Performance analysis and optimized control strategy for a three-shaft, recuperated gas turbine with power turbine variable area nozzle

Applied Thermal Engineering 164 (2020) 114353

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Performance analysis and optimized control strategy for a three-shaft, recuperated gas turbine with power turbine variable area nozzle

T

⁎

Qiao Zhoua,b, Zhao Yina, , Hualiang Zhanga,b, Tao Wanga,b, Wenchao Suna, Chunqing Tana,b a b

Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China University of Chinese Academy of Sciences, Beijing, China

H I GH L IG H T S

part-load performance analysis and optimization for a three-shaft, recuperated gas turbine is conducted. • Detailed modified recuperator simulation model is proposed based on experiment. • AEffect of power turbine shaft speed and variable area nozzle (VAN) angle on engine performance improvement is analyzed. • Influence temperature limits and ambient temperature on optimized control of VAN angle is studied. • Optimizedofcontrol strategy for power turbine shaft speed and VAN angle is proposed. •

A R T I C LE I N FO

A B S T R A C T

Keywords: Recuperated cycle Three-shaft gas turbine Power turbine variable area nozzle Performance analysis Optimized control strategy

Performance analysis and optimization for a three-shaft, recuperated gas turbine with power turbine variable area nozzle (VAN) is conducted in this paper. A modified recuperator simulation model is obtained from recuperator experiment. A gas turbine performance simulation program is established. The influence of power turbine shaft speed on engine performance is studied and optimized control of power turbine shaft speed is proposed. Comparative performance analysis proves VAN angle control improves engine performance, but leads to less compressor surge margin and increased risk of over temperature. The optimized control of VAN angle under different safety operation temperature limits and ambient temperature is investigated. Power turbine shaft speed barely influences the optimized control of VAN angle, but temperature limits and ambient temperature greatly do. By decoupling power turbine shaft speed and VAN angle, an optimized control strategy for power turbine shaft speed and VAN angle is proposed, which brings 6.37%, 15.88%, 47.80% increases in output power, and 10.84%, 25.59%, 64.97% increase in thermal efficiency when relative high-pressure shaft speed is 0.95, 0.90 and 0.85, respectively. Maximum thermal efficiency could be achieved at part-load conditions rather than design-point if temperature limit is relaxed.

1. Introduction Gas turbine is widely applied in power generation, mechanical drive, automotive vehicle and marine vessel [1]. Simulation technique has been studied to investigate gas turbine performance since 1960s. Several programs to analyze steady-state, transient-state performance [2,3] and start-up, shut-down characteristics [4] of turbofan [5,6] and navy engines [7,8] were developed since 1970s. Lots of modified programs were then established based on the programs above. Tsoutsanis et al. developed a novel compressor map generation method to ensure simulation accuracy [9] and analyzed performance of GE LM2500 gas turbine [10]. Song et al. proposed a hybrid turbine cooling modeling

⁎

method and effectively analyzed the performance of a three-shaft gas turbine with turbine cooling [11]. Wang T. et al. established a hybrid model by combining component-level modeling method with neural net algorithm [12], which performed well with high accuracy and brought reduction in simulation time. Haglind and Elmegaard predicted performance with higher accuracy by taking pressure loss, mechanical loss and bleed into consideration [13]. Wei et al. [14] and Duan et al. [15,16] established mathematical model for a single-shaft micro gas turbine and analyzed performance under constant and variable rotational speed control. To meet the increasing demand of energy saving, variable geometry and recuperator have been applied to improve gas turbine performance.

Corresponding author. E-mail address: [email protected] (Z. Yin).

https://doi.org/10.1016/j.applthermaleng.2019.114353 Received 11 January 2019; Received in revised form 23 July 2019; Accepted 6 September 2019 Available online 07 September 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

Applied Thermal Engineering 164 (2020) 114353

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Nomenclature

1 2 21 3 31 4 5 51 6 61 7 71 8 81 9 cool cor d f hot h ht in l lc lt m out pt

Symbols CC f HC HT LC LT m M1 M2 n p PT RE T VAN VGV W

combustor chamber fuel-air ratio high-pressure compressor high-pressure turbine low-pressure compressor low-pressure turbine mass flow rate [kg/s] torque (Y660-SG) [N·m] torque (Y1600-SG) [N·m] relative rotational speed total pressure [kPa] power turbine recuperator temperature [K] power turbine variable area nozzle compressor variable guide vane work [kW]

Greek letters α ε η π

heat transfer coefficient [kW/(m2·K)] recuperator effectiveness efficiency pressure ratio

LC inlet LC outlet HC inlet HC outlet RE cool flow inlet RE cool flow outlet CC outlet HT inlet HT outlet LT inlet LT outlet PT inlet PT outlet RE hot flow inlet RE hot flow outlet recueprator cool flow corrected parameters design point value fuel recueprator hot flow high-pressure shaft high-pressure turbine inlet parameter low-pressure shaft low-pressure compressor low-pressure turbine mechanical efficiency for shafts outlet parameter power turbine

Subscripts 0

ambient conditions

combined cycle part-load performance. Kim S. et al. proposed an analysis method to derive optimized VGV schedule in axial compressor to improve part-load performance of a turbofan engine with required surge margin [21]. To fully improve gas turbine performance, Gu and Wang H. analyzed the influence of VAN control and obtained optimized operating strategy [22,23] based on the work in [11], while Wang T. developed high efficiency mode [24] and high power mode [25] to achieve better part-load performance. Recuperator, or combination of recuperator and variable geometry is another effective way to improve gas turbine performance [26]. The detailed recuperator introduction about the application, type, manufacture and performance was presented in [27,28]. The heat transfer and pressure loss [29,30] are the most vital characteristics in recuperator researches, which are usually obtained by experiment investigation [31–34]. To simulate recuperated gas turbine performance

In open literature and actual applications, there are mainly two traditional ways to achieve variable geometry: compressor variable guide vane (VGV) and power turbine variable area nozzle (VAN). Kim T. S. studied the influence of control strategy and found appropriate VGV control improved part-load performance [17]. Kim J. H. et al. conducted comparative performance analysis of single-shaft and split-shaft industrial gas turbine. The VGV control increased thermal efficiency in the case of single-shaft engine, while that of the split-shaft one didn’t appear to be effectively improved [18]. Haglind quantified the effect of variable geometry under fuel control and VGV control [19]. The VGV control brought small improvement in thermal efficiency within relative output power range of 100%~85%, but it reduced thermal efficiency at lower output power conditions. Haglind further analyzed the effect of VGV and VAN control of combined cycle gas turbine for marine ship use [20]. Conclusion was drawn that variable geometry improved

Fig. 1. The sketch of the recuperated gas turbine. 2

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Table 1 Design parameters of the gas turbine. Parameter

Value

Pressure ratio Turbine inlet temperature Turbine exit temperature Output power Thermal efficiency

Simple cycle

Recuperated cycle

10.0 1261 K 772.4 K 942.6 kW 29.1%

10.2 1261 K 808.3 K 761.6 kW 31.5%

(a) Geometry of the cool flow and hot flow

Table 2 Recuperator effectiveness models. εd εd + (1 − εd)(m / md)1 − z

Model A1: [14,35–37,39]

ε=

Model B1: [38] Model C1: [41] Model D1: [41]

ε = εd (m/md)β ε = 1 − (1 − εd) (m/md) ε = εd

10mm 1.50mm (b) Size of the corrugation

0.15mm

0.07mm

3mm

60°

(c) Size of the triangle-shape fin for cool and hot flow channel Fig. 3. The sketch and geometry of heat transfer unit.

outlet backpressure valve

heater

recuperator

Fig. 2. Effectiveness for different recuperator [35].

Model A2: [35]

Δp Δpd

x

y

p m T =⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎝ md ⎠ ⎝ Td ⎠ ⎝ pd ⎠

Model B2 [38]:

(Δp / pin ) (Δp / pin )d

[36,39]:

=

outlet

z

hot flow

2 ⎛ (m T / p)in ⎞ (m T / p)in, d

⎝

(Δp / pin ) (Δp / pin )d

Model C2: [41]

2

(Δp / pin )hot (Δp / pin )hot , d

Δp p

=

[14,15,35–39], modeling the heat transfer and pressure loss is the first key step since it greatly affects simulation accuracy. Several recuperator models were proposed in [35–41], and revealed mass flow was the dominator factor affecting heat transfer and pressure loss. For recuperated gas turbine, there are two common configurations: singleshaft micro gas turbine, and multi-shaft gas turbine with variable geometry. For the single-shaft one, it’s widely used in power generation and distributed energy system, and has drawn most of the research attentions. With mathematical simulation model, Duan derived the analytical mathematical expression of output power and thermal efficiency, obtained optimized shaft control law and designed the control system [15,16]. Kim et al. developed a novel recuperator model which reducing computing time and improving simulation accuracy, and simulated the dynamic operation [36] and analyzed the effect of internal leakage on a micro gas turbine performance [39]. For the variable geometry one, Kim and Hwang analyzed performance characteristics of split-shaft recuperated gas turbines and proved VGV and VAN operation improved gas turbine part-load performance by maintaining high

⎟

⎟

2

m ·Tin ⎞ = ⎛ 2in ⎝ (min·Tin)d ⎠hot ⎜

⎟

( )

Δp p d

Table 4 Information of the scaled recuperator. Design mass flow rate Design Design Design Design

cool flow inlet temperature hot flow inlet temperature cool flow inlet pressure hot flow inlet pressure

cool flow

Fig. 4. Recuperator experiment test rig.

(m T / p)in (m T / pin )in, d

1.55 ⎞ ⎛ ⎛ m ⎞ ·⎛ Tout ⎞ 0.55 ⎟ ⎟ ⎜ ⎝ p ⎠in ⎜⎝ Tin ⎠ =⎜ 2 1.55 ⎞ ⎟ ⎛ Tout m ⎜ ⎛ p ⎞ ·⎜ 0.55 ⎟ ⎟ ⎝ ⎠in, d ⎝ Tin ⎠d ⎝ ⎠cool ⎜

inlet

⎠

=

⎜

(Δp / pin )cool (Δp / pin )cool, d

Model D2: [41]

inlet

outlet

Table 3 Recuperator pressure loss models.

0.30 kg/s 200 °C 600 °C 0.40 MPa 0.11 MPa

3

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Table 5 Information of the measurement system. Measured parameters

Measuring equipment

Accuracy

Number

Range

Temperature (before heating) Temperature (after heating, cool flow) Temperature (cool flow inlet) Temperature (cool flow outlet) Temperature (after heating, hot flow) Temperature (hot flow inlet) Temperature (hot flow outlet) Pressure (cool flow) Pressure (hot flow) Mass flow

Pt1000 (THPT-231) Pt1000 (THPT-231) Pt1000 (THPT-231) K type Thermocouple (THPT-230) K type Thermocouple (THPT-230) K type Thermocouple (THPT-230) K type Thermocouple (THPT-230) Pressure transmitter (THGP) Pressure transmitter (THGP) Gas turbine meter

± 0.2% ± 0.2% ± 0.2% ± 0.75% ± 0.75% ± 0.75% ± 0.75% ± 0.2% ± 0.2% ± 1.5%

2 1 1 1 1 1 1 3 3 2

0–100 °C 0–300 °C 0–300 °C 0–800 °C 0–800 °C 0–800 °C 0–800 °C 0–0.5 MPa 0–0.1 MPa 65–1300 m3/h

Fig. 6. Recuperator effectiveness.

Relative pressure loss

0.08 Experiment Fitting curve

0.06 0.04 0.02 0.00

0.00 0.01 0.02 0.03 0.04 Semi-corrected mass flow rate (m T / p)

(a) Relative pressure loss

0.12

Cool flow pressure loss Experiment Fitting curve

0.10 0.08 0.06 0.04 0.02 0.00

0.00 0.04 0.08 0.12 0.16 Semi-corrected mass flow rate (m T / p)

(b)

Fig. 5. Experiment data of effectiveness and pressure loss.

Hot flow pressure loss

Fig. 7. Recuperator pressure loss for cool and hot flow. 4

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1.6

Relative pressure ratio

110%

1.2 100%

1.0 0.8

90%

80% 70% 0.6 60%

0.4

0.9

100%

0.8

0.6 0.2

60%

60% 70%

80%

90%

0.4 0.6 0.8 1.0 1.2 Relative corrected mass flow

1.0 0.8

1.0

60%

Relative pressure ratio

1.0 0.8

110% 100%

0.6 80% 60% 70%

90% 100% 105%

0.9

1.8

90%

1.1

Relative efficiency

Relative pressure ratio Relative efficiency

80%

Fig. 10. HT maps.

1.2

1.0 0.9 0.8 0.6 0.2

70%

0.8 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Relative corrected (mass flow* shaft speed)

1.4

1.4

0.7

105%

1.2

Fig. 8. LC maps.

0.4 1.2

70%

90% 100%

1.1

110%

1.0

0.7

1.4

80%

0.6

0.2 1.1

Relative efficiency

Relative efficiency

Relative pressure ratio

1.4

60% 70% 80% 90% 100%110%

0.4 0.6 0.8 1.0 1.2 Relative corrected mass flow

1.6 1.4

70% 60%

1.2

110% 115% 100% 90% 80%

1.0 0.8 0.6 0.4 1.0 60% 70%

0.8

80% 90% 100%

0.6 110%

0.4 115%

0.2 0.2 0.4 0.6 0.8 1.0 1.2 Relative corrected (mass flow* shaft speed)

1.4

Fig. 9. HC maps. Fig. 11. LT maps.

turbine exhaust temperature [35]. From the work above, the combined application of recuperator and variable geometry improves gas turbine performance. But the work is mainly focused on single-shaft and split-shaft gas turbine. For threeshaft recuperated gas turbine with variable geometry, the performance analysis and optimization are not studied in detail. The investigation of the optimized combined control strategy for power turbine shaft speed (npt) and VAN angle is more difficult due to more complex structure and more adjustable control parameters. Thus, the strategy might be different from those of three-shaft simple cycle one and single-shaft recuperated one. Moreover, the recuperator model to predict heat transfer and pressure loss is selected without detailed recuperator experiment support in most researches. To overcome these problems, the paper first conducts recuperator

experiments to study the characteristics of heat transfer and pressure loss, and then obtains a modified recuperator simulation model. Meanwhile, a performance simulation program, which is modularized with object-oriented programming method and of great extensibility to multi gas turbine configurations, is established to conduct performance analysis of three-shaft recuperated gas turbine with VAN. Based on the program, part-load performance analysis and optimization are studied in detail. The effect of npt and VAN angle on gas turbine performance improvement is analyzed. Finally, by decoupling power turbine shaft speed and VAN angle to ease the optimization process, the optimized control strategy for the combined adjustment of npt and VAN angle is evaluated. As a result, significant performance improvement is achieved. 5

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Relative pressure ratio

1.6 1.4 1.2 1.0 0.8 0.6

20°

0.4

25°

30°

35° 40°

30°

35°

50°

0.8

40°

0.6

25° 20°

Fig. 14. Geometry of power turbine.

50°

0.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Relative corrected mass flow

Start Input: ambient conditions (T0 , p0) and component maps

Fig. 12. PT maps (npt,cor = 100%).

Control parameters: nh, npt, VAN

1.4

Independent state parameters initialization:

1.2

X

1.0 0.6 0.4

20°

25°

1.0

Relative efficiency

[nl ,

lc

,

hc

, T5 ,

ht

,

lt

, T4 ]T

Gas turbine thermodynamic calculation (inlet, LC, HC, RE, CC, HT, LT, PT,exhaust)

0.8 30° 35° 40° 50° 30° 25°

0.8

Updating X

Relative pressure ratio

1.6

35°

40°

Error function calculation E(X)=[E1, E2, E3, E4, E5, E6, E7]T Solving dXi with Newton-Raphson iteration method

50°

No

0.6 20°

7

Ei2 10

6

Updating nh , npt , VAN angle

Relative efficiency

1.0

i 1

Yes

0.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Relative corrected mass flow

Performance Analysis Compressor Surge Limit Shaft Speed Limit Temperature Limit

Fig. 13. PT maps (npt,cor = 70%).

2. Gas turbine performance simulation program

Performance Optimization

2.1. Description of the gas turbine Fig. 1 shows the recuperated gas turbine studied in this paper. It’s improved from a simple cycle, three-shaft gas turbine, which contains a two-shaft gas generator and a single-stage power turbine. The gas generator contains a single-stage low-pressure centrifugal compressor, a single-stage high-pressure centrifugal compressor, an annular combustion chamber, a single-stage high-pressure axial turbine, a single-stage low-pressure axial turbine. The power turbine is the variable geometry type and its VAN angle is adjustable. The simple cycle gas turbine is designed for vehicle mechanical drive. Experiments as well as performance simulation [24,25] both

End Fig. 15. Flow chart of the performance simulation process.

indicate thermal efficiency of simple cycle one declines fast at part-load conditions. This cycle modification to combine recuperator and VAN together would give full play to enhance gas turbine thermal efficiency [26]. Table 1 shows the parameters of the simple and recuperated cycle. For the gas turbine, three control parameters, fuel mass flow rate 6

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full-scale experiment. A scaled one is tentatively designed and fabricated for early research purpose to investigate the characteristics of heat transfer and pressure loss. The scaled recuperator is the crosscorrugated, compact plate-fin type. The design parameters are shown in Table 4. The cool and hot flow channels are of same geometry and size. Fig. 3 shows the geometry of the heat transfer unit, including the geometry of the cool and hor flow channel, size of the corrugation and the fin. The test rig shown in Fig. 4 includes air supply blower, air compressor, heating system, flow control system, measurement system and the recuperator. The air supply blower provides low-pressure air for hot flow, while air compressor provides high-pressure air for cool flow. Two heaters with maximum heating power of 240 kW and 80 kW are used to heat hot and cool flow air. The measurement system is shown in Table 5. The required flow could be obtained by automatic control or manual control. With automatic control, flow control system refers to a database to automatically adjust electromagnetic valves and heating load to meet the desired flow conditions. For manual control, the desired flow is obtained by manually adjusting electromagnetic valves, manual valves and heating load.

measurement system gas turbine dynamometer

inlet

dynamometer

Fig. 16. Simple cycle gas turbine test rig. Table 6 Measurement information of the simple cycle test rig. Parameter

Equipment

Accuracy

Number

T0 p0 T9 M1 M2 nl nh np mf

Platinum resistance thermometer Pressure measuring module Thermocouple Hydraulic dynamometer Hydraulic dynamometer Tachometer generator Tachometer generator Tachometer generator Mass flow meter

± 0.5 °C ± 0.3% ± 4 °C ± 0.2% ± 0.2% ± 0.2% ± 0.2% ± 0.2% ± 1%

4 4 6 1 1 1 1 1 1

2.2.3. Modified model With the experiment test rig, the characteristics of heat transfer and pressure loss under different cool flow pressure, hot flow temperature and mass flow rate are investigated. Fig. 5(a) shows ε and relative pressure loss under different cool flow inlet pressure. As cool flow inlet pressure increases, ε and hot flow relative pressure loss are almost unchanged, while cool flow relative pressure loss decreases. The results indicate cool flow inlet pressure has little effect on ε and hot flow pressure loss, but relatively higher effect on cool flow pressure loss. In Fig. 5(b), as hot flow inlet temperature increases, hot flow pressure loss increases, but ε and cool flow pressure loss remain nearly unchanged. That implies hot flow inlet temperature affects hot flow pressure loss, but has negligible influence on ε and cool flow pressure loss. Fig. 5(c) presents the variation of ε and pressure loss under different mass flow rate. As mass flow decreases, ε increases and pressure loss decreases. Moreover, the variation in hot flow relative pressure loss is larger than that of cool flow. This variation trend in Fig. 5(b) and (c) is consistent with the work in [27,34]. From the results above, mass flow rate significant affects ε, while pressure and temperature barely affect it. Similarly to the previous ε models, we establish the relationship between ε and mass flow rate by data fitting. In Fig. 6, fitting coefficients z = 0.22 for Model A1 and β = −0.12 for Model B1 are obtained. Among three ε modeling methods, Model A1 fits best, followed by Model B1 and Model C1. So, Model A1 with coefficient of z = 0.22 is selected to predict ε shown in Eq. (2).

(Wf), npt and VAN angle, are adjustable to change gas turbine working conditions. The only fuel control strategy leads to severe decrease in part-load thermal efficiency, and proper control strategy to adjust npt and VAN angle can reduce the downtrend. To fully utilize the fuelsaving effect of recuperator, the paper focuses on the investigation of the optimized combined control strategy for npt and VAN angle. 2.2. Recuperator simulation model 2.2.1. Previous model Recuperator modeling is to obtain characteristics of heat transfer and pressure loss. Some simulation models are displayed in [14,35–39,41]. Effectiveness (ε) is widely adopted to simulate heat transfer and its definition is shown in Eq. (1). Some ε simulation models are listed in Table 2. In these models, mass flow is the dominating parameter while the influence of temperature and pressure is ignored. In Model A1, B1 and C1, ε increases as mass flow decreases, while ε is modeled constant in Model D1. Model A1 are based on an assumption that heat transfer coefficient α is proportional to the power of mass flow mz. Fig. 2 shows the relationship between ε and mass flow for three different recuperators [35]. It indicates ε of a specific recuperator differs greatly from other recuperator and ε is influenced by the designed value and the exponent z.

ε=

Tcool, out − Tcool, in Thot , in − Tcool, in

ε=

εd εd + (1 − εd )

m 0.78 md

( )

(2)

For recuperator pressure loss, experiment investigation proves the pressure, temperature and mass flow rate affect it. Based the previous work that pressure loss is commonly modeled as the function of corrected mass flow rate, we rearrange the experiment data to obtain the relationship between pressure loss and corrected mass flow rate. In Fig. 7, point (0, 0) is supplemented for the consideration that pressure loss should be zero if there is no working fluid. The fitting equations to predict pressure loss are presented in Eqs. (3) and (4). Compared with Model B2, the power exponent is neither 1 nor 2, but the value between 1 and 2. That indicates the pressure loss in recuperator is different from that in pure pipe flow without heat transfer and also implies pressure loss variation for cool and hot flow is different. Until now, a modified recuperator simulation model is obtained.

(1)

The mass flow, temperature and pressure both affect recuperator pressure loss. Several pressure loss models are listed in Table 3. In most researches, either the pressure loss is modeled constant seen in Table 3, Model D2, or is modeled as duct pressure loss which is determined by corrected mass flow seen in Table 3, Model B2. Moreover, pressure loss for cool and hot flow is usually modeled the same. 2.2.2. Experiment conditions Considering that the designed mass flow rate of the recuperated gas turbine is relatively large, it would be challenging and costly to conduct 7

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(a)

(b)

Fuel mass flow rate – output power

(c)

Fuel mass flow rate – high-pressure shaft speed

Fuel mass flow rate – exhaust temperature

(d)

Fuel mass flow rate – low-pressure shaft speed

Fig. 17. Validation of the performance simulation program. 1.12

(Δp / pin )cool (m T / p)in ⎞ = ⎜⎛ ⎟ (Δp / pin )cool, d ( m T / pin )in, d ⎠ ⎝ cool

2.3.2. Program establishment With the recuperator model and component maps above, an inhouse simulation program is established based on component-level simulation method in MATLAB/Simulink. Each component (compressors, combustor, turbines, and recuperator) is modularized with object-oriented programming method, which benefits multi-goal performance simulation and program extension for multi-configuration gas turbine. The polynomial equations of specific heats of the working fluids are adopted to conduct thermodynamic calculation of enthalpy and entropy in each component to improve calculation accuracy. To conduct performance simulation, the high-pressure shaft speed nh, npt and VAN angle are selected as control parameters, and a set of independent state parameters X are chosen.

(3)

1.28

(Δp / pin )hot (m T / p)in ⎞ = ⎜⎛ ⎟ (Δp / pin )hot , d ⎝ (m T / pin )in, d ⎠hot

(4)

2.3. Performance simulation program 2.3.1. Component characteristics maps Recently, computational fluid dynamics (CFD) has gained great development in grid generation, discretization scheme, turbulence model and computation speed [22], and has been widely applied in turbomachinery to solve problems of fluid-thermal-solid interaction. In this paper, the CFD software Numeca is adopted to obtain maps of LC, HC, HP, LT and PT, which are shown in Figs. 8–13. Figs. 12 and 13 show PT maps under different VAN angles and PT corrected shaft speed (npt,cor). The PT geometry is shown in Fig. 14 and the designed VAN stagger angle is 30°. PT flow capacity decreases as VAN angle decreases. For a given npt,cor and pressure ratio, there is an optimal VAN angle to obtain highest turbine efficiency. If the angle deviates from the optimal one, the efficiency falls. For a given VAN angle, the change in npt,cor has small effect on corrected mass flow, but larger effect on efficiency.

X = [nl , πlc , πhc , T5, πht , πlt , T4 ]T

(5)

Fig. 15 is the flow chart of the performance simulation process. After state parameters X are given initial value to start thermodynamic calculation, we obtain the error function E(X) = [E1, E2, E3, E4, E5, E6, E7]T, which represents the balance of mass flow, power and recuperator heat transfer. The E(X) is defined as:

8

E1 = (mhc − mlc )/ mhc

(6)

E2 = (mht − mhc − mf )/ mht

(7)

E3 = (mlt − mht )/ mlt

(8)

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31 30

1.0 0.8

VAN angle / °

Relative thermal efficiency

1.2

npt=100%

0.6

npt=90% npt=80%

0.4 0.2

0.4 0.6 0.8 Relative output power

1.0

Relative value

Șpt

İ Wf

nl

0.4

npt=100% npt=90% npt=80% nl nl nl

0.2 0.0

Relative thermal efficiency

nl Șpt

Wf

0.80

Șpt

Șpt

Șpt

İ Wf

İ Wf

İ Wf

0.6 Fuel control only Conbined control

0.4 0.2 0.0

0.2 0.4 0.6 0.8 Relative output power

Fuel control only

1.2 1.0

Optimized npt control

T5

T81

Șpt

İ f

0.8

0.90

T51 Șpt

0.6

0.85 0.80

T81

0.4

0.75

0.2

0.70

İ f

Combined control

0.80 0.85 0.90 0.95 1.00 Relative high-pressure shaft speed

(c) Component parameters

0.80 0.85 0.90 0.95 1.00 1.05 Relative high-pressure shaft speed

1.4

Relative pressure ratio

(c) Optimized control of npt Fig. 18. Effect of npt on gas turbine performance improvement.

E4 = (mpt − mlt )/ mpt

1.0

(b) Engine performance

0.95

0.65 0.75

0.8

Relative value

Relative power turbine shaft speed

1.00

1.0

0.85 0.90 0.95 1.00 High-pressure shaft speed

(b) Component parameters 1.05

Fuel control only Conbined control

1.2 npt=70%

İ

0.6

26

(a) VAN angle control law

1.0 0.8

27

24 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Relative high-pressure shaft speed

(a) Engine performance 1.2

28

25

npt=70%

0.2 0.0

29

(9)

E5 = (Wlt ·ηl, m − Wlc )/(Wlt ·ηl, m)

(10)

E6 = (Wht ·ηh, m − Whc )/(Wht ·ηh, m)

(11)

E5 = (T81 − T81 −)/ T81

(12)

1.2 1.0 0.8 0.6 0.4 0.2 0.2

where T81 and T81−_ are recuperator hot flow inlet temperature obtained from Eq. (1) and from component thermodynamic calculation, respectively; ηl,m and ηh,m are mechanical efficiencies of low-pressure and high-pressure shaft. In this paper, Newton-Raphson iteration method is applied to numerically solve the error function. After gas turbine performance calculation is done, performance optimization is undergoing with safety operation limits, such as compressor surge limit, shaft speed and temperature limits.

Fuel control only Combined control

60%

70%

80%

105% 100% 90%

0.4 0.6 0.8 1.0 1.2 Relative corrected mass flow

1.4

(d) LC working line Fig. 19. Comparison between fuel flow control and combined control.

2.3.3. Program validation To validate program accuracy, simulation results should be compared with that of experiment and software GasTurb. However, the recuperator has not been installed on the gas turbine and the 9

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30°, npt=100%

24.5°, npt=100%

1.2

30°, npt=70%

24.5°, npt=70%

1.0

27.25°, npt=70%

30

27.25°, npt=100%

0.8 0.6

31

30°

27.25° 24.5°

0.4

60%

0.2 0.2

VAN angle /°

Relative pressure ratio

1.4

70%

80%

90%

100%

24.5°, npt=100%

1.2

30°, npt=70%

24.5°, npt=70%

1.1

27.25°, npt=70%

27.25°

0.6 0.8 1.0 1.2 Relative corrected mass flow

1.4

npt=70%

27 26

Limited by T5 and T81 Safety operation

0.80 0.85 0.90 0.95 1.00 Relative high-pressure shaft speed

30

VAN angle /°

ΔT = 0 K ΔT = 50 K ΔT = 65 K

not exceeding design value exceeding design value 50 K not exceeding 873.15 K

Relative high-pressure shaft speed

npt=80%

28

(b) ǻT=50K

Table 7 Recuperator hot flow inlet temperature T81 limits.

1.00

npt=90%

29

31

Fig. 20. Compressor working lines under different constant VAN angle and npt.

1.05

npt=100%

24

(b) HC working line

limit A limit B limit C

0.80 0.85 0.90 0.95 1.00 Relative high-pressure shaft speed

25 105% 100% 95% 85%

0.6 0.4

26

30

30°

0.7

27

(a) ǻT=0K

24.5°

0.8

npt=70%

31

27.25°, npt=100%

0.9

28

1.4

VAN angle /°

Relative pressure ratio

30°, npt=100%

1.0

npt=80%

25

(a) LC working line 1.3

npt=90%

29

24

0.4 0.6 0.8 1.0 1.2 Relative corrected mass flow

npt=100%

29 28 27

npt=100% npt=90% npt=80% npt=70%

26 25 24

Limited by T81

0.80 0.85 0.90 0.95 1.00 Relative high-pressure shaft speed

(c) ǻT=65K Fig. 22. Optimized control of VAN angle.

0.95 information of the measurement system. During the experiment, VAN angle is fixed at 30°, PT works at designed speed, and ambient condition is 289.15 K, 102.275 kPa. The validation results are shown in Fig. 17. The simulation results of the proposed program are almost consistent with GasTurb. Compared with experiment, there is some deviation in exhaust temperature, while other parameters match well. The deviation is caused by the different temperature location of simulated point and measured point. The simulated temperature is PT outlet temperature, while the measured one is that of exhaust pipe where some exhaust heat is leaked to ambient. Taken together, the simulation program is credible and can give reliable performance prediction.

0.90 0.85 0.80 0.75

30

29

28 27 26 VAN angle / °

25

24

Fig. 21. Temperature limits at different operation conditions.

recuperated cycle experiment data is not available. Besides, the modified recuperator model cannot be migrated to GasTurb since there is no interface to import external user-defined models. Considering these facts, we carry out program validation for simple cycle simulation program by comparing with simple cycle experiment data and GasTurb. Fig. 16 shows simple cycle gas turbine test rig and Table 6 shows the

3. Performance analysis and optimization Considering the complex working conditions in mechanical drive, the demanded output power is always changing. To ensure better partload performance, the optimized control strategy of power turbine shaft speed and VAN angle is studied. 10

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VAN angle /°

30 29 28 27

Relative thermal efficiency

31 T0=273.15K T0=288.15K T0=303.15K

26 25 24 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Relative high-pressure shaft speed

(a) ǻT=0K 31

VAN angle /°

30 29 28

T0=273.15K

24 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Relative high-pressure shaft speed

(b) ǻT=50K

VAN angle /°

28

T0=273.15K T0=288.15K T0=303.15K

27 26 25 24 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Relative high-pressure shaft speed

31 30 29 28 27 26

VAN angle /°

Relative power turbine shaft speed

VAN angle control law

25 0.80 0.85 0.90 0.95 1.00 Relative high-pressure shaft speed

1.2

3.2.1. Explanation for the improvement In this part, the ambient condition is 288.15 K, 101.325 kPa, and PT works at designed speed. With the combined control of fuel mass flow and VAN angle, performance analysis is conducted to figure out why VAN control help improve gas turbine performance. The combined control is to control fuel mass flow and VAN angle simultaneously. Fig. 19 shows the comparison between fuel control only and combined control. Fig. 19(a) shows the VAN angle control law. The VAN angle remains unchanged in the fuel control only, while VAN angle is in a linear relationship with nh in the combined control. Fig. 19(b) proves VAN control improves gas turbine performance. By decreasing VAN angle, the thermal efficiency is quite higher than that of fuel control only when output power is the same. Fig. 19(c) explains the reason of the performance improvement. In Fig. 19(c), the solid lines represent data of fuel control only and dash lines represent data of combined control. By decreasing VAN angle, gas turbine mass flow decreases more rapidly and ε increases faster. Compared with fuel control only, higher ε of the combined control enhances recuperator heat transfer to increase combustor inlet temperature T4 to benefit thermal efficiency. The improvement in recuperator also leads to higher HT inlet temperature T51 and less fuel-air ratio f. As T51 increases, recuperator hot flow inlet temperature T81 also increases.

Fig. 23. Optimized control of VAN angle under different ambient temperature.

npt control law

0.4 0.6 0.8 1.0 Relative output power

3.2. Effect of VAN angle control

(c) ǻT=65K

1.05 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65

Fuel control with npt and VAN, limit B

0.2

In this part, the ambient condition is 288.15 K, 101.325 kPa, and VAN angle is fixed at 30°. Steady state performance simulation is conducted to study the effect of npt control on gas turbine performance. Fig. 18 is the simulation results under different constant npt control. In Fig. 18(a), when relative output power is about at the range of 100–50%, higher npt helps obtain better thermal efficiency even though the improvement is quite small. As output power declines, lower npt is beneficial for performance improvement. Fig. 18(b) is the comparison of component parameters. The curves of ε, Wf and low-pressure shaft speed nl under different npt are almost overlapping. Since ε is modeled as the function of mass flow rate, it indicates engine mass flow rate is almost unchanged under different npt. Then, the same nh, almost unchanged mass flow, Wf and nl imply gas generator working lines are almost the same even npt changes. So, npt has little effect on gas generator performance, but great effect on PT efficiency ηpt. This could be explained by PT maps in Figs. 12 and 13. When VAN angle is fixed, the change in npt has small influence on mass flow. When output power is at high level, πpt is relative high, and the change in ηpt is not remarkable to result in huge performance difference even though npt changes greatly. As output power declines to a low level, πpt decreases and lower npt helps remain higher ηpt leading to performance improvement. To obtain better thermal efficiency, the optimized control of npt is shown in Fig. 18(c).

T0=303.15K

25

29

Fuel control with VAN Fuel control with npt and VAN, limit A

3.1. Effect of npt control

T0=288.15K

26

30

Fuel control Fuel control with npt

Fig. 25. Overall performance under different control strategy.

27

31

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.0

24

Fig. 24. Optimized control strategy for npt and VAN angle with limit A.

11

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Referring to PT maps, decreasing VAN angle is beneficial for ηpt improvement when mass flow decreases. Therefore, by decreasing VAN angle to adjust PT mass flow, higher T51, T81, ηpt, ε, and less f finally lead to better gas turbine part-load performance. However, VAN angle control also brings negative effects on gas turbine safety operation. In Fig. 19(c), too much decrease in VAN angle at lower working conditions leads to over temperature in T81. Due to the decreased mass flow, LC is also forced to work at lower shaft speed and less surge margin shown in Fig. 19(d). Besides, the output power range is also narrowed if nh is the same.

turbine exhaust temperature are selected as control parameter and maintained unchanged at designed value or a certain high value to improve part-load thermal efficiency [16,39].

3.2.4. Influence of T0 on optimized control of VAN angle Considering that gas turbine usually works at different ambient temperature T0, influence of T0 on the optimized control of VAN angle is studied. Fig. 23 shows the optimized control of VAN angle for different T0 when PT works at designed speed. As T0 increases, PT outlet temperature T8 increases. Taking the results in Fig. 23(a) for example, with T0 of 303.15 K, VAN angle should be increased to prevent over temperature when nh is at designed speed. For T0 of 273.15 K, even though T8 is lower and VAN angle could be decreased to obtain better performance, the decreased VAN angle would lead to low-pressure shaft over speed and VAN angle might as well remains unchanged for safety operation. As T0 decreases, decrease in VAN angle is larger since lower T0 is corresponding to lower T8. In Fig. 23(a)–(c), as temperature limits loosen, the angle decline speed increases and the angle meets the minimum value (24.5°). Besides, the VAN angle for lower T0 declines faster and reaches the minimum value with higher nh. It means lower T0 is of faster decline speed in optimized VAN angle adjustment.

3.2.2. Safety operation limits To ensure safety operation, limits on compressor surge margin, shaft speed and temperature are considered. Fig. 20 shows LC and HC working lines under different constant VAN angle and npt. As VAN angle decreases, compressor surge margin decreases, but is still of sufficient surge margin. The decreased VAN angle also makes LC shaft speed lower. For the concern of compressor surge margin and proper LC working speed, the adjustment of VAN angle is set at the range of 30°~24.5°. High-pressure shaft speed is set as control parameter and over speed wouldn’t occur in high-pressure components. For temperature limits, limits on T5 and T81 are considered. T5 is set to not exceed design point value to ensure turbine safety operation. For limit of T81, there is some temperature margin between designed value (808 K) and recuperator material limited temperature (873.15 K), and three limits are considered shown in Table 7.

3.3. Optimized control strategy for npt and VAN angle Based on the work above that npt has little influence of the optimized control of VAN angle, the combined optimized control of npt and VAN angle could be decoupled. So, we propose an optimized control strategy for npt and VAN angle by combining the optimized control of npt in Fig. 18(c) and optimized control of VAN angle in Fig. 22 together. Fig. 24 shows the optimized control strategy with temperature limit A under ambient condition of 288.15 K, 101.325 kPa. By optimizing control strategy, gas turbine part-load thermal efficiency is significantly improved shown in Fig. 25. The optimized strategy to control both npt and VAN angle brings most significant performance improvement, followed by only controlling VAN angle and only controlling npt. Compared with only fuel control, the optimized control strategy shown in Fig. 24 brings 6.37%, 15.88%, 47.80% increases in output power, and 10.84%, 25.59%, 64.97% increase in thermal efficiency when nh is 0.95, 0.90 and 0.85, respectively. This performance variation trend within the proposed control strategy is similar to WR-21 marine gas turbine. Moreover, the maximum thermal efficiency of recuperated gas turbine could be achieved at part-load working conditions, rather than the design point condition if recuperator outlet temperature limit is relaxed (seen in the optimized control strategy with temperature limit B in Fig. 25). The simulation results could be proved by performance data of Honeywell AGT-1500 gas turbine. The AGT-1500 gas turbine is of a same structure with the objective gas turbine and its maximum thermal efficiency is obtained at part-load condition. For simple-cycle three-shaft gas turbine, the optimized control of npt and VAN angle slightly eases the downtrend of engine part-load performance, and brings no more than 1.7% [23] and 4.0% [24] increase in thermal efficiency. However, the performance improvement of the recuperated one is quite larger in this paper and the capacity of VAN angle adjustment to improve gas turbine performance is fully utilized. With the optimized control of VAN angle, not only power turbine efficiency but also recuperator effectiveness is greatly improved in recuperated gas turbine, while the improvement is mainly achieved in power turbine efficiency for simple cycle gas turbine. Thus, the combined application of VAN and recuperator is a promising way to fully enhance gas turbine performance and could bring great economic benefits in energy saving.

3.2.3. Optimized control of VAN angle In this part, the optimization goal is to obtain higher thermal efficiency at part-load conditions. Before investigating the optimized control of VAN angle, we propose an assumption that npt has little influence on the optimized control of VAN angle based on the analysis that npt has small influence on gas generator working line when VAN angle is unchanged. If the assumption is proved true, it would greatly simplify the investigation of optimized combined control of both npt and VAN angle. With temperature limit A, Fig. 21 shows the temperature limits to ensure safety operation when PT works at designed speed and VAN angle varies from 30° to 24.5°. In Fig. 21, gas turbine works safely in the black zone with improved thermal efficiency. For the red zone, gas turbine works at high load conditions, and decreased VAN angle leads over temperature in both T5 and T81. For the blue zone, over temperature only occurs in T81. The phenomenon that over temperature only happens in T5 doesn’t occur; that means it’s the increased T5 gradually lead to over temperature in T81. This results offer a suggestion in VAN angle operation to improve engine performance: the VAN angle adjustment at high-load conditions should be quite careful and the decrease in VAN angle should be small to avoid over temperature, while the angle adjustment could be relatively larger at lower load conditions. Fig. 22 shows the optimized control of VAN angle under different npt and temperature limits. Under ambient condition of 288.15 K and 101.325 K, the optimized control of VAN angle for different constant npt is almost the same. The assumption above is then proved true and this discovery could simplify the optimized control strategy investigation on npt and VAN angle. More precisely, the optimized control strategy could be decoupled to obtain the optimized control of npt and optimized control of VAN angle, separately. Fig. 22 tells the temperature limits significantly affect the optimized control of VAN angle and changes angle decline speed. Besides, the value and location of the minimum VAN angle are also determined by temperature limits. As temperature limits gradually loosen, minimum VAN angle value declines and is obtained at higher high-pressure shaft speed. Together with the work in Fig. 21, it indicates maintaining T81 as high as possible is an effective method to obtain considerable part-load performance. This performance optimization strategy has been used in single-shaft recuperated gas turbine, such as Capstone C30, which the 12

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4. Conclusions

References

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(1) Recuperator mass flow is the key parameter affecting recuperator effectiveness and pressure loss. Pressure loss variation for hot and cool flow at part-load conditions is not the same. Pressure loss for hot and cool flow should be modeled separately. (2) With a fixed VAN angle, power turbine shaft speed has little influence on gas generator working line, but it changes power turbine efficiency to affect gas turbine performance. Higher power turbine speed is beneficial for better gas turbine performance when output power is relatively high, while lower power turbine speed improves thermal efficiency at low output power conditions. (3) VAN control indeed enhances gas turbine part-load performance. Decreasing VAN angle results in less mass flow and leads to improvement in temperatures of combustor outlet and recuperator hot flow inlet, recuperator effectiveness and power turbine efficiency, which all contribute to this enhancement. However, too much decrease in VAN angle might leads to over temperature in combustor outlet and recuperator hot flow inlet. Besides, it also leads to less compressor surge margin. (4) The power turbine shaft speed has little influence on the optimized control of VAN angle. It is the temperature limit that affects the optimized control of VAN angle. As temperature limit loosens, the decline speed of VAN angle could be larger to obtain higher thermal efficiency. Maintaining power turbine outlet temperature as high as possible is an effective method to achieve considerable performance improvement. (5) Ambient temperature also affects the optimized control of VAN angle. With same temperature limit, VAN angle could be decreased more rapidly as ambient temperature declines, since lower ambient temperature results in lower power turbine outlet temperature. (6) The optimized control strategy for power turbine shaft speed and VAN angle could be decoupled to investigate the optimized control of power turbine shaft speed and optimized control of VAN angle, separately. With the proposed optimized control strategy, 6.37%, 15.88%, 47.80% increases in output power, and 10.84%, 25.59%, 64.97% increase in thermal efficiency are gained when relative high-pressure shaft speed is 0.95, 0.90 and 0.85, respectively. The maximum thermal efficiency could also be achieved at part-load conditions rather than design-point if temperature limit is relaxed. The performance improvement effect of VAN angle control in recuperated gas turbine is quite higher than that in simple cycle gas turbine. Acknowledgments The research is supported by National Science Foundation of China (No. 51806216) and National Key R&D Program of China (2018YFB0905101). The authors are grateful for the support, encouragement, constructive suggestions and assistance from all the research team members. The author especially wants to thank Shiqing Chen, Hualiang Zhang, Tao Wang, Wenchao Sun, for their sincere help in the research idea, simulation program and recuperator experiment. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114353. 13

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