Evaluation of Best Weight Pattern for Multiple Criteria Load Dispatch

Electric Power Components and Systems, 34:21–35, 2006 Copyright © Taylor & Francis Inc. ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325000691001520

Evaluation of Best Weight Pattern for Multiple Criteria Load Dispatch LAKHWINDER SINGH Department of Electrical and Electronics Engineering Baba Banda Singh Bahadur Engineering College Fatehgarh Sahib, Punjab, India

J. S. DHILLON Department of Electrical and Instrumentation Engineering Sant Longowal Institute of Engineering and Technology Longowal, Punjab, India

R. C. CHAUHAN Sant Longowal Institute of Engineering and Technology Longowal, Punjab, India The multiple criteria load dispatch (MCLD) problem is a multiple noncommensurable objective problem that minimizes both cost and emission together subject to physical and technological constraints. In this article, a solution methodology is proposed for the determination of best objective weights to solve multiobjective thermal power dispatch problem. The weights are calculated by conventional statistical measures, which characterize the correlation coefficients matrix evolution. Two fundamental concepts of multiple criteria decision method are undertaken, which are the contrast intensity of the alternative’s performance in each single objective and the conflicting nature of the objectives with each other. The extraction and exploitation of these two features is beneficial in the decision making. The proposed method needs few noninferior solutions and, hence, gives computational efficiency. The validity of the proposed method is demonstrated on IEEE 11-bus, 17-lines system, comprising three generators. Keywords economic load dispatch, best weight pattern, decision making, noninferior solutions

1.

Introduction

In a large number of real-life decision problems, a decision maker is faced with difficulties to take a decision, especially when multiple objectives are present in the decision Manuscript received in final form on 21 March 2005. Address correspondence to Lakhwinder Singh, Assistant Professor, Department of Electrical and Electronics Engineering, Baba Banda Singh Bahadur Engineering College, Fatehgarh Sahib, Punjab 140 407, India. E-mail: [email protected]

21

22

L. Singh et al.

and these are to be simultaneously achieved. This problem becomes more complicated when the objectives are conflicting, noncommensurable, and imprecise in nature. The situation is formulated as a multiple criteria optimization problem in which the goal is to maximize or minimize several objective functions, simultaneously. Many approaches and methods have been proposed to solve such multiobjective problems [1–3]. The solution methods are broadly grouped under two major categories: noninteractive and interactive methods. In the noninteractive method, a global preference function of the objectives is identified and optimized with respect to the constraints. In the interactive method, a local preference function or trade off among objectives is identified by interacting with the decision maker and the solution process gradually proceeds towards a globally satisfactory solution. Mostly, the interactive methods are used to find noninferior solutions. So far, the main purpose of the optimal power scheduling problem has mainly confined to minimize the generation cost of a power system. However, to meet the environmental regulations enforced in recent years, emission control has become one of the important operational objectives. The pollution of the earth’s atmosphere caused by three principal gaseous pollutants, oxides of carbon (COx ), oxides of sulphur (SOx ), and oxides of nitrogen (NOx ) from thermal generator plants, is of greater concern to power utility and communities. For the solution of such multiobjective problems different techniques have been reported in literature pertaining to environmental-economic dispatch problem. Ramanathan [4] has presented a methodology to include emission constraints in classical economic dispatch, which contains an efficient weights estimation technique. Talaq et al. [5] have given a summary of work in the area of environmental/ economic dispatch which includes several techniques intended to reduce emissions into the atmosphere due to electric power generation. Hota et al. [6] have solved the economic emission load dispatch through an interactive fuzzy satisfying method. Basu [7] has used the Hopfield neural networks to solve fuel constrained economic emission load dispatch problem. Brar et al. [8] have used fuzzy logic based weightage pattern searching to obtain the solution of multiobjective load dispatch problem. The evolutionary optimization technique has been employed in which the “preferred” weightage pattern is searched to get the “best” optimal solution in noninferior domain. Chen et al. [9] have presented a direct NewtonRaphson economic emission dispatch method which considers the line flow constraints by evaluating the B-coefficients from the sensitivity factors with dc load flow. Abido [10] has presented a novel approach based on the strength Pareto evolutionary algorithm to solve environmental/economic power dispatch optimization problem. Fuzzy based mechanism is employed to extract the best compromise solution over the trade off curve. Yang et al. [11] have developed a new explicit interactive trade off analysis method, which is based on identification of normal vectors on a non-inferior frontier. The intent of this article is to solve multiple criteria electric power load dispatch problem having four objectives, that is, the economic index and impact on environment due to NOx , SOx , and COx gaseous pollutants interactively. The multiobjective optimization problem is converted into single objective optimization problem by applying weighting method and few noninferior solutions are obtained by simulating arbitrary weights. Two fundamental concepts of multiple criteria decision methodology are undertaken which are the contrast intensity of the alternative’s performance in each single objective and the conflicting nature of the objectives with each other. The extraction and exploitation of these two features is beneficial in the decision making. The proposed method gives computational efficiency and validity of the method is demonstrated on IEEE 11-bus, 17-lines system, comprising three generators. The results are compared with Brar et al. [8].

Best Weight Pattern for MCLD

2.

23

Problem Formulation

The problem formulation treats multiple criteria load dispatch as a multiobjective mathematical programming problem in which the attempt is to minimize conflicting objective functions simultaneously, while satisfying several equality and inequality constraints. 2.1.

Economy Objective

The fuel cost of a thermal unit is regarded as an essential criterion for economic feasibility. The fuel cost curve is assumed to be approximated by a quadratic function of generator power output PGi as F1 (PGi ) =

Ng i=1

2 (ai PGi + bi PGi + ci ) $/h,

(1a)

where ai , bi , and ci are the cost coefficients of ith generator. Ng is number of generators. PGi is the real power output of ith generator and acts as a control variable. 2.2.

Emission Objectives

The atmospheric pollutants such as NOx , SOx , and COx caused by fossil fueled thermal generating units can be modeled separately and they are directly related to the cost curve through emission rate per Mcal, which is a constant factor for given type of fuel. Therefore, the amount of pollutants emission is given as a quadratic function of generator power output PGi as NOx emission: F2 (PGi ) =

Ng i=1

SOx emission: F3 (PGi ) =

Ng i=1

COx emission: F4 (PGi ) =

Ng i=1

2 (d2i PGi + e2i PGi + f2i ) Kg/h

(1b)

2 (d3i PGi + e3i PGi + f3i ) Kg/h

(1c)

2 (d4i PGi + e4i PGi + f4i ) Kg/h,

(1d)

where d2i , e2i , and f2i are the NOx emission coefficients of ith generator. d3i , e3i , and f3i are the SOx emission coefficients of ith generator. d4i , e4i , and f4i are the COx emission coefficients of ith generator. 2.3.

Objective Constraints

To ensure a real and reactive power balance, the following equality constraints are imposed: Ng

PGi =

Nb

i=1

i=1

Ng

Nb

i=1

QGi =

i=1

PDi + Ploss

(1e)

QDi + Qloss ,

(1f)

24

L. Singh et al.

where N b are the number of buses in the system. PGi and QGi are the real and reactive powers of ith generator, respectively. PDi and QDi are real and reactive demands at ith bus, respectively. Ploss and Qloss are real and reactive power losses in the transmission lines, respectively. The inequality constraints are imposed on generator output are: min max ≤ PGi ≤ PGi ; PGi max Qmin Gi ≤ QGi ≤ QGi ;

i = 1, 2, . . . , Ng

(1g)

i = 1, 2, . . . , Ng,

(1h)

min and P max are the minimum and maximum values of real power output of where PGi Gi max ith unit, respectively. Qmin Gi and QGi are the minimum and maximum values of reactive power output of ith unit, respectively.

2.4.

Power Transmission Losses

The real and reactive power transmission losses Ploss and Qloss are given below and the loss coefficients Aij , Bij , Cij , Dij are evaluated from line data by performing decoupled load flow analysis. Ploss =

Nb Nb

[Aij (Pi Pj + Qi Qj ) + Bij (Qi Pj − Pi Qj )]

(2)

[Cij (Pi Pj + Qi Qj ) + Dij (Qi Pj − Pi Qj )],

(3)

i=1 j =1

Qloss =

Nb Nb i=1 j =1

where Pi = PGi − PDi , Qi = QGi − QDi Aij =

Rij cos(δi − δj ); |Vi ||Vj |

Bij =

Rij sin(δi − δj ) |Vi ||Vj |

Cij =

Xij cos(δi − δj ); |Vi ||Vj |

Dij =

Xij sin(δi − δj ) |Vi ||Vj |

where δi and δj are load angles at ith and j th buses, respectively. Vi and Vj are voltage magnitude at ith and j th buses, respectively. Rij is the real component of impedance bus matrix. Xij is the reactive component of impedance bus matrix. Aggregating the objectives and constraints, the problem can be mathematically stated as a nonlinear multiple criteria constrained optimization problem as follows. Minimize

[F1 (PGi ), F2 (PGi ), F3 (PGi ), F4 (PGi )]T

Subject to

Eq. (1e) to (1h),

(4)

where (Fj (PGi ); j = 1, 2, . . . , 4) are the objective functions to be minimized over the set of admissible decision vector, PGi QGi ; i = 1, 2, . . . , Ng.


3.

25

Weighting Method

To generate the noninferior solutions, the multiple criteria problem is converted in scalar optimization problem and is given as: L

Minimize

wk F k

(5)

k=1

Subject to

i)

L

wk = 1.0,

wk ≥ 0.0

(6)

k=1

ii) Eq. (1e) to (1h), where wk are the levels of the weighting coefficients. L is the total number of objectives. This approach yields meaningful results to the decision maker when solved many times for different values of wk ; k = 1, 2, . . . , L. The values of the weighting coefficients vary from 0 to 1. To find the solution, constrained problem is converted into an unconstrained problem. Equality and inequality constraints are clubbed with objective function using Lagrangian multipliers and penalty method, respectively. The generalized augmented function is formed as: L(PGi , QGi , λp , λq ) =

L

  Ng Nb w j Fj − λ p  PGi − PDi − Ploss 

j =1

i=1

i=1

  Ng Nb − λq  QGi − QDi − Qloss  i=1

i=1

(7)

 Ng 1  + k {(Pi − Pimin )2 + (Pimax − Pi )2 r i=1

 2 max + (Qi − Qmin − Qi )2 } i ) + (Qi

where λp , λq are Lagrangian multipliers, r k is penalty factor and should have small value. The Newton-Raphson algorithm is applied to obtain the noninferior solutions for arbitrary weight combinations, to achieve the necessary conditions.

4.

Evaluation of Best Weight Pattern

The fuzzy sets are defined by membership functions. These functions represent the degree of membership in certain fuzzy sets using values from 0 to 1. By taking account of the minimum and maximum values of each objective function together with the rate

26

L. Singh et al.

of increase of membership satisfaction, the decision maker must determine the membership function µ(Fi ) in a subjective manner. Here it is assumed that µ(Fi ) is a strictly monotonic decreasing function defined as:

µ(Fi ) =

 1;         F max − Fi i

Fi ≤ Fimin

 Fimax − Fimin        0;

;

Fimin ≤ Fi ≤ Fimax

(8)

Fi ≥ Fimax

where Fimin and Fimax are minimum and maximum values of ith objective function in which the solution is expected. For every objective, Fi , of this multiple criteria problem defined by Eq. (4), a membership function µ(Fi ) is defined which maps the values of Fi within the interval [0, 1]. This transformation is based on the concept of the ideal point, Fimin . So, the value µ(Fi ) expresses the degree to which the alternative is close to the ideal value Fimin , which is the best performance of ith objective and far from the anti-ideal value Fimax , which is the worst performance of ith objective. Fimin is achieved by performing minimization of ith scalar objective with respect to constraints of the problem. Fimax is achieved by exploiting conflicting nature of the objectives, that is, if one objective will have minimum value, other objectives will have maximum values [12]. The initial matrix of evaluations is converted into a matrix of relative scores with membership function, µ(Fi ), and standard deviation, σµ(Fi ) , which quantifies the contrast intensity of the ith objective. So, the standard deviation of µ(Fi ) is a measure of the value of that objective to the decision making process. A symmetric matrix of correlation coefficients is constructed, with dimension L × L and a generic element r[µ(Fi ), µ(Fj )], which is the linear correlation coefficient between the membership functions µ(Fi ) and µ(Fj ) is r[µ(Fi ), µ(Fj )] = i = 1, 2, . . . , L;

Cov[µ(Fi ), µ(Fj )] ; σµ(Fi ) σµ(Fj )

(9)

j = 1, 2, . . . , L

where Cov[µ(Fi ), µ(Fj )] is covariance between µ(Fi ) and µ(Fj ). µ(Fi ) and µ(Fj ) are the membership functions of ith and j th objectives, respectively. σµ(Fi ) and σµ(Fj ) are the standard deviations of membership functions, µ(Fi ) and µ(Fj ), respectively. It has been observed that the more discordant the scores of the alternatives in objective µ(Fi ) and µ(Fj ), lower are the values of r[µ(Fi ), µ(Fj )]. In this sense, the sum L j =1 [1 − r{µ(Fi ), µ(Fj )}] represents a measure of the conflict created by objective µ(Fj ) with respect to the decision situation defined by the rest of the objectives. Information contained in multiple criteria decision method problems is related to both contrast intensity and conflict of the decision objective. Hence, the amount of information Ci emitted by the ith objective can be determined by composing the measures which


27

quantify the two notions through the following multiplicative aggregation formula: Ci = σµ(Fi )

L

[1 − r{µ(Fi ), µ(Fj )}]; (10)

j =1

i = 1, 2, . . . , L;

j = 1, 2, . . . , L.

According to the previous analysis, the higher the value Ci , larger is the amount of information transmitted by the corresponding objective and the higher is its relative importance for the decision making process. Objective weights are evaluated by normalizing these values to unity according to the following equation: Wi∗ =

Ci L

;

i = 1, 2, . . . , L.

(11)

Cj

j =1

5.

Computational Procedure

Stepwise procedure to compute best compromising solution by evaluating weights is given below: 1. Input the system data consisting of line data, fuel cost curve, and emission curve coefficients, limits on active and reactive power generations and demand etc. 2. Compute loss coefficients Aij , Bij , Cij , Dij , real and reactive power losses Ploss and Qloss by performing load flow using decoupled method. 3. Find the minimum and maximum values for each of the multiple objectives Fjmin , Fjmax ; j = 1, 2, . . . , L. This is carried out by giving full weightage to one objective and neglecting other objectives. 4. Simulate arbitrary weight combinations varying in a step size of 0.1, 0.05, and 0.01, such that their sum remains one. 5. Generate noninferior solutions by solving Eq. (7), using Newton-Raphson Method for all weight combinations in Step 4. 6. Compute membership functions of the obtained noninferior solutions using Eq. (8). 7. Calculate the correlation coefficients using Eq. (9). 8. Calculate Ci using Eq. (10) and optimal values of the normalized weights using Eq. (11). 9. The best set of decision vector is found by solving the following problem   4 Minimize  wj∗ Fj  j =1

Subject to

6.

Eq. (1e) to (1h).

Test System and Results

The validity of the proposed method is illustrated on 11-bus, 17-lines IEEE system comprising three generators. The fuel cost, NOx emission, SOx emission, COx emission

28

L. Singh et al. Table 1 Fuel cost ($/h) equations F11 = 21.82P12 + 742.890P1 + 847.1484 F12 = 13.45P22 + 830.154P2 + 274.2241 F13 = 59.63P32 + 691.559P3 + 202.0258

Table 2 NOx emission (Kg/h) equations F21 = 31.74P12 − 136.061P1 + 324.1775 F22 = 67.32P22 − 239.928P2 + 610.2535 F23 = 61.81P32 − 039.077P3 + 050.3808

Table 3 SOx emission (Kg/h) equations F31 = 12.84P12 + 445.647P1 + 508.5207 F32 = 08.13P22 + 497.641P2 + 165.3433 F33 = 35.78P32 + 414.938P3 + 121.2133

Table 4 COx emission (Kg/h) equations F41 = 1.05929P12 − 00.9552794P1 + 01.342851 F42 = 1.06409P22 − 01.2736420P2 + 01.819625 F43 = 4.03144P32 − 12.1981200P3 + 11.381070

equations are given in Tables 1–4. Line data and load data is depicted in Tables 5 and 6. Minimum and maximum values of the functions are shown in Table 7. Minimum and maximum values of the objectives are obtained by using Step 3 of the computational procedure. Owing to conflicting nature of the cost and emissions, objectives F2 , F3 , and F4 will have maximum values when the value of F1 objective, is minimum. The problem is solved for three cases undertaken as below: Case I: Arbitrary weights combinations varying in step of 0.1 Case II: Arbitrary weights combinations varying in step of 0.05 Case III: Arbitrary weights combinations varying in step of 0.01 In all the three cases weights are arbitrarily varying with the given step sizes, so that their sum remains equal to one. Noninferior solutions obtained from all the weight combinations for Case I, Case II, and Case III are shown in Tables 8–10.


29

Table 5 Line data Line

From

To

R (p.u.)

X (p.u.)

B (p.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 1 2 2 2 3 4 4 4 5 5 7 7 8 8 8 10

9 11 3 7 10 4 6 8 9 6 9 8 10 9 10 11 11

0.15 0.05 0.15 0.10 0.05 0.08 0.10 0.10 0.15 0.12 0.05 0.05 0.08 0.12 0.08 0.10 0.12

0.50 0.16 0.50 0.28 0.16 0.24 0.28 0.28 0.50 0.36 0.16 0.16 0.24 0.36 0.24 0.28 0.36

0.030 0.010 0.030 0.020 0.010 0.015 0.020 0.020 0.030 0.025 0.010 0.010 0.015 0.025 0.015 0.020 0.025

Table 6 Load data Bus

P (MW)

Q (MVAR)

1 2 3 4 5 6 7 8 9 10 11

0.25 0.25 0.25 0.25 0.25 0.10 0.40 0.90 0.70 0.25 0.25

0.05 0.05 0.05 0.05 0.05 0.02 0.10 0.45 0.35 0.05 0.05

Table 7 Minimum and maximum values of objectives F1min = 4584.7830 $/h

F1max = 4703.9410 $/h

F2min = 619.1288 Kg/h

F2max = 780.0698 Kg/h

F3min = 2750.5600 Kg/h

F3max = 2821.4900 Kg/h

F4min = 5.887253 Kg/h

F4max = 15.05706 Kg/h

30 0.3 0.1 0.1 0.2 0.1 0.1

0.2

0.4

0.6

0.4

0.4

0.7

0.2

0.2

0.1

0.1

0.2

0.4

0.1

0.2

0.2

0.1

0.1

0.3

w2

w1

0.1

0.4

0.2

0.1

0.2

0.2

0.1

0.1

0.2

0.2

0.1

0.1

w3

Weights

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.4

0.5

0.6

0.7

w4 Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership Objectives Membership

Objectives Membership 4590.0080 0.9561538 4587.7290 0.9752781 4587.1240 0.9803553 4593.1250 0.9299855 4591.8700 0.9405249 4586.1420 0.9885959 4599.9400 0.8728010 4585.9740 0.9900137 4585.5970 0.9931690 4589.2860 0.9622104 4585.7430 0.9919438 4585.4500 0.9944025

F1 $/h µ(F1 ) 660.96250 0.7400679 667.21970 0.7011890 669.44880 0.6873389 655.09110 0.7765496 657.31070 0.7627582 673.86760 0.6598827 646.13680 0.8321870 674.90370 0.6534451 677.34300 0.6382884 663.02900 0.7272280 676.53280 0.6433228 678.52420 0.6309494

F2 Kg/h µ(F2 ) 2753.6500 0.9564381 2752.2920 0.9755825 2751.9310 0.9806732 2755.4850 0.9305646 2754.7400 0.9410695 2751.3500 0.9888617 2759.5320 0.8735101 2751.2480 0.9902867 2751.0260 0.9934326 2753.2050 0.9627094 2751.1100 0.9922452 2750.9390 0.9946580

F3 Kg/h µ(F3 )

Objectives and membership functions

Table 8 Noninferior solutions for Case I

8.2572000 0.7415488 7.9952460 0.7701159 7.9184600 0.7784896 8.6072260 0.7033772 8.4827500 0.7169518 7.7664660 0.7950651 9.2089040 0.6377622 7.7382220 0.7981455 7.6634310 0.8063015 8.2081770 0.7468950 7.6956310 0.8027899 7.6306820 0.8098729

F4 Kg/h µ(F4 )

31 0.10 0.10 0.05 0.05 0.05 0.05

0.70

0.75

0.80

0.00

0.10

0.90

0.05

0.40

0.10

0.05

0.25

0.40

0.05

0.30

0.25

0.00

0.05

0.30

w2

w1

0.00

0.80

0.90

0.05

0.05

0.10

0.10

0.05

0.10

0.15

0.05

0.05

w3

Weights

0.05

0.05

0.05

0.10

0.10

0.10

0.40

0.40

0.45

0.55

0.60

0.90



F1 $/h µ(F1 ) 712.30960 0.4210253 680.96160 0.6158045 680.80550 0.6167746 682.52880 0.6060668 682.04800 0.6090544 673.86760 0.6598827 678.52420 0.6309494 678.80540 0.6292019 684.90910 0.5912768 731.13650 0.3040452 682.14490 0.6084522 685.21610 0.5893692

F2 Kg/h µ(F2 ) 2752.1790 0.9771761 2750.7280 0.9976319 2750.7430 0.9974219 2750.6700 0.9984511 2750.6980 0.9980553 2751.3500 0.9888617 2750.9390 0.9946580 2750.9180 0.9949540 2750.6140 0.9992393 2756.7950 0.9120983 2750.7410 0.9974495 2750.6070 0.9993392

F3 Kg/h µ(F3 )


Table 9 Non-inferior solutions for Case II

6.7127670 0.9099748 7.5222630 0.8216963 7.5327770 0.8205498 7.4960580 0.8245541 7.5138230 0.8226168 7.7664660 0.7950651 7.6306840 0.8098727 7.6217670 0.8108451 7.4515520 0.8294076 6.3248930 0.9522738 7.5361910 0.8201774 7.4448310 0.8301406

F4 Kg/h µ(F4 )

32 0.01 0.03 0.02 0.01 0.01 0.02

0.49

0.48

0.91

0.01

0.92

0.94

0.01

0.04

0.02

0.02

0.02

0.04

0.01

0.01

0.01

0.00

0.01

0.03

w2

w1

0.02

0.04

0.95

0.03

0.02

0.01

0.04

0.05

0.03

0.02

0.01

0.01

w3

Weights

0.02

0.03

0.03

0.04

0.47

0.49

0.90

0.91

0.92

0.94

0.97

0.98



F1 $/h µ(F1 ) 746.3849 0.2092999 718.1047 0.3850177 695.0566 0.5282260 703.9601 0.4729045 703.4026 0.4763685 692.1300 0.5464101 691.1785 0.5523221 686.9037 0.5788836 688.6962 0.5677458 688.9153 0.5663847 689.8676 0.5604674 688.7009 0.5677166

F2 Kg/h µ(F2 ) 2763.1220 0.8228955 2756.8940 0.9107008 2752.0830 0.9785288 2751.7390 0.9833786 2751.6300 0.9849172 2750.9590 0.9943758 2750.5230 1.0000000 2750.5360 1.0000000 2750.5540 1.0000000 2750.5790 0.9997315 2750.5530 1.0000000 2750.5550 1.0000000

F3 Kg/h µ(F3 )


Table 10 Noninferior solutions for Case III

6.094937 0.9773513 6.381523 0.9460981 6.899249 0.8896382 6.793436 0.9011775 6.811592 0.8991975 7.060969 0.8720021 7.279467 0.8481741 7.380732 0.8371308 7.360234 0.8393662 7.367848 0.8385358 7.333614 0.8422692 7.360615 0.8393247

F4 Kg/h µ(F4 )


33

Table 11 Correlation coefficients for Case I 1.0 −0.9602153 0.9999986 0.9921809

−0.9602153 1.0 −0.9601532 −0.9870314

0.9999986 −0.9601532 1.0 0.9921168

0.9921809 −0.9870314 0.9921168 1.0

Table 12 Correlation coefficients for Case II 1.0 0.8905969 0.9999064 −0.8649421

0.8905969 1.0 0.8965653 −0.9980310

0.9999064 0.8965653 1.0 −0.8716264

−0.8649421 −0.9980310 −0.8716264 1.0

Table 13 Correlation coefficients for Case III 1.0 0.9741936 0.9999371 −0.8941231

0.9741936 1.0 0.9755495 −0.9472796

0.9999371 0.97556495 1.0 −0.8988999

−0.8941231 −0.9472796 −0.8988999 1.0

Correlation coefficients of the membership functions are achieved from Eq. (9) and are given in Tables 11–13. Comparison of results is shown in Table 14. It has been observed from Case I, Case II, and Case III that the minimum value of the membership functions of the objectives is improved by decreasing the step size of the weights. The membership function increases from 0.497247 to 0.689364 with the decrease in step size from 0.1 to 0.05 and further 0.689364 to 0.703725 with the decrease in step size from 0.05 to 0.01. Further it has been observed that the noninferior solutions given in Table 8 has more membership function than the final membership function obtained by evaluating the weights. Then that set is recommended to be discarded and the process is repeated with smaller step of variation. Results of the proposed method are also compared with Brar et al. [8] and membership function is improved from 0.702979 to 0.703725. It is clear from all the three cases that by reducing the step size of the weights better satisfaction is achieved and when the overall outcome is less than some of the noninferior solutions try with the reduced step size which will improve the overall outcome. The “best” power generation schedule and voltage profile obtained at each bus by the proposed method are given in Table 15.

7.

Conclusions

In the multiobjective framework it is realized that cost and emissions are conflicting objectives and are subjected to mutual interface. The solution set of the formulated problem

34

L. Singh et al. Table 14 Comparison of results Cost ($/h)

Case I Weights Objectives Membership Case II Weights Objectives Membership Case III Weights Objectives Membership Brar et al. [8] Weights Objectives Membership

NOx emission (Kg/h)

SOx emission (Kg/h)

COx emission (Kg/h)

0.1653 4617.7840 0.723048

0.4991 633.2728 0.912117

0.1653 2770.1480 0.723842

0.1703 10.49740 0.497247

0.1607 4594.5260 0.918237

0.1896 652.8414 0.790528

0.1615 2756.3190 0.918807

0.4882 8.73572 0.689364

0.1639 4593.1370 0.929891

0.1745 655.0237 0.776969

0.1645 2755.4950 0.930424

0.4971 8.604041 0.703725

0.4700 4587.9210 0.973668

0.1770 666.9317 0.702979

0.1770 2752.396 0.974116

0.1760 8.039393 0.765302

is noninferior due to contradictions among objectives taken and has been obtained through evaluation of best objective weights. The novel formulation as multiobjective optimization problem has made it possible to quantitatively grasp trade off relations among conflicting objectives. Objective weights are evaluated by conventional statistical measures, which characterize the correlation coefficients matrix evolution. Two fundamental concepts of multiple criteria decision method are considered which are the contrast intensity of the alternative’s Table 15 “Best” power schedule and voltage profile of proposed method Generation

Load

Injected power

Voltage profile

Bus

PG

QG

PD

QD

P

Q

V

δ

1 2 3 4 5 6 7 8 9 10 11

1.7276 1.4476 0.9257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.5872 0.5224 0.3240 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.25 0.25 0.25 0.25 0.25 0.10 0.40 0.90 0.70 0.25 0.25

0.05 0.05 0.05 0.05 0.05 0.02 0.10 0.45 0.35 0.05 0.05

1.4776 1.1976 0.6757 −0.2500 −0.2500 −0.1000 −0.4000 −0.9000 −0.7000 −0.2500 −0.2500

0.0000 0.0000 0.0000 −0.0500 −0.0500 −0.0200 −0.1000 −0.4500 −0.3500 −0.0500 −0.0500

1.07000 1.08800 1.06200 0.95153 0.87598 0.91366 0.97645 0.93226 0.87888 1.00568 1.00708

0.00000 −0.08664 −0.08530 −0.22538 −0.32343 −0.28598 −0.20045 −0.23566 −0.29051 −0.16574 −0.11737


35

performance with respect to single objective and the conflicting nature of the objectives with each other. The extraction and exploitation of these two features is beneficial in the decision making. Objective weights derived in this paper are found to embody the information which is transmitted from all the objectives participating in the multiple criteria problem. The proposed method needs few noninferior solutions and hence gives computational efficiency. The validity of the method is demonstrated on IEEE 11-bus, 17-lines system comprising three generators. Results of the proposed method are compared with Brar et al. [8] which are better in terms of membership satisfaction achieved.

References 1. A. A. El-Keib, H. Ma, and J. L. Hart, “Environmentally constrained economic dispatch using the LaGrangian relaxation method,” IEEE Trans. on Power Systems, vol. 9, no. 4, pp. 1723– 1729, 1994. 2. J. S. Dhillon, S. C. Parti, and D. P. Kothari, “Stochastic economic emission load dispatch,” Electric Power Systems Research, vol. 26, pp. 179–186, 1993. 3. T. J. Manetsch, “An approach to optimal planning for SO2 emission compliance,” IEEE Trans. on Power Systems, vol. 9, no. 4, pp. 1921–1926, 1994. 4. R. Ramanathan, “Emission constrained economic dispatch,” IEEE Trans. on Power Systems, vol. 9, no. 4, pp. 1994–2000, 1994. 5. J. H. Talaq, F. El-Hawary, and M. E. El-Hawary, “A summary of environmental/economic dispatch algorithms,” IEEE Trans. on Power Systems, vol. 9, pp. 1508–1516, 1994. 6. P. K. Hota, R. Chakrabarti, and P. K. Chattopadhyay, “Economic emission load dispatch through an interactive fuzzy satisfying method,” Electric Power Systems Research, vol. 54, pp. 151–157, 2000. 7. M. Basu, “Fuel constrained economic emission load dispatch using Hopfield neural networks,” Electric Power Systems Research, vol. 63, pp. 51–57, 2002. 8. Y. S. Brar, J. S. Dhillon, and D. P. Kothari, “Multiobjective load dispatch by fuzzy logic searching weightage pattern,” Electric Power Systems Research, vol. 63, pp. 149–160, 2002. 9. S.-D. Chen and J.-F. Chen, “A direct Newton–Raphson economic emission dispatch,” Electrical Power & Energy Systems, vol. 25, pp. 411–417, 2003. 10. M. A. Abido, “Environmental/economic power dispatch using multiobjective evolutionary algorithms,” IEEE Trans. on Power Systems, vol. 18, no. 4, pp. 1529–1537, 2003. 11. J. B. Yang and D. Li, “Normal vector identification and interactive tradeoff analysis using minimax formulation in multiobjective optimization,” IEEE Trans. on System, Man and Cybernetics—Part A: Systems and Humans, vol. 32, no. 3, pp. 305–319, 2002. 12. D. Diakoulaki, G. Mavrotas, and L. Papayannakis, Determing objective weights in multiple criteria problems: The critic method, Computer Ops. Res., vol. 22, no. 7, pp. 763–770, 1995.

Evaluation of Best Weight Pattern for Multiple Criteria Load Dispatch

Evaluation of Best Weight Pattern for Multiple Criteria Load Dispatch

Suggest Documents

multiple criteria evaluation of entrepreneurship

MULTIPLE CRITERIA ANALYSIS FOR EVALUATION OF ... - CiteSeerX

Evaluation of Economic Load Dispatch Problem in ... - ARC Journals

multiple criteria evaluation of buildings with ...

A multiple-criteria framework for evaluation of ... - Semantic Scholar

multiple criteria decision making multiple criteria ...

multiple criteria decision making multiple criteria

Modular Multiple Dispatch with Multiple Inheritance - CiteSeerX

Modular Multiple Dispatch with Multiple Inheritance - CiteSeerX

artikel dispatch criteria - RePub, Erasmus University Repository

Economic Load Dispatch Using Population ... - Semantic Scholar

Multiple Dispatch as Dispatch on Tuples - Semantic Scholar

Multiple Criteria Evaluation of Quality and Optimisation of e-Learning

Economic Efficiency, Economic Load Dispatch ...

Solving Economic Emissions Load Dispatch ... - Semantic Scholar

Economic Load Dispatch Problem with Valve

Economic load dispatch using bacterial foraging ... - CiteSeerX

Economic Efficiency, Economic Load Dispatch, Incremental Cost

economic load dispatch using particle swarm

Multiple Criteria Evaluation and Ranking of Social Penetration of ...

Solving Economic Emissions Load Dispatch ... - Semantic Scholar

Economic load dispatch using bacterial foraging ... - CiteSeerX

Multiple criteria assessment of alternatives for built

multiple criteria evaluation of suppliers in company operating in ...