ED with Emission Objective

)

( ^(2.94)

x

x SO

BG i

Gi

SO P EC

E d

¦



)

( (2.95)

2

2( ) _CO

BG i

Gi

CO P EC

E d

¦



(2.96)

where

EC_NOx, EC_SO2, EC_CO2 are total system NO_x, SO₂ and CO₂ emissions, respectively (tons/h),

EC_NOx(P_Gi), EC_SO2(P_Gi), EC_CO2 (P_Gi) are NO_x, SO₂, and CO₂ emissions of the generator connected to bus i, respectively (tons/h),

F_T total system operating cost ($/h),

F(P_Gi) operating cost of the generator connected to bus i ($/h), BG set of buses connected with generators,

max

PGi maximum real power generation at bus i (MW),

min

PGi minimum real power generation at bus i (MW).

2.7.2 ED with Emission Objective

The emission objective is combined with the total cost objective using a weight factor as shown in equation (2.97).

( ) ( )

T Gi Gi

i BG i BG

F F P W E P

Œ Œ

=

Â

+ ◊

Â

^(2.97)

where W is the weight factor and E(P_Gi) is the emission function of unit i.

Note that the weight factors are in fact the emission factors. With the confl icting nature of these objectives, the multi-objective optimization technique can help handle the power dispatch optimization problem under the requirement of emission consideration. The multi-objective function can be expressed as follows:

Minimizing the generator fuel cost:

F_T

¦

BG i

PGi

F( ) ^(2.98)

and minimizing total NO_x emissions:

¦

BG i

Gi NO

NO E P

EC x x( ) (2.99)

and minimizing total SO_x emissions:



¦

BG i

Gi SO

SO E P

EC x x( ) (2.100)

and minimizing total CO₂ emissions:

¦

BG i

Gi CO

CO E P

EC ( )

2

2 (2.101)

subject to the power balance constraints in equations (2.17) and (2.18), and the generator minimum and maximum limit constraints:

Fig. 2.16 Membership function of total operating cost.

1

0.8

0.6

0.4

0.2

0

Total fuel cost

90 95 100 105 110 115

ȕ₁ Į₁

ȝ₁

max min

Gi Gi

Gi P P

P d d ^(2.102)

In this section, the solution technique used is fuzzy linear programming (FLP).

In FLP [20], the goal of decision-maker can be expressed as a fuzzy set and the solution space is defi ned by constraints that can be modeled by a fuzzy set. The multi-objective fuzzy minimization problem can be formulated as

Maximize{—₁ (x),—₂ (x),—₃ (x),—₄ (x)} (2.103) subject to B˜P_Gid~d

(2.104) and power balance constraints in (2.17) and (2.18), and low and high limits of P_Gi in (2.93).

PGi is the column matrix representing the set of real power generation of the generator connected to bus i; d is the vector representing a fuzzy objective function.

Each row of B in (2.99) is represented by a fuzzy set with the membership functions of—_i(x).—_i(x) that can be interpreted as the degree to which P_Gisatisfi es the fuzzy objective function. Here, —₁(x) is the degree of satisfaction of P_Gi for the objective function, whereas —₂(x) to —₄(x) are the degrees of satisfaction of P_Gi for the total system’s NO_x, SO_x and CO₂ emissions, respectively. In this chapter, the hyperbolic function is used to represent the nonlinear, S-shaped membership function. The function can be expressed as:

( ) tanh ^{i i}

i x a b i

m = ◊1 ÊÁËÊÁËB Pⁱ◊ ^Gij- + ˆ˜¯◊g ˆ˜¯+1

2 2 2 ^(2.105)

whereĮ_i,ȕ_i and Ȗ_i are the parameters representing the shape of —_i(x) depending on the decision maker and B_i is the row i of B.

To obtain the membership function of the objective function, Į₁ is defi ned as the minimum total generator fuel cost as solved by the linear programming (LP) ignoring emissions. On the other hand, ȕ₁ is the maximum total fuel cost found among the minimization solutions of other fuzzy objective functions. Ȗ₁ is obtained by Į₁/ȕ₁. Similarly, Į₂,Į₃, and Į₄ are the minimum SO₂, NO_x and CO₂ emissions determined by LP without considering total generator fuel cost and any other fuzzy objective functions; ȕ₂, ȕ₃, and ȕ₄ are the maximum NO_x, SO_x and CO₂ emissions found among the total fuel cost minimization solutions and fuzzy objective functions.

With the defi ned membership functions of objective functions and fuzzy constraints, the fuzzy optimization problem can be reformulated as

Maximize—' (2.106)

subject to —' ” —_i(x), for i = 1,…, 4 (2.107)

and 0 ” —' ” 1 (2.108)

and power balance constraints as well as low and high limits of P_Gi in (2.17), (2.18) and (2.93).

The FLP computational procedure is as described below:

Step 1: Solve the linear programming for individual objective functions.

Step 2: Compute the individual objective value of each case.

Step 3: Obtain Į_iandȕ_i from minimum and maximum of all objective values computed in Step 2.

Step 4: Solve the fuzzy linear programming of the multi-objective problem using Į_i and ȕ_i from Step 3.

Example 2.6

IEEE 30-bus system data are used as test data. The diagram of the network is depicted in Fig. 2.17. Generator fuel cost and SO₂, NO_x and CO₂ emission functions are given in Tables 2.2 and 2.3.

Fig. 2.17 IEEE 30 bus test system.

~ ~

~

1

2 8

3

9 8

6

11

7 5

4 2

15 14

12

18 19

13 17 16

20 23

24

30 10

27 29 25 26

22 21

~

~

~

For the implementation of the linear fuzzy programming method, these functions are linearized into 5 piece-wise linear functions.

Tables 2.4–2.7 address the dispatch results of minimum operating cost, SO₂, NO_x and CO₂ emissions, respectively. Table 2.8 shows the dispatch results of the multi-objective solution. The results show that the single objective approaches result in inferior results in the other objectives and a lower degree of satisfaction.

For example, the total cost minimization solution carries high SO₂, NO_X, and CO₂ emission values.

In this test case, the minimum operating cost solution results in the highest SO₂ emission of 7035.94 kg/h. On the other hand, the minimum NO_x emission results in the highest total operating cost and CO₂ emissions, of 6161.4 $/h and 6104.2 kg/h, respectively, while the minimum CO₂ solution results in the highest NO_x emission of 5187.67 kg/h.

By contrast, the proposed FMOPD effectively trades off between the objectives of total system operating cost, SO₂, NO_x and CO₂ emissions in a fuzzy reasoning sense leading to the best compromise solution while satisfying transmission line limits and transformer loading constraints. Note that the FMOPD results in a satisfaction degree of 0.881.

Table 2.2 Generator cost function of the IEEE 30 bus system.

Gen bus Min

(MW) Table 2.3 Emissions of the IEEE 30 bus system.

2 2 2 2 2

Table 2.4 Dispatch results for the minimum total operating cost condition.

** ** Generation Cost ** **

BUS P_GEN Cost Inc-Cost

(MW) ($/h) ($/MWh)

1 50.00 944.00015 21.60000

2 68.00 1733.87260 25.54960

5 36.00 872.19306 26.47760

8 50.00 935.49803 18.99999

11 43.07 812.15973 19.08103

13 40.00 735.59962 16.50000

Total Cost = 6033.32319 $/h Total SO2 = 7035.94301 kg/h Total NOX = 5060.62788 kg/h Total CO2 = 5917.43863 kg/h

Table 2.5 Dispatch results for the minimum SO2 emission condition.

** ** Generation Cost ** **

BUS P_GEN Cost Inc-Cost

(MW) ($/h) ($/MWh)

1 50.00 944.00013 21.60000

2 68.00 1733.87267 25.54960

5 50.00 1331.49953 28.25000

8 36.68 625.17418 17.43752

11 42.00 781.24482 18.83320

13 40.00 735.59969 16.50000

Total Cost = 6151.39102 $/h Total SO2 = 6709.76760 kg/h Total NOX = 5009.04237 kg/h Total CO2 = 5976.29335 kg/h

Table 2.6 Dispatch results for the minimum NOx emission condition.

** ** Generation Cost ** **

BUS P_GEN Cost Inc-Cost

(MW) ($/h) ($/MWh)

1 50.00 944.00000 21.60000

2 69.86 1791.70996 25.69841

5 43.00 1094.37920 27.33440

8 34.00 567.96080 17.13120

11 50.00 1027.75000 20.75000

13 40.00 735.60000 16.50000

Total Cost = 6161.39996 $/h Total SO2 = 6874.94078 kg/h Total NOX = 4897.48710 kg/h Total CO2 = 6104.20084 kg/h

Thus, FMOPD potentially can be applied to overcome the shortcomings of the method using emission factors in the combined objective.

2.8 ECONOMIC DISPATCH WITH TRANSMISSION

In document artificial-intelligence-in-power-system-optimization.pdf (Page 51-57)