125
59.24418
20
15
10
10
12
1.24617
307.93807
150
85.18758
20
15
10
10
12
2.191507
373.23132
175
100.89183
30.32740
15
10
10
12
3.22341
442.25017
217.07993
200
119.67087
26.06898
18.87652
13.78662
13.66851
12.08733795
4.16655
522.81766
256.80624
627.22085
225
118.92056
57.53339
21.93812
10
10
12
5.38559
603.54686
302.32113
719.20861
250
143.52965
40.77984
26.99656
17.34251
11.142420
16.67902326
6.33603
678.70047
341.78078
810.29037
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PD [MW]
ET [Kg/hr]
FT [$/hr]
P
161.27002
A
185.01992
380.95321 453.72197
P
R
E
532.53666
TABLE III. SOLUTION OF ECED PROBLEM USING “ MAX-MAX ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND PD [MW]
P1 [MW]
P2 [MW]
P3 [MW]
P4 [MW]
P5 [MW]
P6 [MW]
PL [MW]
FC [$/hr]
ET [Kg/hr]
FT [$/hr]
125
56.86102
20.06372
15.047792
10.03186
10.03186
14.23964
1.17732
308.14541
161.03957
641.03807
150
72.75913
26.18850
15
10
15.91995
12
1.86978
377.98216
180.92521
744.68174
175
93.16211
31.34994
21.40836
10.00000
10.00000
12.00000
2.93164
445.62887
212.76567
864.714943
200
99.30213
36.50068
20.85520
19.25324
11.89337
15.67300
3.47954
526.94579
239.38682
995.04231
225
107.83022
43.289227
19.47893
14.51787
21.53742
22.69070
4.31053
611.78293
273.11198
1142.54779
250
116.88350
46.059563
33.09354
23.47180
23.18714
12.05904
4.71450
681.28151
305.43312
1272.58868
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TABLE IV. SOLUTION OF ECED PROBLEM USING “ MIN-MIN ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND
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125
P1 [MW] 50
P2 [MW] 20
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PD [MW]
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P3 [MW] 24.23410
P4 [MW] 10
P5 [MW] 10
P6 [MW] 12
PL [MW] 1.157822
FC [$/hr] 347.27731
ET [Kg/hr] 176.53295
FT [$/hr] 650.50410 784.01709
56.40699
20
17.84798
35
10
12
1.448447
411.49699
197.41393
175
50.47174
60.59487
15.25515
25.03881
13.08341
12.68332
2.127860
479.42649
236.06647
867.91785
200
54.22001
41.52906
44.06467
27.14251
10.87374
24.37622
2.263236
617.83020
265.25749
1044.82175
225
51.69781
79.99986
30.60762
11.83091
27.09253
3.240044
679.61300
325.98955
1185.53346
250
51.16725
66.59178
33.61245
29.99998
39.99997
3.248752
789.57070
346.64448
1315.22078
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150
27.01040
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31.87319
125
P1 [MW] 50
P2 [MW] 20
P3 [MW] 15
150
50.19205
20.07682
24.81668
175
50.03497
33.14022
22.20011
200
50.74571
28.03500
37.46397
225
51.50238
36.72258
250
50
59.68597
P4 [MW] 14.49196
P5 [MW] 14.64726
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TABLE V. SOLUTION OF ECED PROBLEM USING “ MAX-MIN ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND P6 [MW] 12
PL [MW] 1.10535
FC [$/hr] 362.50492
ET [Kg/hr] 175.58672
1843.93631 2008.36079
FT [$/hr]
19.92442
12.06751
1.19782
412.24160
190.99146
34.99981
16.46368
19.76566
1.60276
485.08990
219.89801
2230.79140
33.22445
27.27866
25.06728
1.79950
620.75557
262.16425
2610.159305
49.99942
34.99959
29.99965
23.94385
2.17961
728.15300
303.01587
3004.76540
40.13582
35
30
38.13364
2.96184
801.20611
344.21610
3349.47296
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24.02089
P3 [MW]
P4 [MW]
P5 [MW]
PL [MW]
FC [$/hr]
ET [Kg/hr]
FT [$/hr]
125
50
24.9889
15.18250
10
13.90827
12
1.080855
311.49243
163.40551
848.74527
150
50
32.86385
21.58919
24.89281
10
12
1.349023
393.00387
190.47636
995.28557
175
50.01372
30.60851
22.17764
19.94887
27.36502
26.48449
1.602481
496.78281
215.91303
1155.06425
200
55.59804
43.65718
22.73687
24.48946
29.17241
26.52964
2.189019
576.13809
245.97362
1348.16807
225
56.47744
47.69916
35.98409
33.79507
16.08599
37.61918
2.638009
680.52638
290.26304
1570.22562
250
50
60.46457
38.80995
35
28.74108
40
3.019302
799.69421
344.90276
1783.39658
P
P2 [MW]
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P1 [MW]
N
PD [MW]
E
P6 [MW]
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TABLE VI. SOLUTION OF ECED PROBLEM USING “ AVERAGE ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND
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TABLE VII. SOLUTION OF ECED PROBLEM USING “ COMMON ” PRICE PENALTY FACTOR WITH DIFFERENT POWER DEMAND PD [MW]
P1 [MW]
P2 [MW]
P3 [MW]
P4 [MW]
P5 [MW]
P6 [MW]
PL [MW]
FC [$/hr]
ET [Kg/hr]
125
50
29.15293
15
10
10
12
1.153414
308.15719
164.70723
398.65777
150
50.05627
44.01325
19.40510
11.80851
14.26846
12.00521
1.548407
386.22854
192.51063
489.02103
175
50
42.65671
23.10016
23.18883
19.90007
17.86280
1.709441
480.88097
216.87242
592.42020
200
60.21483
52.77214
25.68620
20.50336
24.94807
18.27944
2.402566
562.59383
247.00588
700.05337
225
67.34043
52.23848
31.58388
28.34019
27.76907
20.56920
2.847481
656.39000
278.25186
818.72990
250
80.31849
51.30956
29.08470
35
26.38334
31.59217
3.691293
740.35495
310.20077
946.86882
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FT [$/hr]
