90

CHAPTER 7

STOCHASTIC ECONOMIC EMISSION

DISPATCH-MODELED USING WEIGHTING METHOD

7.1 INTRODUCTION

Nearly 70% of electric power produced in the world is by means of

thermal plants. Thermal power stations are the major causes of atmospheric

pollution because of the high concentration of pollutants they cause. The main

draw back of generating electricity from fossil fuel releases several

contaminants such as Sulphur Oxides, Nitrogen Oxides, and Carbon-di-Oxide

into the atmosphere and generate particulates. The contaminants cause

atmospheric pollution. The pollution minimization has attracted a lot of

attention due to the public demand for clean air.

In recent years rigid environmental regulation forced the utility

planners to consider emission control as an important objective. In this work

Economic Emission Dispatch (EED) is considered as a stochastic problem.

The cost and emission coefficients are considered as random variables and a

stochastic model is developed. Due to consideration of the problem as

stochastic there are three objectives to be minimized, cost, emission, variance

of power from its expected value. A multi-objective problem is formulated

considering all the three objectives.

91

7.2 PROBLEM FORMULATION

The first objective function to be minimized is the total operating

cost for thermal generating units in the system. A quadratic operating cost

curve is assumed.

N

21 i i i i i

i 1

F a p b p c

(7.1)

where F1 is the cost function to be minimized

ai,bi,ci are the cost coefficients of the ith generator

N is the total number of generators

Pi is the power output of ith generator.

7.2.1 Stochastic Model

A Stochastic model of function F1 is formulated by considering cost

coefficients and load demand as random variables. By taking expectation the

stochastic model can be converted into its deterministic equivalent. The

random variables are assumed to be normally distributed and statistically

dependent on each other. As the random variables are statically dependent

both variance and covariance of generated power exists. The expected value

of operating cost is obtained by expanding the operating cost function using

Taylor’s series, about mean. The expected cost is:

N

21 i i i i i

i 1

E(F ) E a P b P c

(7.2)

N

21 i i i i i

i 1

F [(E(a P ) E(b P ) E(c )]

(7.3)

N

21 i i i i i

i 1

F [(E(a )E(P ) E(b )E(P ) E(c )]

(7.4)

92

N 2

1 i i ii i ii i i i ii 1

F (a [P var(P )] 2P cov(a ,P )) b (P cov(b ,P )) c

(7.5)

where i i ia ,b ,c are the expected cost coefficient of the ith generator

iP is the expected value of power generated by ith generator

1F is expected cost function to be minimized

ivar(P ) is variance of power Pi

i icov(a ,P ) is the covariance of random variable ai and Pi

i icov(b ,P ) is the covariance of random variable bi and Pi.

The expected operating cost function is represented as:

N 2

1 ii i i i i i i i, ii ii 1

F a p b p c a var(P ) 2P cov(a ,p ) cov(b p ) (7.6)

By substituting for variance and covariance by its coefficient of variation and

correlation coefficient the equation (7.6) can be rewritten as:

i i i i i i i i i

N 221 i ii i iP a P a p b P b P

i 1

F (1 C 2R C C )a P (1 R C C )b P c

(7.7)

where a b Pi i iC ,C ,C is coefficient of variation of random variables ai, bi, Pi

a Pi iR is correlation coefficient of random variable ai and Pi

b Pi iR is correlation coefficient of random variable bi and Pi.

7.2.2 Expected NOx Emission

The amount of NOx emission is given as a function of generator

power output Pi which is quadratic.

93

N

22 i i i i i

i 1

F d p e p f

(7.8)

where F2 is the emission function to be minimized

di, ei, fi are the emission coefficients of the ith generator

N is the total number of generators

Pi is the generator power output of ith generator.

A stochastic model of function F2 is formulated by considering

emission coefficients and load demand as random variables.

. N 2

2 i i i i ii 1

E(F ) E d P e P f

(7.9)

N 2

2 i i i i ii 1

E(F ) E(d P ) E(e P ) E(f ) (7.10)

N 2

2 i ii i i i ii 1

F [d P d var(P ) 2P cov(d ,P )

i ii i ie P cov(e ,P ) f ] (7.11)

where ii id ,e ,f are the expected emission coefficient of the ith generator

iP is the expected value of power generated by ith generator

2F is emission function to be minimized

ivar(P ) is variance of power Pi

i icov(d ,P ) is the covariance of random variable di and Pi

i icov(e ,P ) is the covariance of random variable ei and Pi .

94

i i i i i i i i i

N 222 i i ii iP d P d P e P e P

i 1

F (1 C 2R C C )d P (1 R C C )e P f

(7.12)

where d e Pi i iC ,C ,C is coefficient of variation of random variables di, ei, Pi

d Pi iR is correlation coefficient of random variable di and Pi

e Pi iR is correlation coefficient of random variable ei and Pi.

Covariance of bivariate random variables is considered positive or

negative. Covariance is represented by correlation coefficient; it is varied

from –1.0 to 1.0. One pair of random variable is considered at a time while

the rest of random variables are considered independent of each other

(uncorrelated).

7.2.3 Expected power deviation

Generator outputs Pi are treated as random variables. There is

variation in random variable from their expected value which results in

surplus or deficit power. Expected deviations are proportional to the

expectation of the square of the unsatisfied load demand.

2N N

3 D Li ii 1 i 1

F var P E P (P P )

(7.13)

where DP is the expected power demand

LP is expected transmission loss.

Substituting power balance equation (7.24) in (7.13) we get

N N

23 ii

i 1 i 1

F E[ P P ]

(7.14)

95

2N

3 iii 1

F E (P P )

(7.15)

N N N

23 i i ji i j

i 1 i 1 j 1i j

F E (P P ) 2 (P P )(P P )

(7.16)

As random variables are assumed as normally distributed and

statistically dependent expected deviations are given by equation (7.18)

N N N

3 i i ji 1 i 1 j 1

i j

F var(P ) 2cov(P ,P )

(7.17)

i i j i j

N N N223 i i jP P P P P

i 1 i 1 j 1j i

F C P R C C P P

(7.18)

where P Pi jR is correlation coefficient of random variable Pi and Pj

7.2.4 Expected Transmission Loss

The transmission power loss expressed through the simplified well

known loss formula expression as a quadratic function of the power

generation are given by equation (7.19) as:

N N

L i ij ji 1 j 1

P PB P

(7.19)

where PL is the transmission loss

Bij is the loss coefficient.

96

Taking expected value of transmission loss

N N

L i ij ji 1 j 1

E[P ] E PB P

(7.20)

N N N

2L i ii i ij j

i 1 i 1 j 1j i

P E P B E PB P

(7.21)

The expected transmission loss is represented as equation (7.22)

N N N N N N2

i ii ijL ii i i ij j i ji 1 i 1 i 1 j 1 i 1 j 1

j i j i

P P B B var P P B P B cov(P ,P )

(7.22)

where LP is the expected transmission loss

ijB is the expected loss coefficient.

Substituting for variance and covariance equation (7.22) can be

written as equation (7.23).

i P i ji j

N N N22L ii i i ij jP p P P

i 1 i 1 j 1j i

P (1 C )B P (1 R C C )P B P

(7.23)

7.2.5 Equality and Inequality Constraints

Real power balance is the equality constraint to be satisfied is given

by equation (7.24).

N

i D Li 1

P P P

(7.24)

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Expected power generation limits are the inequality constraints to be satisfied.

min maxi i iP P P (7.25)

where miniP and max

iP are expected lower and upper limits of ith generator

power output.

The multi-objective problem is formulated as follows:

i i i i i i i i i

i i i i i i i i i

i i j i j

N 221 i ii i iP a P a P b P b P

i 1N 22

2 i i ii iP d P d P e P e Pi 1N N N22

3 i i jP P P P Pi 1 i 1 j i

j i

Minimize

F (1 C 2R C C )a P (1 R C C )b P c

F (1 C 2R C C )d P (1 R C C )e P f

F C P R C C P P

N

i D Li 1min maxi i i

3

k kk 1

Subject to

P P P

P P P (i 1.......N)

w 1 w 0

To generate the non-inferior solution of multi objective optimization problem,

PSO is used. wK is the levels of the weighting coefficients. The values of

weighting coefficients vary from 0 to 1 for each objective. The weight

w1, w2, w3 are varied in the range 0 to 1 in such a way that their sum is 1.0.

This approach yields meaningful result to the decision maker when solved

many times for different values of wk, (k = 1,2,3). This provides K number of

non-inferior solution which are pareto optimal non dominating. To identify

one optimal solution out of K solution fuzzy membership satisfaction index kD is used.

(7.26)

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7.3 PSO ALGORITHM FOR ECONOMIC EMISSION

DISPATCH

The following steps are involved in solving the deterministic and

stochastic model of economic and emission dispatch problem.

Step 1: Read total number of thermal units, cost coefficients,

emission coefficients, B coefficients, coefficient of variation of each plant,

correlation coefficients of random variables, maximum number of iterations,

population size, acceleration constants c1 and c2, inertia weight wmin and wmax

Step 2: Confine the search space. Specify the lower and upper limits

of each decision variable.

Step 3: Initialize the individual of population. The velocity and

position of each particle should be initialized with in the feasible decision

variable space Xi=[X1, X2, X3……XN].

Step 4: Feed or generate the weight (wk=1,2…n) where n is the

number of objectives. Here n=3. Sum of w1, w2, w3 must be equal to one.

Step 5: For each individual Xi of the population the transmission

loss PLi is calculated using B-coefficients.

Step 6: Evaluate the fitness of each individual Xi in terms of pareto

dominance.

Step 7: Record the non dominated solutions found sofar and save

them in archive.

Step 8: Initialize the memory of each individual where the

personnel best position ( t )best idp is stored.

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Step 9: Find the best particle out of the population and store its

position as (t )best dg .

Step 10: Set iteration count t=1

Step 11: Update the velocity of each particle Xi using the

equation (4.1).

Step 12: Update the position of each particle Xi using the

equation (4.2).

Step 13: Check whether the new particles are with in the feasible

region. If any element violates inequality constraints then the position of the

individual is fixed to its minimum/maximum limits according to

equation (4.5).

Step 14: Check the power balance constraints. Any violations are

penalized by adding penalty.

Step 15: Calculate the fitness function of the new individual.

Step 16: Update the archive which stores the non dominated

solution.

Step 17: Update memory of each particle using the equation (4.6).

Compare new particles fitness (t 1)idX with particles ( t )

best idp . If the current value

is better then the previous value then set (t 1)best idp to the new value and its

location equal to current location in d dimensional space. Compare new

particle’s fitness with the population’s overall best (t )best dg particle fitness. If the

fitness value of the new particle is better, reset (t 1)best dg to current particle array

index and value.

100

Step 18: Iteration = Iteration +1

Step 19: The algorithm repeats Step 11 to Step 18 until a sufficient

good fitness or a maximum number of iterations/epochs are reached. Once

terminated, the algorithm outputs the points of (t)best dg and f( (t)

best dg ) as its

solution.

7.4 TEST SYSTEM AND RESULTS

The validity of the proposed method is illustrated on a six generator

sample system Dhillon et. al (1993) and the result obtained from the proposed

algorithm is compared with that of Newton-Raphson iterative method. For

deterministic case coefficient of variation and correlation coefficient is zero.

The coefficient of variation and correlation coefficient assumed are:

Cai = Cbi = Cdi = Cei = Cpi = 0.1 (i = 1...6).

Two cases are considered:

Case 1: With dependent variables: In this case all the random

variables are considered dependent on each other

a P b P c P d P P Pji i i i i i i i iR R R R R 1.0 (i=1,2….6,j=1,2,….6)

Case 2: With independent variables: In this case all the random

variables are considered independent of each other

a P b P c P d P P Pji i i i i i i i iR R R R R 0.0 (i =1,2…6, j = 1,2…6).

The best solution obtained for expected demand of 500 MW, 700

MW, 900 MW for both the cases is shown in Table 7.4.

101

Table 7.1 gives the fuel cost coefficient and the capacity limits of

the thermal generating units. Table 7.2 gives the emission coefficients for the

thermal generating units. Table 7.3 gives the loss coefficients of the generator.

Table 7.1 Expected fuel cost coefficients and capacity limits

Generator No. ia ib ic maxiP (MW)

miniP (MW)

1 0.15274 38.53973 756.79886 10 125

2 0.10587 46.15916 451.32513 10 150

3 0.02803 40.39655 1049.99770 35 225

4 0.03546 38.30553 1243.53110 35 210

5 0.02111 36.32728 1658.56960 130 325

6 0.01799 38.27041 1356.65920 125 315

Table 7.2 Expected NOx emission coefficients

Generator No. id ie if

1 0.00419 0.32767 13.85932

2 0.00419 0.32767 13.85932

3 0.00683 0.54551 40.2669

4 0.00683 0.54551 40.2669

5 0.00461 0.51116 42.89553

6 0.00461 0.51116 42.89553

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Table 7.3 Expected loss coefficients

0.002022 -0.000286 -0.000534 -0.000565 -0.000454 0.000103

-0.000286 0.003243 0.000016 -0.000307 -0.000422 -0.000147

-0.000533 0.000016 0.002085 0.000831 0.000023 -0.000270

-0.000565 -0.000307 0.000831 0.001129 0.000113 -0.000295

-0.000454 -0.000422 0.000023 0.000113 0.000460 -0.000153

0.000103 -0.000147 -0.000270 -0.000295 -0.000153 0.000898

Table 7.4 Comparison of best optimal solution

Load MW

PSO Newton Raphson Cost

(Rs/hr) NOx

(kg/hr) Risk

(MW)2 Cost

(Rs/hr) NOx

(kg/hr) Risk

(MW)2

Case 1

500 28525.49 304.10 2688.05 28550.15 312.51 2674.56

700 38951.57 506.6 5420.01 39070.74 528.44 5401.18

900 50634.68 798.38 9225.77 50807.24 864.06 9110.65

Case 2

500 28430.29 286.68 566.01 28476.63 287.48 558.75

700 38910.66 492.80 1117.33 39010.74 493.97 1114.28

900 50620.7 798.11 1862.71 50854.86 800.62 1861.07

103

Table 7.5 Expected optimal generation schedule using PSO

Load (MW)

1P (MW)

2P (MW)

3P (MW)

4P (MW)

5P (MW)

6P (MW)

LP (MW)

Case 1 500.0 57.34 37.99 41.03 74.67 179.2 128.31 18.5 700.0 88.72 58.51 66.55 110.00 240.28 172.39 36.48 900.0 124.67 92.89 82.07 153.85 286.94 220.09 60.8

Case 2 500.0 63.82 44.63 35.00 86.43 158.42 129.40 17.6 700.0 88.72 58.51 66.55 110.0 240.28 172.39 36.48 900.0 123.51 89.49 88.49 151.09 288.76 220.65 63.08

Table 7.6 Expected optimal generation schedules using NR method

Load (MW)

1P (MW)

2P (MW)

3P (MW)

4P (MW)

5P (MW)

6P (MW)

LP (MW)

Case 1 500.0 59.87 39.65 35.00 72.39 185.24 125.00 17.16 700.0 85.92 60.96 53.90 107.12 250.50 176.50 34.92 900.0 122.00 86.52 59.94 140.95 325.00 220.06 54.49

Case 2 500.0 59.67 41.41 51.87 83.26 157.83 126.75 20.79 700.0 89.09 65.47 69.71 116.07 223.59 175.70 39.66 900.0 124.61 92.96 87.09 152.74 286.84 220.77 65.03

Table 7.4 gives the comparison of best optimal results obtained

using PSO and NR method. Table 7.5 gives the expected optimal generation

schedule using PSO. Table 7.6 gives the expected optimal generation

schedule obtained using NR method.

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Table 7.7 gives the deterministic results obtained using PSO. The

corresponding generation schedule is given in Table 7.8.

Table 7.7 Deterministic results using PSO

Load (MW) Cost (Rs/hr) NOx (kg/hr) Transmission loss (MW)

500 28289.99 286.14 17.67

700 38744.2 491.94 36.1

900 50510.3 790.2 64.0

Table 7.8 Deterministic generation schedule using PSO

Load (MW)

P1 (MW)

P2 (MW)

P3 (MW)

P4 (MW)

P5 (MW)

P6 (MW)

PL (MW)

500.0 61.68 38.79 39.11 77.41 172.65 127.67 17.67

700.0 87.2 61.23 68.98 108.0 240.22 171.42 36.1 900.0 121.1 89.19 89.68 151.87 289.09 222.41 64.0

Table 7.9 shows the Minimum, maximum values of each objective

for Case1. Table 7.10 shows the weight vector for the best optimal solution in

PSO and the corresponding kD . min

1F is obtained by giving full weightage to 1F

and neglecting other objectives. This is minimum cost dispatch. min2F is

obtained by giving full weightage to 2F . This is minimum emission dispatch. min3F is obtained by giving full weigtage to 3F . This is minimum risk dispatch.

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Table 7.9 Minimum and maximum values of objective function (Case 1)

Load (MW)

min1F

(Rs/hr)

max1F

(Rs/hr)

min2F

(kg/hr)

max2F

(kg/hr)

min3F

(MW)2

max3F

(MW)2

500 28172.25 29428.39 267.5 354.2 2576.2 2881.46

700 38775.48 39762.64 469.3 556.7 5317.6 5623.78

900 50470.65 51434.49 766.3 854.3 9122.7 9430.77

Table 7.10 Values of weight and kDµ (Case 1)

Load (MW) w1,w2,w3 kDµ

500 0.5,0.3,0.2 0.01621

700 0.4,0.3,0.3 0.01546

900 0.4,0.4,0.2 0.01557

By taking the weight w1,w2,w3 as 0.4,0.3,0.3 the percentage relative

deviation in 1F from their deterministic value with respect to

a P b P P Pi i i i i jR ,R ,R (i j) is calculated and shown in Figure 7.1. The percentage

deviation in cost increases as RPiPj is varied from positive to negative values.

There is a decrease in percentage deviation of cost when RaiPi, RbiPi is varied

from positive to negative value.

With the same weight percentage relative deviation in 2F from their

deterministic value with respect to d P e P P Pi i i i i jR ,R ,R (i j) is calculated and

shown in Figure 7.2. The percentage deviation in emission increases as RPiPj is

varied from positive to negative values. There is a decrease in percentage

deviation of emission when RdiPi, ReiPi is varied from positive to negative

value.

106

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

-1 0 0 0.5 1

Correlation coefficient

Perc

enta

ge v

aria

tion

in e

xpec

ted

cost

variations in Rpipj

variation in Rbipi

variation in Raipi

Figure 7.1 Percentage variations in expected cost with respect to

correlation coefficients

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1 0 0 0.5 1Correlation coefficients

Perc

enta

ge v

aria

tion

emiss

ion

variation in Rpipj

variation in Rdipi

variation in Reipi

Figure 7.2 Percentage variations in expected emission with respect to

correlation coefficients

107

Figure 7.3 shows the variation of the cost with iteration. Figure 7.4

shows the variation of emission with iteration. The trade of curve between

cost and emission is shown in Figure 7.5.

With various combination of learning factors the convergence

characteristic of the swarm is studied. It is found that the learning factor of

c1 = 2, c2 = 2 gives good convergence. Figure 7.6 shows the convergence for

different sets of learning factor. Figure 7.7 shows the convergence

characteristic for different population size. The population size of 50 gives

good convergence.

Figure 7.3 Fuel cost characteristic of economic emission dispatch using

PSO

104

Cos

t

Iteration

108

Figure 7.4 Emission characteristic of economic emission dispatch using

PSO

Figure 7.5 Trade off curve between cost and emission using PSO

Emis

sion

Iteration

Emis

sion

Cost

109

Figure 7.6 Convergence characteristic for different learning factor

using PSO

Figure 7.7 Convergence characteristic of various population size for

economic emission dispatch

104

Iteration

Iteration

Fitn

ess

Fitn

ess

104

110

7.5 CONCLUSION

PSO is capable of reducing the cost and emission, but there is

increase in risk. Compared to the saving in cost and reduction in the emission

level the increase in risk is tolerable. So the proposed algorithm is efficient in

finding the best optimal solution. For the loads considered in this problem

stochastic cost on an average is 0.3% higher and emission is 0.865% higher

compared to deterministic results. Table 7.11 gives the average % variation of

stochastic results obtained using PSO with NR method. The execution time is

21 seconds in PIV 3GHz system.

Table 7.11 Average results for all the three loads using PSO

Details Case 1 Case 2

Cost reduction (Rs/h) 0.30% 0.25%

NOx reduction (kg/h) 4.71% 0.177%

Risk increase (MW2) 0.59% 0.606%