Time Dependent Production

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Three different application examples involving time dependence in different ways are analyzed using the executable tool "VarloadTL.exe." They all start from a known design point in the steady state. The first predicts the detailed performance of a power system at a sought load ratio (load/design load). The second estimates the penalty of off-design performance of a cogeneration system given demand profiles. The third computes the optimal operation of a facility of a number of single-product systems producing the same product. Only the highlights of the systems and their results are presented. Running the executable tools and referring to detailed flow diagrams obtain the detailed results.

7.2.1 Predicting the part-load of a simple combined cycle

The combined cycle considered is shown in Figure 7.9 from three viewpoints. In Figure 7.9a, the cycle is viewed as a connectivity structure of 10 components and 17 states. In Figure 7.9b, it is viewed from its control strategy. A simple control philosophy for part-load operation is assumed. The driving shaft speed, the pressure at exit of the gas turbine, the condenser pressure, a state of saturated liquid at exit of condenser and a state of saturated vapor at exit of the boiler section are set at the full load values (design point). A shaft speed sensor adjusts the compressor inlet guide vanes IGV (Sehra et al., 1992). A firing temperature sensor adjusts the fuel flow to combustor. The set value of the firing temperature may deviate from the design point to match the gas turbine speed with that of the compressor. The water level in boiler separating drum controls the rate of steam flow in the bottoming cycle. The condensate level in the condenser controls the rate of cooling water. Figure 7.9c views the computational scheme for part-load operation.

Table 7.3 The influence of each of the five system parameters

Parameter Parameter First Law Second Law Co-Generated

Value Efficiency Efficiency (Net Work + Heat (kW)

Net Work/Fuel (hhv) Heat Exergy)/Fuel Exergy

1) Extent of fuel cell reaction XTR9

2) Fuel cell efficiency ETA9

3) Extent of reformer reaction XTR7

4) Reformer excess steam ratio EXCS7

5) Subcooling by control exchanger 17 DTSUB

4 0.4581 0.5241 89.8

8 0.4588 0.5252 89.8

12 0.4590 0.5255 89.8

16 0.4591 0.5257 89.8

20 0.4591 0.5258 89.8

A sequence of iteration-free computation was not possible. Tearing at four locations has to be introduced with four variables to be computed simultaneously.

7.2.1.1 Convergence: Referring to Figure 7.9, four masses are manipulated until the changes of four variables converge to zero simultaneously. The masses are those of air at 1, of combustion gas at 3, of steam of the bottoming cycle at 6 and of cooling water at 15. Tearing at 18, 3, 10, and 8 are introduced. The difference between load and power AW at 18, AM at 3, AH at 10 and AH at 8 are converged to allowable deviations. The deviations are set at iO.Ol the values at full load. Allowable deviations were reached in 3 or 4 iterations. The sets of manipulated variables, the tearing and the stm turbine stm turbine

Figure a Components and States

Figure a Components and States

Figure c

Computational

Strategy

Figure c

Computational

Strategy

Ojj~j inlet guide vanes IGV

Figure b The Control Philosophy

Figure b The Control Philosophy

Figure 7.9 The combined cycle for off-design analysis.

deviations are not unique but their number is. The selection depends on the computational procedure. The tool "VarloadTL.exe" shows how to describe "Tearing" and how convergence progresses for a given load ratio.

7.2.1.2 Design and off-design results: Table 7.4 captures some of the features of a feasible design, an optimal version and the part-load performance of the optimal version. The general trend is reduction of loadings and efficiencies with the reduction of load. However, few irregularities do occur. In order that the gas turbine speed matches that of the compressor, the firing temperature increases from the design value of 871°C to 906-908°C. The excess air ratio decreases from 3.06 to 2.73-2.81. The loading increases for the cooling water pump and for the superheater. The effectiveness of heat exchangers may increase or decrease. The effectiveness of the boiler increases, that of the economizer decreases, those of the superheater and the condenser decrease then increase.

Table 7.4 Design vs. off-design performance features.

Run Item

Parameter Design Load/Full Optimized Case Feasible Optimized 0.9

Loadings (Heat Q, Power P (MW), Mass M (kg/s), Cost C (S/h))

Devices:

Loadings (Heat Q, Power P (MW), Mass M (kg/s), Cost C (S/h))

Devices:

1) Compn (air) P

96.3

59.7

47.0

42.7

39.4

36.8

34.8

2) Expn (gas) P

153.3

116.6

94.1

82.1

71.1

6.9

51.4

3) Expn (stm) P

26.3

26.4

28.1

27.5

26.9

26.2

25.3

4) Pmp (feed) P

0.155

0.17

0.181

0.178

0.175

0.172

0.169

5) Pmp (c.w.) P

0.17

0.10

0.125

0.128

0.131

0.134

0.137

6) Combn M

4.0

3.50

3.18

2.95

2.36

2.41

2.18

7) Hx (suphtr) Q

12

14.4

16.0

16.2

16.4

16.6

16.7

8) Hx (blr) Q

46.1

43.0

44.2

43.1

41.9

4.6

39.1

9) Hx (econzr) Q

25.5

21.3

2.8

19.7

18.7

17.6

16.5

10) Hx (condnr) Q

57.5

56.4

53.2

51.7

5.3

48.7

47.1

System:

Fuel Q

212.2

186.4

17.0

155.8

141.9

128.2

114.6 [Q

Net pwr P

82.97

82.97

74.9

66.6

58.3

5.0

41.6

Compnts C

718

707

707

707

707

707

707

Profit C

940

1121

949

748

543

336

128

Efficiencies

Devices:

1) Compn rju

0.82

0.892

0.84

0.79

0.73

0.66

0.59

2) Expn gas na

0.87

0.907

0.91

0.90

0.89

0.88

System:

First law r) Second law rj Production cost cPP in c/kWh

0.85

0.935

0.75

0.945

0.75

0.667

3.52

3.06

0.010

0.002

0.851

0.987

0.005

0.001

0.010

0.011

0.926

0.983

0.005

0.005

0.010

0.087

0.668

0.986

0.005

0.002

0.010

0.473

0.940

0.835

0.005

0.010

0.500

0.4247 0.4835

0.94 0.93 0.63 2.72 0.001 0.952 0.001 0.012 0.992 0.004 0.090 0.960 0.001 0.503 0.820 0.010 0.426

0.4408 0.4789 3.22

0.93 0.92 0.63 2.72 0.001 0.984 0.001 0.010 0.996 0.003 0.080 0.949 0.001 0.481 0.835 0.010 0.402

0.4275 0.4644 3.40

0.92 0.90 0.62 2.73 0.001 0.928 0.001 0.009 0.999 0.003 0.072 0.938 0.001 0.459 0.835 0.010 0.430

0.4106 0.4461 3.65

0.91 0.89 0.62 2.76 0.001 0.916 0.001 0.008 0.999 0.002 0.064 0.927 0.001 0.436 0.866 0.009 0.358

0.3896 0.4233 3.98

0.90 0.86 0.61 2.81 0.001 0.905 0.001 0.007 0.999 0.002 0.056 0.915 0.001 0.411 0.885 0.008 0.335

0.3633 0.3946 4.45

Figure 7.10 Flowsheet of first configuration: thermally-driven cooling and heating.

7.2.2 A gas turbine system for power, cooling and heating

7.2.2.1 The investigated systems: Two main configurations, Figures 7.10 and 7.11 are considered. The first configuration, Figure 7.10, provides cooling and heating thermally. The second configuration, Figure 7.11, provides them mechanically. The numbers assigned to the devices and to the states for the purpose of computation are removed for clarity. Both configurations have the same ground rules. They both burn natural gas. They both have the same fuel and product prices and the same capital recovery rate.

The first configuration (Figure 7.10) consists of a simple gas turbine power unit, heat recovery steam generator, a single stage LiBr/H20 absorption refrigeration subsystem for cooling, a steam heating coil for heating and a water heater for domestic use. For this cooling water dump condenser

R142I) VC refrgn unit evaporator t chilled water heating watei fuel compr combstr

stm turbine gas turbine unit

firing fj|

dumpingj j superht t chilled water superht

Active states

May not be active

Components

33

boiler

Process devices

28

States

60

Active states

48

economzr

Decision variables

90

(Efficiency decns

51)

condenser

R142b HP unit

Si evapr/condnsr dump nden!

steam power bottoming unit

cooling water feed pump

Figure 7.11 Flowsheet of second configuration: mechanically-driven cooling and heating.

configuration two recovery ideas are introduced: a gas turbine regenerator and a vapor-compression cooling unit. The configuration has in total 33 devices, 60 states and 106 decision variables. The decision variables consist of 53 efficiency parameters (local decisions), 5 global. The rest are boundary and ground-rules decisions.

The second configuration (Figure 7.11) consists of a gas turbine operating as a combined cycle producing only power. Cooling is supplied by a mechanical vapor-compression system and heating is provided by a heat pump. Refrigerant R142b is the working fluid in both mechanical systems. The heat pump is fed by part of the combined cycle condenser heat. The configuration has 28 process devices, 48 active states and 90 decision variables. The decision variables are 51 efficiency decisions and 7 global. The rest are boundary and ground-rules decisions.

403530252015105

power cooling

;ooling power heatine cooling power pooling power

.heating hot wtr heating heating

Demand Profiles Main Features Of Demand Profiles

Max and Min Power MW 30 & 8 Load Factor 0.6778

Max and Min Cooling MW 36& 20 Load Factor 0.7940

Max and Min Heating MW 15& 3 Load Factor 0.4389

Max and Min Hot wtr MW 10& 2 Load Factor 0.6000

Average delivery MW of MW Exergy

61.49 22.96

403530252015-

Power 2

Coolirfg 1

Heating 1

12 16

20 24

Production Profiles Of The Reference Cases Of Configurations 1 And 2

Figure 7.12 Demand and available profiles: reference cases of configurations 1 and 2.

The local and the global decisions can be handled by automated optimization. All the decisions can be changed manually. However, the boundary and the fixed decisions are kept constant throughout the investigation.

Figure 7.12 shows the demand profiles of power, cooling, heating and hot water assumed as given and the values that characterize their variability. The figure also shows the available power-matching cooling, heating and hot water profiles for the reference case of the first configuration and the power profile for that of the second configuration.

The period r of repeatable pattern is taken 24 h. The minimum duration of a load is taken as 1 h. Neither the period nor the duration put any limitation to the method as described in the section of time-dependent production. In evaluating the ideal off-design operation, it is helpful to think of the load factors of the demand profiles as being also time duration ratios of on-off operation at design conditions. The power-to-fuel efficiency obeys a relation described by Equation 6.4. The extrapolated zero-load efficiency a is set at 0.2 that of the design efficiency and the design efficiency is set at the peak power load.

Each configuration is first treated as base-load production system at four efficiency levels. The highest efficiency assumed high firing temperature using cooled-blade turbine. At each level the system is run through the repeatable pattern of the load profiles to compute the off-design fuel penalty for the assumed system efficiency-to-load relation. The objective function J is the production cost and is computed on two steps according to the Equation 6.3.

For the first configuration, mismatches are simply handled by dumping and re-firing. Hot gases are vented at the exit of the gas turbine when heat available is more than demanded and re-firing more fuel at the exit of the gas turbine when heat available is less than demanded. When heating and cooling demands are satisfied and excess heat goes to the domestic hot water, hot water is dumped.

A regenerator is proposed to recover heat from the vented gas by heating the air before entering the gas turbine combustion chamber.

A vapor-compression refrigeration cycle for cooling (R12) is proposed to reduce re-firing when the absorption refrigeration cooling is less than demanded.

No measures are taken to recover heat from the dumped domestic hot water.

For the second configuration, the fuel input is controlled to meet the sum of the electrical power, the power for the mechanical vapor compressor and the power for the heat pump. No dumping or re-firing is needed for this configuration. Another advantage is that the sum of the three loads tends to have less variability than the individual demands.

The efficiency levels of the second configuration are higher than the first and so is the cost.

7.2.2.2 Results using the simplified off-design system-efficiency: Several runs were made to investigate the influence of a number of factors on off-design performance. These factors include: changing the design points of each configuration, changing the design points of the added off-design recovery devices and changing the quoted overall system performance equation as function of the power loading ratio. The results are summarized in the cost-efficiency diagram of Figure 7.13. The results are also tabulated in Table 7.5. The following observations may be noted:

• The handling of the time-dependent problem as a base-load problem and an off-design penalty problem appears encouraging. In this application the two problems can be assumed decomposed.

• Variable operation reduces efficiency and increases cost per unit product or unit reference product in more than one product system.

• Higher base-load design efficiency allows higher operation efficiency.

• Added recovery devices to reduce the fuel penalty of dumping and re-firing of the first configuration depend on the system design efficiency. This complexity emphasizes the importance of a screening method.

- For example, for the reference design efficiency of about 30%, the regenerator and the vapor compression solutions were cost-ineffective. Dumping and re-firing seem to be the answer.

- For the lower efficiency of about 20%, the VC solution was not needed because of the absence of re-firing, and the use of the regenerator became cost-effective

Efficiency = Sum of Energies Delivered / Fuel Exergy

Figure 7.13 The cost-efficiency diagram: variable-load cost penalty vs. base-load cost.

Efficiency = Sum of Energies Delivered / Fuel Exergy

Figure 7.13 The cost-efficiency diagram: variable-load cost penalty vs. base-load cost.

Table 7.5 Penalties of time-dependent production systems.

Parameter Configuration 1 (Venting/Refiring) Configuration 2

Reference Optimum Low Eff HiFire Optm Reference: Optimum Low Eff HiFire Optm

Reference Optimum Low Eff HiFire Optm Reference: Optimum Low Eff HiFire Optm

Design

1551

1279

2597

1085

1515

1434

2036

1194

Ideal operation

1182

997

1958

819

1237

1176

1636

975

Expected operation

1309

1221

2109

1040

1299

1233

1723

1023

2. Fuel rate (MW)

Design

114.4

87.5

198.2

82.5

101.5

92.0

147.8

78.4

Ideal operation

77.6

59.3

134.3

55.9

73.7

66.0

107.8

56.5

Expected operation

90.3

81.7

149.4

78.0

79.9

71.9

116.4

61.2

3. Devices ($/h)

(design, operation)

407

404

615

260

500

514

558

410

4. Products exergy cost ($/kWh)

Design

0.0453

0.0364

0.0759

0.0309

0.0449

0.0425

0.0603

0.0354

Ideal operation

0.0515

0.0425

0.0853

0.0349

0.0546

0.0519

0.0722

0.0430

Expected operation

0.0570

0.0520

0.0918

0.0443

0.0573

0.0544

0.0760

0.0451

5. Second law efficiency

Design

0.3179

0.4072

0.2046

0.4320

0.3507

0.3874

0.2412

0.4550

Expected operation

0.2763

0.3120

0.1670

0.3271

0.3079

0.3426

0.2115

0.4024

6. Time-dependence penalty

Fuel penalty (MW)

7.540

17.40

43.80

17.40

6.124

5.667

8.714

4.825

Cost penalty (%)

25.8

42.9

21.0

43.5

4.043

3.952

4.280

(icontinued)

Table 7.5 (Continued)

Configuration 1 with energy recovery

Expected operation with a regenerator

Regenerator cost (S/h)

14.40

-

103.7

-

Fuel saving ($/h)

6.864

-

136.4

-

Net saving ($/h)

-7.551

-

32.8

-

Products exergy cost ($/kWh)

0.0573

-

0.0904

-

Second law efficiency

0.2784

-

0.1838

-

Expected operation with a VC chiller

VC cost (S/h)

41.40

68.0

-

69.0

Fuel saving ($/h)

13.90

80.9

-

84.0

Net saving (S/h)

-27.54

12.9

-

15.0

Products exergy cost ($/kWh)

0.0582

0.0515

-

0.0437

Second law efficiency

0.2806

0.3426

-

0.3666

Design exergy delivered = 34 MW, exergy to be delivered = 23 MW, revenues = 1964 S/h. Main ground rules

Fuel price 0.01 $/kWh (higher heating value).

Products market values $/kWh: power 0.045, chilled water 0.03, Heating water 0.02, Hot water 0.01. Common capital costing equations (common design models). Common load profiles, no power exchange with the grid.

Common off-design to design efficiency profile (quadratic efficiency vs. power load ratio equation). Common temperatures for chilled water, heating water, hot water and ambient.

350"

Performance eqn: Qj=

ai+b,*XLi2

b; Mwpwr

eff %

1 90.0 96.0 83

44.5

2 60.7 21.3 27

33.0

3 186.7 53.3 120

50.0

4 224.0 171.0 150

Figure 7.14 Performance equations of a mix of plants.

after allowing for storing the energy of the vented gas. Designing on mean load, the cost-effectiveness of the regenerator improved.

- For the higher efficiency of about 40%, the regenerator solution was not needed because of the absence of gas dumping and the VC solution became more cost effective than re-firing. Designing the VC on mean load and allowing for an ice-making storage unit improved further the cost effectiveness.

• The sensitivity of the quoted overall system performance equation from similar existing plants does not seem to have relevant effect on the reliability of the screening method.

- The extrapolated zero load efficiency of Equation 6.4 set at 0.2 was changed from 1 (ideal control) to —0.2. The deviation factors (Equations 11 and 12) from ideally controlled performance were less than 1.5. The cost increase from the ideally controlled performance case was only 6%.

• Runs assuming constant power production by allowing buying and selling transactions with the grid at prevailing power market price showed in general higher efficiency and lower cost than those in the stand-alone case. The value of the produced power influences both cost and efficiency.

7.2.3 The optimal operation of a mix of power plants

The application considers a group of single purpose power plants available for operation. The overall performance equation of each plant can be satisfactorily

Table 7.6 Load assignment for optimal operation of a facility of plants 380 MW.

Run

Load P,

Qf

Xu

Xl2

*L3

XL4

MW

MW

_

_

_

1

380

903.0

1.000

1.000

1.000

1.000

2

360

859.4

0.915

0.978

1.000

0.917

3

340

819.8

0.830

0.940

1.000

0.836

4

320

783.7

0.752

0.891

1.000

0.757

5

300

751.5

0.673

0.832

1.000

0.678

6

280

722.6

0.594

0.767

1.000

0.600

7

260

697.3

0.517

0.696

1.000

0.522

8

240

675.3

0.440

0.620

1.000

0.445

Optimal and arbitrary assignment

280

722.6

0.594

0.767

1.000

0.600

280

726.7

0.494

0.561

1.000

0.691

280

726.3

0.644

0.717

0.950

0.621

280

750.6

0.794

0.867

0.700

0.711

280

780.3

0.894

0.967

0.500

0.798

expressed by a quadratic equation between load ratio 1 and a minimum allowable load ratio around 0.3. A facility of four available power plants is considered. The quadratic form assumed is Q = a + bX2, where Q = fuel consumption rate in kW and X = the load fraction. The performance equations are shown in Figure 7.14 and the results of arbitrary and optimal operation are given in Table 7.6. The loading of the plants is such that total fuel consumption is minimized. The minimization procedure is that of the Lagrange multiplier using only one multiplier. The fuel consumption equation is augmented by a constraint that expresses the sum of partial loads to be equal to the demand, for a time period of constant demand, along with an undetermined multiplier to the constraint to treat the load fractions as decision variables. This ties the constant b of a load to the Lagrange multiplier. The procedure is detailed in Section 6.4.

It may be noted that the overall performance equation can be fuel consumption per unit product or product per unit fuel (efficiency). The first form is to be minimized. The second is to be maximized. The analysis is also applicable to two product plants if the demand for one of the products is constant.

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