Arab J Sci Eng (2013) 38:1415–1425 DOI 10.1007/s13369-013-0560-y
RESEARCH ARTICLE - ELECTRICAL ENGINEERING
Swarm Intelligence to the Solution of Profit-Based Unit Commitment Problem with Emission Limitations D. Sam Harison · T. Sreerengaraja
Received: 2 February 2011 / Accepted: 13 July 2011 / Published online: 15 February 2013 © King Fahd University of Petroleum and Minerals 2013
Abstract As the electrical industry restructures, many of the traditional algorithms for controlling generating units need modification or replacement. In the past, utilities had to produce power to satisfy their customers with objectives to minimize costs, and all demand/reserve was met. However, it is not necessary in restructured system. In the restructured environment, generation companies (GENCOS) schedule their generators with objective to maximize their own profit without regard for system social benefits. This leads to profit based unit commitment (PBUC) problem. One of the main contributions to the emission of greenhouse gases into the atmosphere, which is thought to be responsible on our environment, is through the use of fossil-fuelled power plants. As a consequence of growing environmental concern, governments are acting in the way to regulate greenhouse gas emission. A major step in this direction is the Kyoto Protocol, which is with the objective of “stabilization and reconstruction of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system”. However, the recent advent of emission allowance trading has renewed interest in the environmentally constrained UC problem. In the new emission-constrained competitive environment, a GENCO with thermoelectric facilities faces the optimal trade-off problem of how to make the present profit by the management of the energy available in fossil fuels for power generation without
excessive emission. Since maximizing profit and minimizing emission are conflicting objectives, a swarm intelligence approach is proposed in this paper to obtain compromised solutions. The binary particle swarm optimization is used to solve the PBUC problem and real-valued particle swarm optimization (RPSO) is used to solve the economic load dispatch which is a sub problem of PBUC. A six generating unit system and a eleven generating unit system have been taken, and the proposed algorithm is applied to solve it for the PBUC with emission limitations. From the comparison of results, the ability of the proposed algorithm is demonstrated in the aspects of solution quality and computational efficiency. Keywords Swarm intelligence · Profit based unit commitment · Emission limitations · Particle swarm optimization
D. S. Harison (B) Department of EEE, C.S.I Institute of Technology, Thovalai, Kanyakumari 629302, Tamil Nadu, India e-mail:
[email protected] T. Sreerengaraja Department of EEE, Anna University of Technology Tiruchirappalli, Tiruchirappalli 620 024, Tamil Nadu, India e-mail:
[email protected]
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1 Introduction In the design, construction and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to maximize the desired benefit or to minimize the effort or time required which ultimately optimizes the solution of the problem undertaken by taking into account the various constraints. Restructuring of the electricity supply industries is a very complex exercise based on national energy strategies and policies, macroeconomic developments and national conditions and application varies from country to country. It is important to point out that there is no single solution applicable to all countries and there is a broad range of diverse trends. Liberalization, deregulation (or reregulation) and privatization are all processes under the general label of market reform. Liberalization refers to the introduction of a less restrictive regulatory framework for companies within a power sector. This could imply deregulation, which is the modification of existing regulation. Ideally, then, a true liberalized energy market would work within a set regulatory framework, overseen by a regulator and with no external political influence upon the participants regarding plant size or fuel choice. Deregulation in general often involves unbundling, in the case of electric utility services represents disaggregating into the basic parts of generation, transmission and distribution. These operations are being responsibilities of generation companies (GENCOs), transmission companies (TRANSCOs) and distribution companies (DISCOs), with a central coordinator called independent system operator (ISO), to balance supply and demand in real time and maintain system reliability and security. In the short-term, typically considered to run from 24 h to 1 week, the solution of the unit commitment problem (UCP) is used to assist decisions regarding generating unit operations. In a regulated market, a power generating utility solves the UCP to obtain an optimal schedule of its units to have
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enough capacity to supply the electricity demanded by its customers. The optimal schedule is found by minimizing the production cost over the time interval while satisfying the demand and a set of operating constraints. The minimization of the production costs assures maximum profits because the power generating utility has no option but to reliably supply the prevailing load [1]. The price of electricity over this period is predetermined and unchanging; therefore, the decisions on the operation of the units have no effect on the firm’s revenues. As deregulation is being implemented in various countries across the globe, the traditional unit commitment problem continues to remain applicable for the commitment decisions made by the ISO. The ISO resembles very much the operation of a power generating utility under regulation. The ISO manages the transmission grid, controls the dispatch of generation, oversees the reliability of the system, and administers congestion protocols. The ISO is a non-profit organization. Its economic objective is to maximize social welfare, which is obtained by minimizing the costs of reliably supplying the aggregate load. Under deregulation, the UCP for an electric power producer will require a new formulation that includes the electricity market in the model. The main difficulty here is that the spot price of electricity is no longer predetermined but set by open competition. Thus far, the hourly spot prices of electricity have shown evidence of being highly volatile. The unit commitment decisions are now harder and the modeling of spot prices becomes very important in this new operating environment. The new formulation of UCP suitable for an electric power producer in a deregulated market can be defined as profit based unit commitment (PBUC) problem. The PBUC is a stochastic optimization problem in which the objective is to maximize expected profits and the decisions are required to meet operating constraints such as capacity limits and minimum up and down time requirements [2]. A careful consideration of how the electric energy that we consume should be obtained from the energy sources available has gained considerable debate. Electric power sector deregulation brought to the electric power business competitions through biding to win the best profit in the electric energy market. The decision of management is in a way to reduce costs and increase income, toning to the best economic perspective. But electric energy conversion companies should be responsible for the pollutant emissions and should take into account constraints to ensure admissible levels of emissions in the environment [3]. The Intergovernmental Panel on Climate Change in 1988 organized by the United Nations Environment Program and the World Meteorological Organization lead the way for the organized international community concern about the environment pollutant emissions due to human activity [4]. The world encounters at Rio de Janeiro in Brazil, 1992, at Kyoto
Arab J Sci Eng (2013) 38:1415–1425
in Japan, 1997, and at Copenhagen in Denmark, 2009, are evidence of the concern that the international community has to act in an organized way in response to the increasing pollution in our environment, resulting from human activity. Protocol of Kyoto signed by a large number of nations requires industrialized countries to limit their emissions of the following gases: carbon dioxide, methane, nitrous oxide, hydro fluorocarbons, perfluoro carbons, sulphur hexafluoride, collectively known as greenhouse gases. If the present trend continues, the augmentation of carbon dioxide in the atmosphere will become considerably higher by the year 2100. Hence, protocol of Kyoto establishes as the reference year for measuring improvement on emission pollution the year 1990. More precisely, the protocol requires that the aggregate anthropogenic carbon dioxide equivalent emissions of the greenhouse gases be at least 5 % below the level of world’s emissions in the year 1990 for the commitment period 2008 to 2012. Nevertheless, the protocol stated that it does not enter into force until rectified by 55 countries, contributing at least with 55 % of the world’s emissions in 1990. United States of America, having a level of emissions of 36.1 % of the world’s emissions in the reference year, withdrawal from Kyoto Protocol in March 2001 leads to the idea of whether or not to go on with the protocol, causing considerable debate. In Bonn July 2001, the protocol of Kyoto was reconsidered and in Marrakesh November of the same year the outcome of Bonn was confirmed, and technical and legal details concerning emission trading as well as concrete sanctions mechanisms in the case of non-compliance were treated. In this paper, the PBUC problem is solved by the proposed algorithm with an objective of maximizing the profit after considering the emission limitations imposed by the Kyoto Protocol and the results have been analyzed.
2 Problem Formulation 2.1 Formulation of Profit Based Unit Commitment The objective of the PBUC problem is to maximize the total profits of GENCOs subjected to a set of system and unit constraints over the scheduling horizon. It is different from minimizing the fuel cost as in Unit commitment problem because GENCOs no longer have the obligation to serve the whole demand. They may choose to generate less than demand, which allows more flexibility in UC schedules. Hence, the decision to commit a generating unit is price based and the bidding strategies need to be made accordingly. So, redefining the UC problem for competitive environment involves changing the demand constraints from equality to less than or equal to and the objective function is modified from cost minimization to profit maximization [5,6] PBUC based on
1417
forecasted market price with profit maximizing objective can be represented as, Max PF = RV − TC
(1)
The revenue and total costs can be calculated from the following equations RV =
T N SPi ∗ P ji ∗ U ij i
TC =
T N i
(2)
j
F(P ji ) + ST j ∗ [1 − U ij − 1]U ij ,
(3)
j
where a j , b j , c j are the unit cost coefficients. The generator start-up cost depends on the time the unit has been switched off prior to the start up, T joff . The overall objective is to maximize PF subject to a number of system and unit constraints. 2.2 Formulation of Emission Modeling The emission control cost results from the requirement for power utilities to reduce their pollutant levels below the annual emission allowances assigned for the affected fossil units. The total emission can be reduced by minimizing the three major pollutants: oxides of nitrogen (NOx ), oxides of sulphur (SOx ) and carbon dioxide (CO2 ). The objective function that minimizes the total emissions can be expressed in a linear equation as the sum of all the three pollutants resulting from generator. The emission cost can be represented as E(P ji ) = h ∗ (α j (P ji )∧ 2 + β j (P ji ) + γ j ),
(4)
where P ji is the Power generation (in MW) of unit j, at hour i, α j, β j , γ j the emission coefficients and h is the price penalty factor. The objective function to minimize emission can be represented as Min EM =
T N i
E(P ji )
(5)
j
2.3 Total Objective Function The total objective function considers at the same time the profits of GENCOs and the cost of pollution level control. These objectives have complicated natures and conflict in some points (the maximization of the profits can maximize the emission cost and vice versa). However, the solutions may be obtained in which profits and emission are combined in a single function with a difference weighting factor. This objective function can be defined by: Max G = λ ∗ PF + (λ − 1) ∗ EM,
(6)
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Table 1 Parameter selection for BPSO Parameter
Choosen value
Number of particles
20
Particle size
Table 5 Solution of PBUC with emission limitations by swarm intelligence method Sl. no.
λ
24 (h) × 11 (generators)
1
0.1
Number of iterations
200
2
0.2
Inertia weight factor
Wmax = 0.9 and Wmin = 0.4
3
0.3
Velocity limit
Vmax = 4 and Vmin = −4
4
0.4
Acceleration constants
C1 = 2 and C2 = 2
5
Profit
Emissions
4,285.61
Total generation
1,298.19
2,320.14
4,789.52
1,675.98
2,564.28
5,292.23
2,083.61
2,824.28
5,557.11
2,380.32
2,996.75
0.5
6,550.17
3,473.30
3,641.05
6
0.6
7,775.92
4,956.58
4,589.33
7
0.7
8,564.31
6,647.18
5,471.10
Table 2 Parameter selection for RPSO
8
0.8
9,718.28
9,332.97
6,723.04
Parameter
9
0.9
10,648.44
12,738.75
7,712.03
10
1
10,755.90
14,069.63
7,947.15
Choosen value
Number of particles
20
Particle size
24 (h) × 11 (generators)
Number of iterations
50
Inertia weight factor
Wmax = 0.9 and Wmin = 0.4
Velocity limits
Vmax = 25 and Vmin = −25
Acceleration constants
C1 = 2 and C2 = 2
2.4 Constraints 2.4.1 Power Balance Constraint The total generated power at each hour must be less than or equal to the load of the corresponding hour, PDi N
Table 3 System data Gen 1
Gen 2
Gen 3
Gen 4
Gen 5
Gen 6
P ji ∗ U ij ≤ PDi
i = 1, 2, 3, . . . , T
(7)
J
C
0
0
0
0
0
0
B
2
1.7
1
3.25
3
3
A
0.00375 0.0175 0.0625 0.0083 0.0250 0.0250
22.983
B
−0.90
−0.10 −0.01 −0.005 −0.004 −0.0055
A
0.0126
0.0200 0.0270 0.0291 0.0290 0.0271
Pmin (MW)
50
20
The generation of the unit is under its minimum and maximum limit
25.313 25.505 24.900 24.700 25.300
15
10
10
P j min ≤ P ji ≤ P j max
12
Pmax (MW)
200
80
50
35
30
40
Start up ($)
40
20
20
10
10
10
Min up (h)
2
2
2
2
2
2
Min down (h) 2
2
2
2
2
2
Initial status
−2
2
2
−2
2
2
2.4.2 Power Generation Limits
(8)
2.4.3 Minimum Up Time This constraint signifies the minimum time for which a committed unit should be turned off and removed from online. X i,onj ≥ MUTi
where λ is a weighting factor that satisfies 0 ≤ λ ≤ 1. The boundary values λ = 1 and λ = 0 give the conditions for the pure maximization of the profit function and the pure minimization of the pollution control level.
(9)
2.4.4 Minimum Down Time This constraint signifies the minimum time for which a de-committed unit should be turned on and brought on-line.
Table 4 Load pattern and spot prices Hours
1
Load
250
Spot price
2
3.78
3
270 3.94
4
290 4.03
5
320 4.10
6
360 4.24
7
390 4.27
8
420 4.34
9
420 4.30
430 4.52
10
11
12
420
360
330
4.53
4.08
4.12
Hours
13
14
15
16
17
18
19
20
21
22
23
24
Load
350
340
330
380
400
420
410
400
390
350
300
260
Spot price
4.46
123
4.04
4.18
4.51
4.33
4.29
4.36
4.05
4.28
4.14
3.92
3.77
Arab J Sci Eng (2013) 38:1415–1425
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Table 6 Best solution of PBUC with emission limitations by binary particle swarm optimization
Table 7 Economic dispatch of the best solution by real-valued particle swarm optimization
Hour
Gen 1
Gen 2
Gen 3
Gen 4
Gen 5
Gen 6
Hour
Gen 1
Gen 2
Gen 3
Gen 4
1
1
1
1
0
1
1
2
1
1
1
0
1
1
1
150.40
50.19
21.64
1
2
163.37
54.46
23.08
1
3
174.70
58.16
15.00
179.59
59.77
3
1
4
1
1
1
1
1
1 1
1 1
Gen 5
Gen 6
Total generation
0
15.78
12.00
270
0
10.00
19.09
250
10.00
10.00
12.00
279.86
24.89
10.00
10.00
12.00
296.25
5
1
1
1
1
1
1
4
6
1
1
1
1
1
1
5
183.38
62.99
25.98
10.00
10.00
12.00
310.34
1
6
191.47
63.68
26.21
10.00
25.18
12.00
328.54
1
7
200.00
65.29
26.75
35.00
26.30
26.80
380.14
193.57
64.37
26.44
35.00
10.00
12.00
341.38
7
1
8
1
1
1
1
1
1 1
1 1
9
1
1
1
1
1
1
8
10
1
1
1
1
1
1
9
200.00
69.43
28.15
10.00
30.00
12.00
349.58
1
10
200.00
69.99
28.23
35.00
10.00
12.00
354.88
1
11
178.20
59.31
24.73
10.00
22.13
12.00
306.37
180.99
60.23
25.04
10.00
10.00
12.00
298.26
11 12
1 1
1
1
1
1
1 1
1 1
13
1
1
1
1
1
1
12
14
1
1
1
1
1
1
13
199.57
66.32
27.10
35.00
10.00
12.00
350
1
14
175.40
58.39
24.42
35.00
10.00
12.00
315.21
1
15
185.19
61.61
25.51
10.00
10.00
24.19
316.49
200.00
69.20
28.07
35.00
10.00
12.00
354.27
15 16
1 1
1
1
1
1
1 1
1 1
17
1
1
1
1
1
1
16
18
1
1
1
1
1
1
17
195.67
65.06
26.67
10.00
26.14
26.63
350.18
1
18
200.00
64.14
26.36
35.00
30.00
25.98
381.48
1
19
200.00
65.75
26.91
10.00
26.62
12.00
341.28
20
176.10
58.62
24.50
35.00
21.65
12.00
327.87
19 20
1 1
1
1
1
1
1 1
1 1
21
1
1
1
1
1
1
22
1
1
1
1
1
1
21
192.17
63.91
26.29
35.00
10.00
25.00
353.19
182.39
60.69
25.20
35.00
10.00
12.00
325.28
10.00
12.00
278.14
12.00
253.04
23
1
1
1
1
1
1
22
24
1
1
1
1
0
1
23
167.02
55.63
23.49
10.00
24
156.53
52.15
22.32
10.00
X i,offj ≥ MDTi
(10)
2.4.5 Spinning Reserve Constraints Spinning reserve is the term used to describe the total amount of generation available from all the units synchronized on the system minus the present load plus losses being incurred. Spinning reserve must be carried so that the loss of one or more units does not cause too far a drop in system frequency. N
R ij ∗ U ij ≤ SR j
0
i = 1, 2, 3 . . . , T
(11)
j
3 Swarm Intelligence 3.1 Overview of the PSO Particle swarm optimization (PSO), first introduced by Kennedy and Eberhart, is one of the modem heuristic optimization algorithms. PSO provides a population-based search procedure in which individuals called particles change their
positions with time. The PSO can generate high quality solutions within shorter calculation time and stable convergence characteristic than other stochastic optimization methods [7]. A swarm consists of a set of particles, where each particle represents a potential solution [8]. Particles are then flown through the hyperspace, where the position of each particle is changed according to its own experience and that of its neighbours. Let xi (t) denotes the position of particle pi in search space, at time step t. The position of pi is then changed by adding a velocity vi (t + 1) to the current position as follows. xi (t + 1) = xi (t) + vi (t + 1)
(12)
The update of the velocity from the previous velocity to the new velocity is determined by: vi (t + 1) = w × vi (t)+C1 × rand(0, 1) × (xpbesti −xi (t)) +C2 × rand(0, 1) × (xgbest−xi (t))
(13)
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Table 8 Total profit and emissions in swarm intelligence method Hour
Total generation (MW)
1
250.00
2 3
Fuel cost ($)
Start up cost ($)
Total fuel cost ($)
659.07
30.00
689.07
270.00
726.57
0
279.86
756.42
10.00
4
296.25
813.30
0
5
310.34
863.33
6
328.54
7
380.14
8
341.38
9
Revenue ($)
Profit ($)
Total emissions (kg)
945.00
255.93
342.23
726.57
1,063.80
337.23
393.71
766.42
1,127.85
361.43
453.35
813.30
1,214.63
401.34
484.97
0
863.33
1,315.85
452.52
530.71
933.10
0
933.10
1,402.87
469.76
556.32
1,125.53
0
1,125.53
1,649.80
524.28
645.22
975.74
0
975.74
1,467.93
492.19
583.81
349.58
1,015.48
0
1,015.48
1,580.08
564.60
615.95
10
354.88
1,027.37
0
1,027.37
1,607.62
580.25
626.17
11
306.37
852.40
0
852.40
1,250.01
397.60
489.93
12
298.26
820.37
0
820.37
1,228.85
408.48
491.33
13
350.00
1,007.27
0
1,007.27
1,561.00
553.73
614.01
14
315.21
882.82
0
882.82
1,273.46
390.64
498.87
15
316.49
889.32
0
889.32
1,322.93
433.60
522.64
16
354.27
1,024.77
0
1,024.77
1,597.75
572.98
624.71
17
350.18
1,017.19
0
1,017.19
1,516.26
499.06
593.74
18
381.48
1,132.07
0
1,132.07
1,636.56
504.50
646.67
19
341.28
980.09
0
980.09
1,487.97
507.88
599.00
20
327.87
930.50
0
930.50
1,327.89
397.39
512.60
21
353.19
1,022.96
0
1,022.96
1,511.64
488.67
591.01
22
325.28
918.06
0
918.06
1,346.65
428.59
530.36
23
278.14
750.77
0
750.77
1,090.30
339.52
430.41
24
253.04
667.72
0
667.72
953.96
286.24
361.03
Total profit
10,648.44$
Fig. 1 Convergence characteristics of binary particle swarm optimization
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Total emission
12,738.75 kg
Fig. 2 Trade off curve between decrease in profit (%) and decrease in emissions (%)
Arab J Sci Eng (2013) 38:1415–1425
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Table 9 Best solution of PBUC with emission limitations by binary particle swarm optimization Hour
Gen 1
Gen 2
Gen 3
Gen 4
Gen 5
Gen 6
Gen 7
Gen 8
Gen 9
Gen 10
Gen 11
1
1
1
1
1
1
1
0
0
0
0
0
2
1
1
1
1
1
1
0
0
0
0
0
3
1
1
1
1
1
1
0
0
0
0
0
4
1
1
1
1
1
1
0
0
0
0
0
5
1
1
1
1
1
1
0
0
0
0
0
6
1
1
1
1
1
1
0
0
1
0
0
7
1
1
1
1
1
1
0
1
1
0
0
8
1
1
1
1
1
1
0
1
1
0
0
9
1
1
1
1
1
1
1
1
1
0
0
10
1
1
1
1
1
1
1
1
1
0
0
11
1
1
1
1
1
1
1
1
1
1
0
12
1
1
1
1
1
1
1
1
1
1
0
13
1
1
1
1
1
1
1
1
1
1
0
14
1
1
1
1
1
1
1
1
1
1
0
15
1
1
1
1
1
1
0
0
1
1
0
16
1
1
1
1
1
1
0
0
0
0
0
17
1
1
1
1
1
1
0
0
0
0
0
18
1
1
1
1
1
1
0
0
0
0
0
19
1
1
1
1
1
1
0
0
1
0
0
20
1
1
1
1
1
1
0
0
1
1
0
21
1
1
1
1
1
1
0
1
1
1
0
22
1
1
1
1
1
1
0
1
0
0
0
23
1
1
1
1
1
1
0
0
0
0
0
24
1
1
1
1
1
0
0
0
0
0
0
3.2 An Improved PSO to PBUC The original version of PSO operates on real values. However, with a simple modification, the particle swarm algorithm could be made to operate on binary problems, such as those traditionally optimized by genetic algorithm, and this method is called binary particle swarm optimization (BPSO). Kennedy and Eberhart proposed a discrete binary version of PSO for binary problems [9]. In their model, a particle will decide on “yes” or “no”, “true” or “false”, “include” or “not to include”, etc. also this binary values can be a representation of a real value in binary search space. In the binary PSO, the particle’s personal best and global best are updated as in continuous version. The major difference between binary PSO with continuous version is that velocities of the particles are rather defined in terms of probabilities that a bit will change to one. Using this definition, a velocity must be restricted within the range [0, 1]. So a map is introduced to map all real valued numbers of velocity to the range [0, 1]. In binary particle swarm, xi and pbest can take on values of 0 or 1 only. The velocity, vi will determine a probability threshold. If vi is higher, the individual is more likely to choose 1, and lower values favor the 0 choice. Such a threshold needs
to stay in the range [0, 1]. The function is called sigmoid function derived as follows: S(vi (t)) = 1/(1 + e(−vi (t)))
(14)
The new position of the particle is obtained using the equation below: xi (t) = 1 if rand() < S(vi (t)) = 0 otherwise
(15)
4 Example Problem and Simulation Results The efficiency of the proposed method has been demonstrated by solving the PBUC problem for a 24-h scheduling horizon, in two practical power systems. For simplicity, the shutdown cost of the generators has been taken equal to zero for every unit [9,10]. The PSO method is sensitive to the tuning of some weights and parameters. So the parameter selection plays an important role [11,12]. Parameter selection and results are given as:
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Table 10 Best solution of economic dispatch problem by real-valued particle swarm optimization Gen 1
Gen 2
Gen 3
Gen 4
Gen 5
Gen 6
Gen 7
Gen 8
Gen 9
Gen 10
Gen 11
1
220.59
189.38
315.05
291.54
293.87
289.57
0
0
0
0
0
2
236.54
203.86
339.43
316.68
316.66
311.83
0
0
0
0
0
3
265.09
227.52
383.45
360.46
363.06
311.83
0
0
0
0
0
4
289.94
300.00
421.94
399.76
402.20
386.16
0
0
0
0
0
5
300.00
260.18
440.42
418.26
422.85
403.20
0
0
0
0
0
6
300.00
300.00
455.07
432.75
438.28
416.28
0
0
157.61
0
0
7
300.00
264.67
449.25
426.91
432.72
411.15
0
210.00
155.30
0
0
8
300.00
273.53
464.11
441.95
447.71
424.83
0
210.00
160.57
0
0
9
300.00
280.88
477.22
455.01
460.00
436.73
215.00
210.00
165.15
0
0
10
300.00
300.00
500.00
500.00
460.00
477.72
215.00
210.00
180.91
0
0
11
300.00
300.00
500.00
500.00
460.00
472.91
215.00
210.00
179.06
136.96
0
12
300.00
300.00
500.00
500.00
460.00
500.00
215.00
210.00
183.68
140.11
0
13
300.00
300.00
500.00
467.66
460.00
447.74
215.00
210.00
169.28
130.32
0
14
300.00
300.00
500.00
500.00
460.00
481.32
215.00
210.00
182.29
139.17
0
15
300.00
278.72
473.32
451.16
460.00
433.25
0
0
163.81
126.56
0
16
300.00
275.76
468.05
445.89
460.00
428.44
0
0
0
0
0
17
300.00
266.94
452.69
430.54
435.31
414.52
0
0
0
0
0
18
300.00
272.05
461.47
439.32
444.94
422.43
0
0
0
0
0
19
300.00
274.27
465.42
443.26
460.00
426.03
0
0
161.04
0
0
20
300.00
280.95
477.26
455.11
460.00
436.85
0
0
165.19
127.50
0
21
300.00
287.63
489.11
466.95
460.00
447.67
0
210.00
169.35
130.34
0
22
300.00
300.00
441.70
419.86
423.84
404.60
0
210.00
0
0
0
23
271.17
232.56
391.87
370.95
373.37
360.09
0
0
0
0
0
24
300.00
253.47
428.50
407.12
410.91
0
0
0
0
0
0
4.1 Parameter Selection For both six units and eleven units system the parameters chosen are given below (Tables 1, 2).
4.2 Test System I A six unit system has been taken from the Refs. [13,14]. The details of the fuel cost components, initial conditions and load pattern with spot prices are given in Tables 3 and 4. The above six unit system is solved by the proposed method and the results are tabulated in Table 5. The best solution of PBUC by the proposed algorithm is shown in Table 6 and the economic load dispatch (EDL) results of the best solution is given in Table 7. The total profit and emission are tabulated in Table 8. Figure 1 shows the convergence characteristics for six unit system of PBUC problem using proposed PSO algorithm. Figure 2 shows the trade off curve between the decrease in profit (%) and decrease in emissions (%) for six units sys-
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tem. The curve is drawn for ten different solutions obtained by taking different weighting factors. 4.3 Test System II Eleven units are to be committed to serve 24-h load pattern. Unit data and load pattern are taken from Refs. [13,14]. The unit commitment schedule obtained by the proposed algorithm and the EDL of the best solution obtained by the proposed algorithm are given in Tables 9 and 10. The total profit and emission obtained for the above example problem are given in Table 11. 4.4 Comparison of Results 4.4.1 For Six Unit System See Table 12. 4.4.2 For Eleven Unit System See Table 13.
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Table 11 Total profit and emissions in swarm intelligence method Hours
Total generation (MW)
Fuel cost ($)
Start up cost ($)
Total fuel cost ($)
Revenue ($)
Profit ($)
Total emissions (kg)
1
1,600.00
49,033.78
4,200.00
53,233.78
53,040.22
−193.56
1,370.35
2
1,725.00
52,327.15
0
52,327.15
56,925.06
4,597.91
1,632.60
3
1,950.00
58,405.69
0
58,405.69
64,545.09
6,139.40
2,166.29
4
2,200.00
65,415.62
0
65,415.62
74,030.04
8,614.42
2,871.95
5
2,300.00
66,671.34
0
66,671.34
74,643.08
7,971.74
2,983.34
6
2,500.01
75,345.71
400.00
75,745.71
87,375.03
11,629.32
3,319.27
7
2,650.00
80,723.30
400.00
81,123.30
91,425.01
10,301.71
3,423.46
8
2,800.00
82,880.43
0
82,880.43
92,980.06
10,099.63
3,631.67
9
3,000.00
92,006.48
500.00
92,506.48
104,400.01
11,893.53
4,124.07
10
3,200.00
96,456.34
0
96,456.34
114,270.66
17,814.32
4,556.10
11
3,350.00
101,404.84
400.00
101,804.84
118,352.46
16,547.62
4,555.07
12
3,450.00
102,557.11
0
102,557.11
121,267.22
18,710.11
4,651.28
13
3,200.00
99,020.12
0
99,020.12
117,120.06
18,099.94
4,358.63
14
3,300.00
101,877.30
0
101,877.30
120,004.04
18,126.74
4,586.30
15
2,750.00
81,931.54
0
81,931.54
92,694.88
10,763.34
3,492.74
16
2,400.00
70,554.61
0
70,554.61
81,570.26
11,015.65
3,387.58
17
2,300.00
68,268.53
0
68,268.53
78,775.02
10,506.49
3,146.22
18
2,500.00
69,442.24
0
69,442.24
79,684.14
10,241.90
3,268.62
19
2,760.00
76,249.11
400.00
76,649.11
86,526.89
9,877.78
3,393.18
20
3,200.00
82,424.55
400.00
82,824.55
93,654.31
10,829.76
3,537.34
21
3,000.00
91,053.73
400.00
91,453.73
103,932.52
12,478.79
3,966.50
22
2,500.00
75,122.51
0
75,122.51
89,875.16
14,752.65
3,425.81
23
2,000.00
59,784.81
0
59,784.81
71,500.33
11,715.52
2,294.96
24
1,800.00
53,036.95
0
53,036.95
63,990.06
10,953.11
2,399.68
Total profit
273,487.83 $
Total emissions
80,543.01 kg
Table 12 Comparison of results for six unit system Method Traditional method
Profit ($)
Emissions (kg)
Total generation (MW)
9,611.31
14,048.10
8,590.00
10,755.89
14,069.63
7,947.15
10,648.44
12,738.75
7,712.03
Method
Profit ($)
Emissions (kg)
Total generation (MW)
Traditional method
281,834.28
88,425.44
62,435.00
Profit based unit commitment limitations by swarm intelligence method Profit based unit commitment with emission limitations by swarm intelligence method
293,800.22
89,814.38
59,025.08
273,487.83
80,543.01
61,005.85
Profit based unit commitment limitations by swarm intelligence method Profit based unit commitment with emission limitations by swarm intelligence method
Table 13 Comparison of results for eleven unit system
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5 Conclusion
Flow Chart for BPSO
In this paper the PBUC problem with Emission limitations is solved by Swarm Intelligence. Six unit system and eleven unit system from literature have been taken and the proposed algorithm is applied for solving the PBUC. The results obtained by the proposed algorithm for 24 h load pattern have been shown in Tables 5, 6, 7, 8 and 9, 10, 11 for six units and eleven units systems, respectively. From the comparison of results given in Tables 12 and 13, the ability of the proposed algorithm to solve the PBUC problem is demonstrated in the aspects of solution quality. From the convergence characteristics of the proposed algorithm, the capability of the proposed algorithm for handling equality and inequality constraints is established.
Start
Initialize the Parameters C1, C2, Wmax, Wmin,
Generate initial population
Calculate the power of Dependent unit from power balance equation C
Evaluate the objective function
Calculate the velocity of each unit in all particles
Appendix Flow Chart for PBUC
If
NO
Vdmin ≤Vid ≤Vdmax
Vid = -4 Or Vid = +4
Start
Input load, start up cost, no of units
Modify the initial population using sigmoid function
From unit selection list and unit combination list
Discard the combination
If Iter=Itermax
If The combination is feasible
NO
Display the global best set of particles as optimal solution NO
YES Calculate the production cost of each feasible combination
Choose more profit combination
Compute unit commitment scheduling
NO Last Hour
YES
Stop
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Stop
NO C
Arab J Sci Eng (2013) 38:1415–1425
1425
Flow Chart for RPSO
References
Start
Initialize the Parameters C1, C2, Wmax, Wmin,
Generate initial population
Calculate the power of Dependent unit from power balance equation C
Evaluate the objective function
Calculate the velocity of each unit in all particles
If
NO
Vdmin ≤Vid ≤Vdmax
Vid = -0.5Vmin Or Vid = +0.5Vmax
Modify the initial population by adding velocities
NO If Iter=Itermax
Display the global best set of particles as optimal solution
C
1. Passino, K.M.: Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Syst. Mag. 22(3), 52–67 (2002) 2. Venkatesh, P.; Kannan, P.S.; Sudhakaran, M.: Application of computational intelligence to economic load dispatch. J. Inst. Eng. (India) 81(2), 39–43 (2000) 3. Sudhakaran, M.; Sivakumar, G.; Vengatachalapathy, P.; Latchumi, K.: Particle swarm optimization for economic and emission dispatch problems. J. Inst. Eng. (India) 88(2), 39–45 (2007) 4. Barisal, A.K.; Hota, P.K.; Chakrabarti, R.: Economic load dispatch by modified bacteria foraging algorithm. In: Proceedings of International Conference-PSACO-2008. Andhra University, Visakhapatnam, March 13–15 2008 5. Wood, J.; Wollenberg, B.F.: Power Generation, Operation and control. Wiley, New York (1984) 6. Tripathy, M.; Mishra, S.: Bacteria foraging-based solution to optimize both real power loss and voltage stability limit. IEEE Trans. Power Syst. 22(1), 126–131 (2007) 7. Chaturvedi, K.T.; Pandit, M.; Srivastava, L.: Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE Trans. Power Syst. 23(3), 1079–1087 (2008) 8. Nanda, J.; Mishra, S.; Saika, L.C.: Maiden application of Bacteria foraging based optimization technique in multiarea automatic generation control. IEEE Trans. Power Syst. 24(2), 156–166 (2009) 9. Park, J.B.; Jeong, Y.W.; Shin Lee, J.-R.: An improved particle swarm optimization for nonconvex Economic dispatch problems. IEEE Trans. Power Syst. 25(1), 156–166 (2010) 10. Voumvoulakis, E.M.; Hatziaragyriou, N.D.: A PSO method for power system dynamic security control. IEEE Trans. Power Syst. 25(2), 1032–1041 (2010) 11. Lu, B.; Shahidehpour, M.: Unit commitment with flexible generating units. IEEE Trans. Power Syst. 20(2), 1022–1034 (2005) 12. Gaing, Z.-L.: Particle swarm optimization for solving the economic dispatch considering the generator constraints. IEEE Trans. Power Syst. 18(3), 1187–1195 (2003) 13. Shrestha, G.B.; Pokharel, B.K.; Lie, T.T.; Fleten, S.E.: Price-based unit commitment for bidding under price uncertainty. IET Gener. Transm. Distrib. 1(4), 663 (2007) 14. Wu, L.; Shahidehpour, M.: Financial risk evaluation in stochastic PBUC. IEEE Trans. Power Syst. 24(4), 1032–1041 (2009)
Stop
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