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proven technology providing not only power but also heat to customers. For most co-generation units, there is a mutual dependency between heat and power, i.e.,

Table 2.19 Characteristics of the 10-unit system in example 2.10.

the heat production capacity depends on power generation and vice versa. Thus, the combined heat and power economic dispatch (CHPED) problem implies new complexities in the integration of co-generation units into the general power system economic dispatch since both power and heat demand must be satisfi ed. Although the combined heat and power systems are well known, only few research works have been reported in the literature in the area of CHPED problems [54-55].

The objective of the CHPED problem is to minimize the total operation cost of power and heat production while satisfying both power and heat load demands and unit power and heat output limits.

The system has three types of units including pure power, combined power and heat, and pure heat units. The feasible heat-power operation region of a combined power and heat unit is shown in Fig. 2.27, where the boundary curve ABCDEF determines the feasible region. Along the boundary, there is a trade-off between power and heat production. It can be seen that along the curve AB the unit reaches maximum power output. By contrast, the unit reaches maximum heat production along the curves BC and DC.

Table 2.20 Priority index of different fuels in the 10-unit system of example 2.10.

Table 2.21 Solutions for the 10-unit system with different load demands in example 2.10.

Unit no. Load of

1 189.7397 1 206.5188 2 216.5441 2 218.2502 2

2 202.3427 1 206.4573 1 210.9057 1 211.6627 1

3 253.8954 1 265.7392 1 278.5441 1 280.7230 1

4 233.0456 3 235.9531 3 239.0967 3 239.6316 3

5 241.8299 1 258.0178 1 275.5195 1 278.4975 1

6 233.0456 3 235.9531 3 239.0967 3 239.6316 3

7 253.2752 1 268.8636 1 285.7170 1 288.5847 1

8 233.0456 3 235.9531 3 239.0967 3 239.6316 3

9 320.3831 1 331.4876 1 343.4932 1 428.5203 3

10 239.3973 1 255.0564 1 271.9863 1 274.8671 1

Total cost ($) 481.273 536.239 574.381 623.809

Mathematically, the problem is formulated as

Power balance constraint 0

Heat balance constraint 0

Generation limit constraints

max F_i(P_Gi) cost function of pure power generating unit i,

Fig. 2.27 Typical heat-power feasible region for co-genera tion units.

P (MW)

Maximum fuel

Maximum heat

Maximum fuel

Heat extract H (MWth)

Power output

F_j(P_Gj,H_Gj) cost function of co-generating unit j, F_k(H_Gk) cost function of pure heat producing unit k, H_L system heat demand,

H_Gj heat production of co-generation unit j, H_Gk heat production of pure heat production unit k, NC number of co-generation units,

NH number of pure heat production units, NP number of pure power generating units, P_L system power demand, in MW,

P_Gi output power of pure power generating unit i, P_Gj output power of co-generation unit j,

P_Gi^max, P_Gi^min maximum and minimum output of pure power unit i,

H_Gk^max, H_Gk^min maximum and minimum output of pure heat production unit k, H_Gj^max(P_Gj), H_Gj^min(P_Gj) maximum and minimum heat output of co-generation unit

j, a function of output power,

P_Gj^max(H_Gj), P_Gj^min(H_Gj) maximum and minimum output power of co-generation unit j, a function of heat production.

For implementation in ALHN, the augmented Lagrangian function is formulated fi rst:

( ) ( , ) ( )

NP NC NH

i Gi i Gj Gj k Gk

i j k

NP NC NP NC

p D Gi Gj p D Gi Gj

i j i j

NC NH NC NH

h L Gj Gk h L Gj Gk

j k j k

L F P F P H F H

P P P P P P

H H H H H H

l b

l b

= = =

= = = =

= = = =

= + +

Ê ˆ Ê ˆ

+ ÁË - - ˜¯+ ÁË - - ˜¯

Ê ˆ Ê ˆ

+ ÁË - - ˜¯+ ÁË - - ˜¯

  Â

   Â

   Â

1 1 1

2

1 1 1 1

2

1 1 1 1

1 2

1 2

(2.188)

where

ȕ_p,ȕ_h penalty factors for power and heat demand balances, respectively, Ȝ_p,Ȝ_h Lagrange multipliers for power and heat demand balances, respectively.

There will be NP+2*NC+NH continuous neurons and two multiplier neurons required in ALHN. The energy function of ALHN is defi ned as follows:

, , , ,

V_Ȝ,h output of multiplier neuron representing Ȝ_h, V_Ȝ,p output of multiplier neuron representing Ȝ_p, V_j,h output of continuous neuron h(j) representing H_Gj, V_k,h output of continuous neuron h(k) representing H_Gk, V_i,p output of continuous neuron p(i) representing P_Gi, V_j,p output of continuous neuron p(j) representing P_Gj.

The dynamics for updating neuron inputs are derived as follows:

,

,

U_Ȝ,h input of multiplier neuron with corresponding output U_Ȝ,h, V_Ȝ,p input of multiplier neuron with corresponding output V_Ȝ,p, V_j,h input of continuous neuron h(j) with corresponding output V_j,h, V_k,h input of continuous neuron h(k) with corresponding output V_k,h, V_,i,p input of continuous neuron p(i) with corresponding output V_,i,p, V_j,p input of continuous neuron p(j) with corresponding output V_j,p.

Inputs of neurons are updated similarly to ALHN as applied to other ED problems.

Outputs of neurons are calculated by:

(

^{max min}

)

^{, min}

As shown in Fig. 2.27, maximum and minimum output power and maximum and minimum heat production of co-generation units are determined as follows:

{ }

max( ) min ( ) , ( )

Gj Gj Gj Gj AB Gj Gj BC

P H = P H P H _(2.200)

{ }

min( ) max ( ) , ( ) , ( )

Gj Gj Gj Gj CD Gj Gj DE Gj Gj EF

P H = P H P H P H ^(2.201)

{ }

max( ) min ( ) , ( )

Gj Gj Gj Gj BC Gj Gj CD

H P = H P H P (2.202)

0 )

min(

Gj

Gj P

H (2.203)

The initialization of ALHN in this problem is similar to that in the normal ED problem in Section 2.4.

Example 2.11

A system having one pure power generation unit, two co-generation units and one pure heat production unit have to cover a load demand of 200 MW power and 115 MWth heat. The data of system are described below:

F₁(P_G1) = 50P_G1 ($/h)

F₂(P_G2,H_G2) = 2650 + 14.5P_G2 + 0.0345P_G2² + 4.2H_G2 + 0.03H_G2² + 0.031P_G2H_G2 ($/h)

F₃(P_G3,H_G3) = 1250 + 36P_G3 + 0.0435P_G3² + 0.6H_G3 + 0.027H_G3² + 0.011P_G3H_G3 ($/h)

F₄(H_G4) = 23.4H_G4 ($/h) 0” P_G1” 150 MW 0” H_G4” 2695.2 MWth

Output limits of heat production and power generation for co-generation units 2 and 3 are given in Figs. 2.28 and 2.29.

The key factor in this problem is how to handle the power generation and heat production of co-generation units within their feasible operating zone enclosed by the non-convex functions.

For co-generation unit 2, the feasible operating region is enclosed by four lines intersecting at four points A, B, C and D. Therefore, the maximum and minimum power generation and heat production will combine these lines. The maximum and minimum limits of power and heat production are determined as follows:



2



2 2 2 max

2 11115 8

45 ) 1

( )

( _{G G G AB G}

G H P H H

P 

{ }

( ) ( )

min( ) max ( ) , ( )

max , 10354.24 .

. .

G G G G BC G G CD

G G

P H P H P H

H H

=

Ï ¸

= Ì - + - ˝

Ó ˛

2 2 2 2 2 2

2 2

1 2886120 134 1 17 8

75 2 104 8

{ }

( ) ( )

max( ) min ( ) , ( )

min , .

G G G G AB G G BC

G G

H P H P H P

P P

=

Ï ¸

= Ì - - ˝

Ó ˛

2 2 2 2 2 2

2 2

1 11115 45 1 7952 75 2

8 134

0 ) ( ₂

min

2 G

G P

H

The maximum and minimum power and heat generation limits are calculated similarly:

Fig. 2.29 Heat-power feasible operation region for co-generation unit 3 in example 2.11.

Fig. 2.28 Heat-power feasible operation region for co-generation unit 2 in example 2.11.

A

Power output

P (MW)

C D

B

H (MWth) Heat extract

0 104.8 180

247

215

98.8 81

P (MW)

H (MWth) A

C

D B

F E

15.9 32.4 75 135.6

Heat extract 125.8

110.2

44 40

Power output

{ }

ȕ_h = 0.0125, updating step sizes for neurons associated with power generation and heat production are 0.00015, that for multiplier neurons associated with power balance is 0.01, and for heat balance 0.0015.

The obtained solution is: P₁ = 0 MW, P₂ = 160 MW, H₂ = 40 MWth, P₃ = 40 MW, H₃ = 75 MW, and H₄ = 0 MWth with the total cost of $ 9,257.10. This solution is feasible since it satisfi es the constraints and the operating points of co-generation units 2 and 3 are within their feasible operating zones.

In document Dise˜ no de Sistemas Embebidos Reconfigurables Empleando Elementos de Interconexi´ on en Chip (página 147-151)