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Measures to reduce solar energy dumped in a solar aided power generation plant
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Measures to reduce solar energy dumped in a solar aided power generation plant Chang Huanga, Hongjuan Houa, , Eric Hub, Gang Yua, Si Chenc, Yongping Yanga ⁎
a
National Thermal Power Engineering & Research Center, North China Electric Power University, Champing District, Beijing 102206, China School of Mechanical Engineering, the University of Adelaide, SA 5005, Australia c Power China International Group Limited, Haidian District, Beijing 100036,China b
HIGHLIGHTS
surplus solar energy continue heating feedwater are proposed and analysed. • Using maximum surplus solar input to the system is deduced by a safety assessment. • The • Preheating feedwater continuously is superior to being charged. ARTICLE INFO
ABSTRACT
Keywords: Solar heat dumped Thermal energy storage system Preheating continuously Technical and economic performances Levelized cost of electricity
Solar aided power generation technology has been proved to be one of the most efficient ways to integrate solar energy into a coal-fired power plant. In a typical plant, the solar field size is normally designed with a solar multiple greater than one. Therefore, sometimes collected solar heat becomes surplus when the collected solar heat exceeds the heat demand. Other than being dumped, the surplus solar heat could be either charged into a thermal energy storage system if there is one; or used to preheat (feedwater) continuously. In this paper, the impacts of these two measures (being charged and preheating continuously) on the technical and economic performances of the integrated plant are analyzed in detail by undertaking a desktop case study. The results show that both measures could reduce the surplus solar heat dumped, but the economic performances vary. For the measure of preheating continuously, the maximum solar input is determined by a safety assessment to ensure a stable and safe operation. Surplus heat over this maximum amount has to be dumped. The results also show that for the plant located in Tibet China, both measures could reduce the levelized cost of energy and enhance the effective solar-to-electricity efficiency, compared to dumping case; while continuing preheating measure is superior to being charged technically and economically.
1. Introduction Due to the shortage of fossil fuels and its negative effects on the environment, renewable energy has attracted more and more attention, such as hydropower, wind power and solar energy. Hydropower plants are critical to the sustainable development of renewable energy [1], however, the increase in water demand for energy generation may be in conflict with the objectives of river conservation [2]. Wind power is gaining worldwide popularity as a large-scale energy source, yet, as the wind power penetration level increases, operating power systems securely and reliably is a serious challenge due to the intermittent nature of wind power [3]. Solar energy has played an essential role in recent years [4], especially, solar thermal energy has attracted increasing
⁎
social and political attention [5]. Solar thermal systems are advantageous since it is easier to store heat than electricity on a large scale [6]. However, standalone concentrated solar power plants are suffering restriction for further development due to their high cost and low efficiency [7]. Solar aided power generation (SAPG), where solar heat serves as a substitution of the extraction steam to preheat feedwater in a regenerative Rankin power station, has been proved an efficient way to utilize solar energy for power generation purpose [8]. The advantages of SAPG are presented in [9], which are not only contribute to increase the efficiencies and reduce fuel consumption, but also provides a better way to use solar heat for power generation. Most current studies on SAPG systems focus on the performance analysis and evaluation methods. Based on the annual performance
Corresponding author. E-mail address: [email protected] (H. Hou).
https://doi.org/10.1016/j.apenergy.2019.114106 Received 15 July 2019; Received in revised form 14 October 2019; Accepted 8 November 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Chang Huang, et al., Applied Energy, https://doi.org/10.1016/j.apenergy.2019.114106
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DNI (W m−2) IDNI LCOE levelized cost of energy ($ kWh−1, cents·kWh−1) overall power generated in the SAPG plant (MWe) PSAPG Psolar power generated from solar heat (MWe) Psolar,annual annual electricity generated from the solar energy (kWh) Qb,input heat exchanged inside the boiler (MWth) Qcoal calorific value of coal combustion (MWth) Qloss,exhaust heat loss due to exhaust gas in the boiler (MWth) Qloss,other other heat loss in the boiler (MWth) Qsolar,input solar heat input (MWth) ηb boiler efficiency (%) thermal cycle efficiency (%) ηc,t ηeste effective solar-to-electricity efficiency (%) efficiency of solar field (%) ηs
Nomenclature DC DNI FS HTF HP PB SAPG SM TES TWS ESTE Asf
direct cost direct normal irradiance fuel-saving heat transfer fluid high-pressure power-boosting solar aided power generation solar multiple thermal energy storage tempering water spray effective solar-to-electricity efficiency aperture area of the solar field (m2)
analysis in [10], the solar field areas and the capacity of thermal energy storage (TES) of a 330 MWe SAPG unit operated in fuel-saving (FS) mode were optimized. In addition, it was concluded that TES would improve efficiency by smoothing the solar thermal into the plant. In [11], Wu pointed out that using TES to store surplus solar heat would increase the annual solar generation and improve the annual efficiency. However, one of the greatest drawbacks for TES is its high cost. Based on the economic optimization in [12], the optimum solar multiple (SM) and lowest levelized cost of electricity (LCOE) were 1.3 and 7.5 cents kWh−1 for a 300 MWe SAPG plant operated in power-boosting (PB) mode. In [13], the optimized results of a 150 MWe SAPG combined heat and power plant indicated that switching integrate mode would have a lower LCOE. In [14], the impact analysis of station capacities
and solar field size were discussed, and Huang concluded that for different solar field aperture area and capacity of units, the optimization integrated scenario based on different criteria varied. In [15], the optimum turbine exit pressure for a 600 MWe SAPG plant was obtained. The plant had air-cooled condenser operated in PB mode to achieve maximum net power output. An operational mode mixed with the PB mode and FS mode had been demonstrated by Eric, based on the economic performance analysis in [16]. Moreover, Eric analyzed the performance of a 300 MWe SAPG plant with diverse “configuration-operation” combinations operated in PB mode, and pointed out some scenarios with the best annual performance. In [17], Hong gave a theoretically and actually correlation of solar-to-electricity efficiency with the exergy destruction of an SAPG system. Hou [18] analyzed and
Fig. 1. Schematic diagram of an SAPG plant. 2
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discussed the performance of a 330 MWe SAPG operated in FS mode. The results showed that the SAPG plant had the maximum allowed solar thermal output over a range of different safe temperature of primary/ reheated steam. Li analyzed and optimized the annual performance of a solar tower aided power generation plant in [19]. A set of transient models of an SAPG plant had been developed by Huang in [20]. Moreover, an adjusting mechanism was proposed to ensure a safe and stable operation. In [21], Zhang proposed the concept of the solar field equivalent efficiency and obtained the optimal velocity of heat transfer fluid in absorber, and to improve the performance of solar field in a 330 MWe SAPG plant operated in FS mode. In terms of the evaluation methods, in 2013, the power generated from solar energy was defined by Hou [22] to evaluate the benefits in an SAPG plant. Then, in 2016, Hou proposed an evaluation method based on exergy analysis [23]. There was an about 5% difference between two studies’ results to evaluate the power generated from solar energy. In addition, “energy, exergy, economic and environmental” method was used to analyze the SAPG system by Adibhatla in [24]. The life cycle assessment and energy-utilization diagram were used to evaluate the SAPG system by Zhai in [25]. An evaluation methods of solar contribution in an SAPG plant was proposed by Zhu in [26] and was used for optimization operation by Zhai in [27]. In order to improve the performance of an SAPG plant, most studies mainly focused on the optimization, such as optimizing solar field size/ SM, TES size, integration mode and plant capacity. However, few studies had focused on the solar heat dumped previously. Since solar radiation changes continuously during the day, the direct normal irradiance (DNI) deviates from its designed value (under the design condition) most of time. To ensure sufficient solar energy is input most of the day, the solar field size is normally designed with an SM greater than one. On the other hand, SM greater than one may result in solar heat collected becomes surplus. Namely, the solar heat collected exceeds the heat demand (to 100% replace the target extraction steam). Since TES is not a necessary device in an SAPG plant [12], if without TES, the surplus solar heat would be dumped. The solar heat dumped weakens the performance of an SAPG plant. Therefore, two measures
are proposed herein to apply in the SAPG plant to use surplus solar heat, which are: 1) being charged into TES and 2) preheating (feedwater) continuously. However, the impacts of measure 2, using surplus solar heat to preheat continuously, on the boiler remain unknown. In our previous work [20], it was found that the introduction of solar heat would cause the boiler deviating from the design conditions, thus affecting the stable operation. When solar input continues increasing, i.e. the surplus solar heat is used to continue preheating the feedwater, a higher temperature of feedwater entering the boiler would inevitably further affect the boiler’s operation. It is necessary to analyze the impact of preheating continuously and propose an adjusting mechanism correspondingly. However, few studies focus on this issue. In this study, a set of transient models of an SAPG plant and an adjusting mechanism are constructed for simulation, which is closer to the real response. Then, this study analyzes the impacts of three ways to deal with the surplus solar heat, which are being dumped, being charged and preheating continuously. Moreover, the maximum solar energy is determined by a safety assessment to ensure a safe and stable operation. Finally, the economic performances of these three ways are analyzed and compared in this study. The findings in this study supplement the knowledge gap, i.e. the unclear impact of (surplus) solar input on the SAPG plant, especially on the boiler. 2. System description Fig. 1 shows a typical schematic diagram of an SAPG plant in which the 2nd and 1st extracted steam is routed to high-pressure (HP) feed heaters, and are both designed to be replaced orderly by solar thermal energy. Through the HTF/water exchanger, the solar thermal energy carried by a heat transferring fluid (HTF) preheats the feedwater. The saved extraction steam can then continue expanding in the turbine to generate power, which is the most common of all SAPG systems. Depending on the load requirements, the SAPG plant can be operated under power-boosting (PB) mode, where the output increases with the same flow rate of the primary steam outlet boiler; or fuel-saving (FS) mode, where the fuel consumption rate decreases with the same
Fig. 2. Schematic diagram of a boiler: E, P, S and R stand for the evaporation, preheating, superheating and reheating stages, respectively. 3
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generating capacity. Since the impact of solar input on the plant operated in these two modes are obviously different (see Section 5 for detail), both modes are analyzed in this study.
It can be seen from Eq.(5), the correlation between ηeste and Qsolar,input is complicated as ηc,t and ηb also will change if Qsolar,input varies. Moreover, the solar-to-electricity efficiency ηste can be expressed as follows
3. Models The SAPG plant presented in Fig. 1 includes the turbine subsystem, the solar field subsystem and the boiler subsystem, shown in Fig. 2, which used a modular modeling method in [20]. Fig. 2 illustrates the details of the boiler in which the feedwater is heated in the plant. In the boiler, the heat required comes from flue gas generated by the combustion of coal. To control the temperature of the superheated and reheated steam, the burner tilt could be adjusted across ± 30°; the 1st, 2nd superheated and reheated tempering water spray (TWS) flows could also be adjusted through spray attemperators (A1 to A3). Since the temperature of exit flue gas is analyzed and discussed to demonstrate the change of boiler efficiency, the detail of the air preheater model can be seen in Appendix A. A summary of the modules included in each subsystem is given accordingly in Table. 1.
ste
Psolar Psolar = Qsolar , input IDNI Asf S /106
LCOE =
Qcoal
AF =
(1)
PSAPG PSAPG = Qinput Qb, input + Qsolar , input
=
b
Qb, input Qcoal
= 100%
Qloss, exhaust + Qloss, other Qcoal
= =
=
Psolar Qsolar , input
PSAPG
=
(3)
5. The effect analysis and safety requirements
(4)
As our previous work [20] shown, in an SAPG plant, the solar heat input would cause redistribution of the heat transfer in boiler sections, resulting in temperature drop of superheated/reheated steam. Some control measures, e.g. adjusting the burner tilt and the TWS (tempering Table 1 List of main models. Module name Boiler subsystem Burner Drum Evaporation heater: water wall Heaters: superheater, reheater, economizer Air preheater Turbine subsystem Turbine stage Condenser Regenerative heaters and deaerator Solar field subsystem Solar collector unit HTF/water heat exchanger
Qcoal b, ori c, t , ori Qsolar , input
(Qsolar , input + Qcoal b ) c, t Qcoal b, ori c, t , ori Qsolar , input c, t
+
Qcoal Qsolar , input
(
b c, t
b, ori c, t , ori )
(8)
Scenarios 1: the surplus solar energy is dumped. The SAPG unit would be operated under the design condition when all the highpressure regenerative extracted steam is displaced. Scenarios 2, i.e. measure 1: the surplus solar energy is charged into a TES, which will be used to supplement the preheating of feedwater when solar energy is insufficient. Scenarios 3, i.e. measure 2: the surplus solar energy is used to preheat feedwater continuously, which would make the temperature of feedwater entering the boiler higher than the designed value.
where Qb,input is the heat exchanged inside the boiler (MWth), Qloss,exhaust is the heat loss due to exhaust gas, which is the main heat loss in the boiler (MWth). The main influential factors of Qloss,exhaust are the exit flue gas temperature and the volumetric flow rate. Qloss,other is the other heat loss, including the heat lost due to unburned gaseous combustibles, unburned solid combustibles, radiation and convection (MWth). The excess air coefficient is approximate constant of 1.25 in this case. Therefore, the boiler efficiency can be reflected effectively by the temperature of exhaust gases when the other heat lost keep almost unchanged. Combined with Eqs. (1) to (4), the ESTE ηeste can be expressed as follows: este
r (r + 1) D (r + 1) D 1
There are three possible ways/scenarios to deal with the surplus solar energy/heat:
Qcoal = Qb, input + Qloss, exhaust + Qloss, other =
(7)
Psolar , annual
4. Scenarios set
where PSAPG is the overall power generated in the SAPG plant (MWe), Qcoal is the calorific value of coal combustion (MWth), ηc,t and ηb are the thermal cycle efficiency and the boiler efficiency. The subscript ori stands for the original plant, i.e. without solar heat input. Due to the solar input, the changes in steams flows make the turbine and boiler work under off-design conditions. Therefore, the efficiencies would change correspondingly. The thermal cycle efficiency ηc,t and the boiler efficiency ηb can be expressed as follows: c, t
(Ccapital·AF ) + CO & M
where r is discount rate and D is lifetime of the power station (year). In the study, r and D are set 6% and 30 years respectively.
(2)
b, ori c , t , ori
(6)
where Psolar,annual is the annual electricity generated from the solar energy (kWh); Ccapital is the increased total capital cost after the solar heat is introduced into the SAPG system ($); CO&M is annual operating and maintenance expenditure of solar field ($); AF is annuity factor defined as
where IDNI is the DNI (W m−2), Asf is the aperture area of the solar field (m2), Qsolar,input is the solar heat input (MWth), ηs is the efficiency of solar field, Psolar is the power generated from solar heat (MWe), which can be calculated as follows:
Psolar = PSAPG
este · S
In order to investigate the economic effect of an SAPG plant, the levelized cost of energy (LCOE) ($ kWh−1) can be calculated through follows [28].
To evaluate the technical performance of an SAPG, the effective solar-to-electricity efficiency (ESTE) ηeste in an SAPG plant is defined as follows [22]:
=
Psolar = IDNI Asf /106
3.2. Levelized cost of energy
3.1. Effective solar-to-electricity efficiency
este
=
(5) 4
Module order B.1 B.2 B.E1 B.P1, B.S1 to B.S6, B.R1 to B.R3 B.A T.1 T.3 T.4, T.5, T.6 S.1 S.2
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water spray) could restore the steam temperature to a certain degree. When the surplus solar heat is used to preheat feedwater continuously (i.e. the measure 2), it would intensified the redistribution. Namely, the drop of the steam temperature at boiler’s outlets might exceed the acceptable level for safety. Therefore, how much the surplus solar energy could be integrated to the system must be determined. The maximum should meet the following requirements:
remains almost unchanged, the power generated in both modes are kept generally unchanged also. 5.1.2. Impacts on the boiler, fuel consumption and boiler efficiency Fig. 7 shows the temperature of feedwater entering the boiler. The feedwater temperature keeps constant at designed 249 °C in both modes until 100% displacement. It can be seen that when the solar input is less than the given maximum value (99.9 MWth in FS mode and 129.1 MWth in PB mode), the feedwater temperature out of the economizer is lower than 353 °C (i.e. 10 °C below the saturating temperature of 363 °C at 19.56 MPa), thus meeting the Requirement a). Fig. 8 shows the coal consumption vs. the solar input in both modes. In FS mode, the coal consumption declines as the solar input increases and the primary steam flow decreases (shown in Fig. 4). When surplus solar heat is used to continue preheating the feedwater, as suggested by measure 2, the temperature of the feedwater entering the boiler increases, thus reducing the heat required in the boiler, i.e. the fuel consumption. While in PB mode, the coal consumption almost keeps unchanged until surplus solar heat is introduced. Fig. 9 shows the material flows diagram of the boiler subsystem. Based on the parameters analyzed above, Table. 2 summarizes the changes trends of the steam flow rate and temperature at the boiler inlets with the increase of solar input, before and after (as measure 2 suggested) the solar heat demand is reached. In both modes, the increase of reheating steam flow due to the increase of solar input (before the maximum solar input value reaches) requires more heat to be exchanged in the reheating stage to avoid temperature drop of reheated steam. Swings up the burner tilt and reduces the TWS attemperations are two ways adopted to achieve this [20]. When the surplus solar heat is used to heat feedwater, i.e. measure 2, the temperature of feedwater entering boiler (water walls) increases. The steam generated in the water walls (evaporation stage) increases. Thus, to avoid temperature drop in primary steam, more heat to be exchanged in superheating stage is required. Swinging up the burner tilt and reducing the TWS attemperations can also help to avoid this problem, as shown in Fig. 10. However, there are limits in these two ways. If the flow rates of steam attemperations are down to zero and the burners’ tilt angles are adjusted to the maximum (i.e. 30°), the temperatures of the steams are still not fully restored, the (surplus) solar energy input would have to be reduced to ensure the operation safe/stable. Therefore, there is a maximum solar energy input to meet the safety Requirement b), i.e. maintaining temperatures of primary/ reheated steam within ± 5 °C, as shown in Fig. 11. It can be seen from Figs. 10 and 11, the burner tilt has been swung
a) To ensure the economizer safe, the temperature of the feedwater in the economizer outlet must be at least 10 °C lower than the saturation temperature, according to the manufacturer. b) At the outlets of the boiler, the temperature of steam must not differ from the designed temperature by ± 5 °C [29]. c) The temperature of the exhaust flue gas (out of the air heater) should not be lower than the dew temperature to ensure the air heater safe. By a safety assessment, the maximum solar energy is determined to be 99.9 MWth with surplus solar heat of 28.0 MWth in FS mode and 129.1 MWth with surplus solar heat of 50.8 MWth in PB mode. The detail safety assessment will be discussed in Section 5.1.2. 5.1. Impacts of inputting solar energy on an SAPG plant In an SAPG plant, as solar input increases, more and more the 1st and 2nd extraction steam will be displaced until 100% displacement. Depending on the operation mode (PB mode or FS mode), the mass flow rates of primary/reheated steam and feedwater are adjusted accordingly. Therefore, the changes of solar input will affect the operations of boiler and turbines, and then the whole plant. The detailed impacts of the solar input change on the key parameters or the performance of the plant are discussed below. 5.1.1. Impacts on the turbine and thermal cycle efficiency Fig. 3 shows the flow rates of 1st and 2nd extracted steam varies with the changes of the solar input. It can be seen in Fig. 3, the 2nd and 1st extracted steam is gradually reduced and eventually dropped to zero (100% displacement) as solar input increases to 71.9 MWth in FS mode and 78.3 MWth in PB mode. When the amount of solar heat exceeds these values, it becomes surplus. Fig. 4 shows the flow rates of primary and reheated steam varies with solar input. Since the 2nd extraction is located after the reheating stage, thus the flow rate of the reheated steam can be expressed by Eq. (9).
mr = m p
mex,1 + mTWS, r
(9)
where m stands for the mass flow rate, and the subscript r, p, ex,1 and TWS,r represent the reheated steam, primary steam, 1st extracted steam and reheating TWS. As more 2nd and 1st extraction steam is saved with the increase of solar heat input, the quantity of the exhaust steam entering the condenser increases, which increases the heat discharged in the condenser, resulting in a decrease in the thermal cycle efficiency ηc,t, defined in Eq. (3), as shown in Fig. 5. Furthermore, due to the reduction of the steam flow through the turbine (Fig. 4) in FS mode, the isentropic efficiency of the turbine decreases, resulting in a larger reduction in ηc,t than that in PB mode [30]. It is noted that the slight drops at the end of the ηc,t curves in both modes are caused by the reduction in the temperature of reheating steam (detail see Section 5.1.2). Fig. 6 shows the SAPG plant’s power generated in FS mode and PB mode. In FS mode, with the increase of solar heat input, the primary steam flow decreases to keep power generated roughly constant. In PB mode, the primary steam flow remains unchanged, which results in an increasing power generated. If surplus solar heat is used to continue heat the feedwater as the measure 2 suggests, the turbine subsystem
Fig. 3. Flow rates of 1st and 2nd extracted steam in FS mode and PB mode. 71.9 MWth and 78.3 MWth are the heat demands in FS mode and PB mode respectively. 5
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Fig. 7. The temperature of feedwater. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 4. Flow rate of primary/reheated steam in FS mode and PB mode. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 8. The coal consumptions in FS mode and PB mode. Fig. 5. The thermal cycle efficiency ηc,t varies with solar input in FS mode and PB mode.
Fig. 9. The heat transfer fluid flows diagram of the boiler subsystem: TWS is the tempering water spray.
up to the maximum (30°) and the reheated TWS has been reduced to the minimum (zero tons·h−1) when the inputting solar heat reaches 86.1 MWth in FS mode and 121.1MWth in PB mode. Then, if continuously increasing surplus solar heat input, the temperature of the reheated steam starts to deviate from design temperature of 543 °C and eventually drop beyond 5 °C with surplus solar heat input of 99.9 MWth
Fig. 6. Power generated PSAPG varies with solar input in FS mode and PB mode.
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Table 2 The states of the input parameters of boiler subsystem before and after the heat demand is reached: ↑, ↓ and — stand for increase, decrease, unchanged/slightly change respectively; Temp. stands for Temperature. FS mode
PB mode
Before
Feedwater Reheating steam Flue gas
After
Before
After
Flow rate
Temp.
Flow rate
Temp.
Flow rate
Temp.
Flow rate
Temp.
↓ ↑ ↓
— — —
— — ↓
↑ — —
— ↑ —
— — —
— — ↓
↑ — —
In summary, 99.9 MWth with the surplus solar heat of 28.0 MWth in FS mode and 129.1 MWth with the surplus solar heat of 50.8 MWth in PB mode are the maximum surplus solar inputs that meet the Requirement a), b) and c) to ensure the plant safe. Fig. 14 shows the differences of the flue gas temperature between that of the original boiler (i.e. without solar input) and that integrated with various solar input. It reveals the exhaust flue gas temperatures decrease in both modes with various solar inputs. Commonly, the excess air coefficient is approximate constant (of 1.25). The adiabatic flame temperature varies with the temperature of the air out of the air preheater. Owing to the upward swing of the burner tilt would reduce the heat to be transferred in the evaporation stage, thus the temperature of the flue gas out of combustion section is increased. From Fig. 14 it can be seen when solar input is just enough to displace all extraction steam, the temperature of the flue gas leaving the water walls rises with swung up of burner tilt in PB mode, while decreasing in FS mode. The reason is that, in PB mode, the flue gas flow almost remains unchanged, while in FS mode, the flue gas flow reduced due to less coal consumption, as shown in Fig. 8. When the surplus solar heat is used to continue preheating the feedwater (i.e. the measure 2), the temperature of feedwater entering to the boiler increases. Therefore, less heat is required for the evaporation stage, which causes the temperature rising of flue gas out of the water walls in both modes. In the preheating stage, i.e. the economizer, if surplus is used, the temperature of feedwater increases. The temperature difference between flue gas and feedwater decreases. Thus, the heat transfer declines, resulting in a higher temperature of flue gas out of economize increases, as the pentagrams shown. Additionally, the reduction of flue gas flow through the air preheater is also the main factor that contributes to the decrease in the
Fig. 10. The adjustment of burner tilt and TWS in FS mode and PB mode.
Fig. 11. The temperatures of primary/reheated steam in FS mode and PB mode.
in FS mode, and of 129.1 MWth in PB mode. Hence, the maximum solar input to maintain temperatures of primary/reheated steam is determined to meet Requirement b). The introduction of solar heat make the boiler work under off-design conditions, thus changing the boiler efficiency ηb. As shown in Fig. 12, the detailed simulation shows that as solar input increases, the boiler efficiencies increase in both modes. The reason is that, as solar input increases, exhaust gas temperature decreases, as shown in Fig. 13. According to Eq. (4), the definition of boiler efficiency ηb increases. However, for safety purpose, the minimum exhaust temperature should higher than the dew point (about 100 °C) [31], i.e. to meet the Requirement c).
Fig. 12. The boiler efficiency ηb varies with solar input in FS mode and PB mode. 7
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Fig. 15. Power generated from solar Psolar and effective solar-to-electricity efficiency ηeste in FS mode and PB mode. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 13. The temperature of exhaust flue gas (out of air heater) in FS mode and PB mode.
temperature of exhaust flue gas.
with a TES with an enough capacity, the solar heat dumped may be avoided. However, due to the high cost of TES, it may not be an economic choice. If the surplus solar heat is used to continue preheating feedwater (measure 2), the more solar input, the higher benefits would be, as shown in Fig. 15. However, since there is a maximum solar input to ensure operation safe, when the solar heat collected excesses the maximum value, the exceeded solar heat should be dumped, thus affecting the technical performance. To make a comparison among these three ways to deal with the surplus solar heat, i.e. being dumped, being charged and preheating continuously, it is necessary to analyze and optimize the annual economic performances.
5.1.3. Impacts on effective solar-to-electricity efficiency Fig. 15 shows the correlation between solar input Qsolar,input and the ESTE ηeste calculated by Eq. (5). From Fig. 15 it can be seen that generally when the solar input Qsolar,input is in the safety interval, the more Qsolar,input, the more power generated by solar energy Psolar and the higher ESTE ηeste in both modes. In FS mode, when the plant runs under the design condition (with 71.9 MWth solar input) and the maximum solar input condition (with 99.9 MWth solar input), the ESTEs ηeste are 36.9% and 39.5% respectively; while in PB mode, they are 40.0% (with 78.3 MWth solar input) and 42.4% (with 129.1 MWth solar input) respectively. 5.1.4. Summary and discussion Summarily, according to the impact analysis, the conclusion can be drawn that using surplus solar heat to continue preheating feedwater is acceptable, if the adjusting mechanism is adopted correspondingly. By the safe assessment, the maximum (surplus) solar input can be determined, which is 99.9 MWth (SM ≈ 1.4) in FS mode and 129.1 MWth (SM ≈ 1.7) in PB mode to ensure a safe and stable operation. Moreover from Fig. 15 indicates above, as the solar input increases to the maximum value, the effective solar-to-electricity efficiency keeps increasing. If TES is equipped with the SAPG plant, as measure 1 suggests, the solar heat stored in the TES should be used/discharged as soon as the solar input drops below the heat demand level, to bring the solar displacement back to 100% and meet a maximum efficiency. Equipped
6. Economic analysis of the three ways to deal with the surplus solar heat In order to compare the economic performances of the three ways to deal with the surplus solar heat, the SAPG plant described in Section 2 is assumed to be located in the middle of the Tibetan Plateau, Lhasa, one of the most solar resourceful areas in China. The combination of low solar zenith angles with low atmospheric attenuation and good weather contributes an annual radiation of 2600 kWh m−2 y−1 in a typical year. In the solar field subsystem, the mean temperature difference between the heat transfer fluid and feedwater is assumed to be 20 °C [32]. The aperture area of each loop is 1413 m2 (47.1 m × 5 m × 6). The highest DNI value is chosen near the solar noon (1144 W m−2 on 25th
Fig. 14. The differences of the flue gas temperature between that of the original boiler and that integrated with various solar input. 8
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Table 3 Design point parameters for the solar field of an SAPG plant. Design point parameters DNI (W·m−2) Altitude (m) Longitude (°) Latitude (°) Ambient temperature (°C) Incidence angle (°) Solar filed size (m2) Solar heat collected (MWth) Efficiency of solar field (%)
Value 1144 3658 91.1 29.7 26.1 6.4
FS mode 91,845 72.0 68.5
PB mode 100,323 78.6 68.5
Table 4 Main initial investment of an SAPG plant [33]. Items
Unit
Value
Direct cost (DC) Solar field Thermal energy storage
$·m−2 $·kWh−1
242.0 31.4
Indirect cost Contingency Engineer, procure, construct Project, land, management Operating and maintenance expenditure
% % % %
10.0 15.0 3.5 1.7
of of of of
DC DC DC DC
Fig. 17. Levelized cost of energy and solar heat wasted of Scenarios.2, i.e. measure 1, varies with solar field size and TES capacity in FS mode.
Fig. 18. Levelized cost of energy and solar heat wasted of Scenarios.2, i.e. measure 1, varies with solar field size and TES capacity in PB mode.
Fig. 16. Levelized cost of energy and solar heat wasted of Scenarios.1 with solar field size in FS mode and PB mode.
June P.M.14:00) as the design point. It requires 65 loops (91,845 m2) in FS mode and 71 loops (100,323 m2) in PB mode to meet the heat demand of 71.9 MWth and 78.3 MWth respectively. Table. 3 summarizes the parameter values at the design point. The costs of the main components in the study are given in Table 4, based on which the LCOE, defined in Eq. (6), is calculated. Fig. 16 shows the LCOE with different solar field areas if the surplus solar heat is dumped. Figs. 17 and 18 show the LCOE with different solar field areas when the plant is run with measure 1 in FS mode and PB mode, respectively. Fig. 19 shows the LCOE with different solar field areas when the plant is run with measure 2. As Figs. 16–19 show, with solar field size increasing, the LCOE curves drop firstly, then rise. There is a minimum LCOE value in each scenario to obtain the best economic performance. Table. 5 summaries the best performances of three ways to deal with the surplus solar energy. It can be seen by comparing Figs. 16–19, preheating continuously,
Fig. 19. Levelized cost of energy and solar heat wasted of Scenarios.3, i.e. measure 2, varies with solar field size in FS mode and PB mode.
measure 2, is the best way to reduce the solar heat dumped, in terms of LCOE and annual ηste, which can meet 5.2 cents/kWh and 18.2% in FS mode and 4.7 cents/kWh and 20.2% in PB mode, respectively. In this 9
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input by using which to continue preheating feedwater. However, there is a maximum amount surplus solar heat input to make sure the plant, especially the boiler runs safely. In this case, the maximum solar energy is determinedto be 99.9 MWth (solar multiple value ≈ 1.4) in fuel-saving mode and 129.1 MWth (solar multiple value ≈ 1.7) in power-boosting mode. 2) In general, when the solar input is less than the maximum value, the more solar heat input Qsolar,input, the higher the effective solar-toelectricity efficiency ηeste will be. Therefore, measure 2 would have better technical performance than measure 1, and both measures would be superior to dumping surplus solar heat. 3) Due to the high cost of a thermal energy storage system, measure 2 is also more favorable than measure 1, in terms of levelized cost of energy. 4) Compared to the fuel-saving mode, the power-boosting mode is preferable technically and economically. The optimal levelized costs of energy by using measure 2 are 5.2 cents/kWh in fuel-saving mode and 4.7 cents/kWh in power-boosting mode, respectively.
Table 5 The best performances of three ways to deal with the surplus solar energy: Asf is the solar field size and annual ηste is the annual solar-to-electricity efficiency. Three ways
Being dumped Being charged Preheating continuously
FS mode
PB mode
Asf (×103 m2)
LCOE (cents/ kWh)
Annual ηste (%)
Asf (×103 m2)
LCOE (cents/ kWh)
Annual ηste (%)
111.0 124.0 137.0
5.3 5.5 5.2
17.8 17.9 18.2
120.0 133.0 192.0
4.9 5.0 4.7
19.3 19.4 20.2
case, after equipping the plant with a TES, i.e. measure 1, although the technical performance is improved, the economic performance is decreased in either PB or FS mode due to the high cost of TES. 7. Conclusions
Declaration of Competing Interest
This study investigates, thoroughly, the impacts of three ways to deal with the surplus solar heat. The two measures, i.e. being charged and preheating continuously, are compared with being dumped by a desktop case study of a 330 MWe solar aided power generation plant. As a solar aided power generation plant can be operated in either powerboosting or fuel-saving mode, the technical (in terms of effective solarto-electricity efficiency ηeste) and economic (in terms of levelized cost of energy) performances of the three ways in each mode are also studied. The following conclusions are therefore drawn:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Funding: This work was supported by Guangzhou Power Supply Co., Ltd. Science and Technology Program (ZHKJXM20180152), National Natural Science Foundation of China No. 51876057 and the China Scholarship Council No. 201806730031.
1) A solar aided power generation plant is able to accept surplus solar Appendix A Air preheater
A tri-sector air preheater is configured to use the flue gas preheating the primary air and secondary air, as Fig. A.1 shown. The lumped parameter method is used in modeling [34]. It is assumed that ignoring axial heat exchange of the air heater [35]. Due to air leak, the mass flow of flue gas, primary air and secondary air out of the air heater can be expressed as follows.
mG, out = mG, in + mPG, leak + mSG, leak mP, out = mP, in mPG, leak mPS, leak mS, out = mS, in mSG, leak + mPS, leak
(A.1)
where m stands for the mass flow rate (kg s−1), the subscript G, P, S, in, out represent flue gas, primary air, secondary air, inlet and outlet respectively. The subscript type of AB, leak stands for the leak from A to B. In the air preheater, the heat transfer Q (W) can be expressed as follows
Fig. A1. Scheme of an air preheater. 10
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QGM = kG AG (Tavg, G
TM , G ) = mG, in cG, in TG, in
mG, out cG, out TG, out + QPG, leak + QSG, leak
QMP = kP AP (TM , P
Tavg , P ) = mP, in cP, in TP, in
mP, out cP, out TP, out
QPG, leak
QMS = kS AS (TM , S
Tavg, S ) = mP, in cP, in TP , in
mP , out cP, out TP, out
QPG, leak + QP S, leak
QPS, leak (A.2) −2
−1
2
where the subscript M stands for the metal body; k, A, T and c represent heat transfer coefficient (W m K ), heat exchange area (m ), temperature (K) and heat capacity (J kg−1 K−1), Tavg are the average temperatures (K), which can be calculated as follows.
Tavg, G = Tavg , P = Tavg , S =
TG, in + TG, out 2 TP , in + TP , out 2 TS , in + TS , out 2
(A.3)
In Eq. (A.2), TM are the average temperatures of the metal body (K), which can be expressed as follows. G G+ P+ S P G+ P+ S S G+ P+ S
mM cM
dTM , G
mM cM
dTM , P
mM cM
dTM , S
d d d
=
(TM , P
=
(TM , S
=
(TM , G
TM , G )
G+ P+ S
TM , P )
G+ P+ S
TM , S )
G+ P+ S
mM cM + QGM mM cM
QMP
mM cM
QMS
(A.4)
where mM and cM are the mass of the whole metal body (kg) and heat capacity (J kg−1 K−1); θ are the angle of the flow section (°), shown in Fig. A.1; ω is the rotational speed of the metal body in the air preheater (°·s−1), τ is the time step, 1 s. In Eq. (A.2), the heat transfer coefficient k can be calculated as follows.
kG = kP = kS =
G
de P
de S
de
( ) ( ) ( )
VG d e 0.8 0.4 Pr , G CT , G G
VP de 0.8 0.4 Pr , P CT , P P VS de 0.8 0.4 Pr , S CT , S S
(A.5) −1
−1
2 −1
where λ, Pr and v are thermal conductivity (W m k ), Prandtl number and kinematic viscosity (m s ), respectively; de is equivalent diameter (m); V is the flow velocity (m s−1). CT is the temperature correction factor, which can be calculated as follows.
CT , G = 1
( ) =( )
CT , P = CT , S
Tavg , P 0.5 TM
Tavg, S 0.5
(A.6)
TM
where TM is the average temperatures of the whole metal body, which can be expressed as follows.
TM =
TM , G
G
+ TM , P P + TM , S G + P + S
S
(A.7)
Then, the temperature of exhaust gas, primary air and secondary air (out of air preheater) can be calculated as follows.
TG, out = TP, out = TS , out =
mG, in cG, in TG, in + kG AG TM , G
TG, in + QPG, leak + QSG, leak 2
kG AG + mG, out cG, out 2 TP , in QPG, leak 2 kP AP + mP , out cP, out 2
mP , in cP, in TP, in + kP AP TM , P
mS , in cS , in TS, in + kS AS TM , S
TS, in 2
QPS, leak
QSG, leak + QPS, leak
kS A S + mS, out cS, out 2
(A.8)
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Measures to reduce solar energy dumped in a solar aided power generation plant Chang Huanga, Hongjuan Houa, , Eric Hub, Gang Yua, Si Chenc, Yongping Yanga ⁎
a
National Thermal Power Engineering & Research Center, North China Electric Power University, Champing District, Beijing 102206, China School of Mechanical Engineering, the University of Adelaide, SA 5005, Australia c Power China International Group Limited, Haidian District, Beijing 100036,China b
HIGHLIGHTS
surplus solar energy continue heating feedwater are proposed and analysed. • Using maximum surplus solar input to the system is deduced by a safety assessment. • The • Preheating feedwater continuously is superior to being charged. ARTICLE INFO
ABSTRACT
Keywords: Solar heat dumped Thermal energy storage system Preheating continuously Technical and economic performances Levelized cost of electricity
Solar aided power generation technology has been proved to be one of the most efficient ways to integrate solar energy into a coal-fired power plant. In a typical plant, the solar field size is normally designed with a solar multiple greater than one. Therefore, sometimes collected solar heat becomes surplus when the collected solar heat exceeds the heat demand. Other than being dumped, the surplus solar heat could be either charged into a thermal energy storage system if there is one; or used to preheat (feedwater) continuously. In this paper, the impacts of these two measures (being charged and preheating continuously) on the technical and economic performances of the integrated plant are analyzed in detail by undertaking a desktop case study. The results show that both measures could reduce the surplus solar heat dumped, but the economic performances vary. For the measure of preheating continuously, the maximum solar input is determined by a safety assessment to ensure a stable and safe operation. Surplus heat over this maximum amount has to be dumped. The results also show that for the plant located in Tibet China, both measures could reduce the levelized cost of energy and enhance the effective solar-to-electricity efficiency, compared to dumping case; while continuing preheating measure is superior to being charged technically and economically.
1. Introduction Due to the shortage of fossil fuels and its negative effects on the environment, renewable energy has attracted more and more attention, such as hydropower, wind power and solar energy. Hydropower plants are critical to the sustainable development of renewable energy [1], however, the increase in water demand for energy generation may be in conflict with the objectives of river conservation [2]. Wind power is gaining worldwide popularity as a large-scale energy source, yet, as the wind power penetration level increases, operating power systems securely and reliably is a serious challenge due to the intermittent nature of wind power [3]. Solar energy has played an essential role in recent years [4], especially, solar thermal energy has attracted increasing
⁎
social and political attention [5]. Solar thermal systems are advantageous since it is easier to store heat than electricity on a large scale [6]. However, standalone concentrated solar power plants are suffering restriction for further development due to their high cost and low efficiency [7]. Solar aided power generation (SAPG), where solar heat serves as a substitution of the extraction steam to preheat feedwater in a regenerative Rankin power station, has been proved an efficient way to utilize solar energy for power generation purpose [8]. The advantages of SAPG are presented in [9], which are not only contribute to increase the efficiencies and reduce fuel consumption, but also provides a better way to use solar heat for power generation. Most current studies on SAPG systems focus on the performance analysis and evaluation methods. Based on the annual performance
Corresponding author. E-mail address: [email protected] (H. Hou).
https://doi.org/10.1016/j.apenergy.2019.114106 Received 15 July 2019; Received in revised form 14 October 2019; Accepted 8 November 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Chang Huang, et al., Applied Energy, https://doi.org/10.1016/j.apenergy.2019.114106
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DNI (W m−2) IDNI LCOE levelized cost of energy ($ kWh−1, cents·kWh−1) overall power generated in the SAPG plant (MWe) PSAPG Psolar power generated from solar heat (MWe) Psolar,annual annual electricity generated from the solar energy (kWh) Qb,input heat exchanged inside the boiler (MWth) Qcoal calorific value of coal combustion (MWth) Qloss,exhaust heat loss due to exhaust gas in the boiler (MWth) Qloss,other other heat loss in the boiler (MWth) Qsolar,input solar heat input (MWth) ηb boiler efficiency (%) thermal cycle efficiency (%) ηc,t ηeste effective solar-to-electricity efficiency (%) efficiency of solar field (%) ηs
Nomenclature DC DNI FS HTF HP PB SAPG SM TES TWS ESTE Asf
direct cost direct normal irradiance fuel-saving heat transfer fluid high-pressure power-boosting solar aided power generation solar multiple thermal energy storage tempering water spray effective solar-to-electricity efficiency aperture area of the solar field (m2)
analysis in [10], the solar field areas and the capacity of thermal energy storage (TES) of a 330 MWe SAPG unit operated in fuel-saving (FS) mode were optimized. In addition, it was concluded that TES would improve efficiency by smoothing the solar thermal into the plant. In [11], Wu pointed out that using TES to store surplus solar heat would increase the annual solar generation and improve the annual efficiency. However, one of the greatest drawbacks for TES is its high cost. Based on the economic optimization in [12], the optimum solar multiple (SM) and lowest levelized cost of electricity (LCOE) were 1.3 and 7.5 cents kWh−1 for a 300 MWe SAPG plant operated in power-boosting (PB) mode. In [13], the optimized results of a 150 MWe SAPG combined heat and power plant indicated that switching integrate mode would have a lower LCOE. In [14], the impact analysis of station capacities
and solar field size were discussed, and Huang concluded that for different solar field aperture area and capacity of units, the optimization integrated scenario based on different criteria varied. In [15], the optimum turbine exit pressure for a 600 MWe SAPG plant was obtained. The plant had air-cooled condenser operated in PB mode to achieve maximum net power output. An operational mode mixed with the PB mode and FS mode had been demonstrated by Eric, based on the economic performance analysis in [16]. Moreover, Eric analyzed the performance of a 300 MWe SAPG plant with diverse “configuration-operation” combinations operated in PB mode, and pointed out some scenarios with the best annual performance. In [17], Hong gave a theoretically and actually correlation of solar-to-electricity efficiency with the exergy destruction of an SAPG system. Hou [18] analyzed and
Fig. 1. Schematic diagram of an SAPG plant. 2
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discussed the performance of a 330 MWe SAPG operated in FS mode. The results showed that the SAPG plant had the maximum allowed solar thermal output over a range of different safe temperature of primary/ reheated steam. Li analyzed and optimized the annual performance of a solar tower aided power generation plant in [19]. A set of transient models of an SAPG plant had been developed by Huang in [20]. Moreover, an adjusting mechanism was proposed to ensure a safe and stable operation. In [21], Zhang proposed the concept of the solar field equivalent efficiency and obtained the optimal velocity of heat transfer fluid in absorber, and to improve the performance of solar field in a 330 MWe SAPG plant operated in FS mode. In terms of the evaluation methods, in 2013, the power generated from solar energy was defined by Hou [22] to evaluate the benefits in an SAPG plant. Then, in 2016, Hou proposed an evaluation method based on exergy analysis [23]. There was an about 5% difference between two studies’ results to evaluate the power generated from solar energy. In addition, “energy, exergy, economic and environmental” method was used to analyze the SAPG system by Adibhatla in [24]. The life cycle assessment and energy-utilization diagram were used to evaluate the SAPG system by Zhai in [25]. An evaluation methods of solar contribution in an SAPG plant was proposed by Zhu in [26] and was used for optimization operation by Zhai in [27]. In order to improve the performance of an SAPG plant, most studies mainly focused on the optimization, such as optimizing solar field size/ SM, TES size, integration mode and plant capacity. However, few studies had focused on the solar heat dumped previously. Since solar radiation changes continuously during the day, the direct normal irradiance (DNI) deviates from its designed value (under the design condition) most of time. To ensure sufficient solar energy is input most of the day, the solar field size is normally designed with an SM greater than one. On the other hand, SM greater than one may result in solar heat collected becomes surplus. Namely, the solar heat collected exceeds the heat demand (to 100% replace the target extraction steam). Since TES is not a necessary device in an SAPG plant [12], if without TES, the surplus solar heat would be dumped. The solar heat dumped weakens the performance of an SAPG plant. Therefore, two measures
are proposed herein to apply in the SAPG plant to use surplus solar heat, which are: 1) being charged into TES and 2) preheating (feedwater) continuously. However, the impacts of measure 2, using surplus solar heat to preheat continuously, on the boiler remain unknown. In our previous work [20], it was found that the introduction of solar heat would cause the boiler deviating from the design conditions, thus affecting the stable operation. When solar input continues increasing, i.e. the surplus solar heat is used to continue preheating the feedwater, a higher temperature of feedwater entering the boiler would inevitably further affect the boiler’s operation. It is necessary to analyze the impact of preheating continuously and propose an adjusting mechanism correspondingly. However, few studies focus on this issue. In this study, a set of transient models of an SAPG plant and an adjusting mechanism are constructed for simulation, which is closer to the real response. Then, this study analyzes the impacts of three ways to deal with the surplus solar heat, which are being dumped, being charged and preheating continuously. Moreover, the maximum solar energy is determined by a safety assessment to ensure a safe and stable operation. Finally, the economic performances of these three ways are analyzed and compared in this study. The findings in this study supplement the knowledge gap, i.e. the unclear impact of (surplus) solar input on the SAPG plant, especially on the boiler. 2. System description Fig. 1 shows a typical schematic diagram of an SAPG plant in which the 2nd and 1st extracted steam is routed to high-pressure (HP) feed heaters, and are both designed to be replaced orderly by solar thermal energy. Through the HTF/water exchanger, the solar thermal energy carried by a heat transferring fluid (HTF) preheats the feedwater. The saved extraction steam can then continue expanding in the turbine to generate power, which is the most common of all SAPG systems. Depending on the load requirements, the SAPG plant can be operated under power-boosting (PB) mode, where the output increases with the same flow rate of the primary steam outlet boiler; or fuel-saving (FS) mode, where the fuel consumption rate decreases with the same
Fig. 2. Schematic diagram of a boiler: E, P, S and R stand for the evaporation, preheating, superheating and reheating stages, respectively. 3
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generating capacity. Since the impact of solar input on the plant operated in these two modes are obviously different (see Section 5 for detail), both modes are analyzed in this study.
It can be seen from Eq.(5), the correlation between ηeste and Qsolar,input is complicated as ηc,t and ηb also will change if Qsolar,input varies. Moreover, the solar-to-electricity efficiency ηste can be expressed as follows
3. Models The SAPG plant presented in Fig. 1 includes the turbine subsystem, the solar field subsystem and the boiler subsystem, shown in Fig. 2, which used a modular modeling method in [20]. Fig. 2 illustrates the details of the boiler in which the feedwater is heated in the plant. In the boiler, the heat required comes from flue gas generated by the combustion of coal. To control the temperature of the superheated and reheated steam, the burner tilt could be adjusted across ± 30°; the 1st, 2nd superheated and reheated tempering water spray (TWS) flows could also be adjusted through spray attemperators (A1 to A3). Since the temperature of exit flue gas is analyzed and discussed to demonstrate the change of boiler efficiency, the detail of the air preheater model can be seen in Appendix A. A summary of the modules included in each subsystem is given accordingly in Table. 1.
ste
Psolar Psolar = Qsolar , input IDNI Asf S /106
LCOE =
Qcoal
AF =
(1)
PSAPG PSAPG = Qinput Qb, input + Qsolar , input
=
b
Qb, input Qcoal
= 100%
Qloss, exhaust + Qloss, other Qcoal
= =
=
Psolar Qsolar , input
PSAPG
=
(3)
5. The effect analysis and safety requirements
(4)
As our previous work [20] shown, in an SAPG plant, the solar heat input would cause redistribution of the heat transfer in boiler sections, resulting in temperature drop of superheated/reheated steam. Some control measures, e.g. adjusting the burner tilt and the TWS (tempering Table 1 List of main models. Module name Boiler subsystem Burner Drum Evaporation heater: water wall Heaters: superheater, reheater, economizer Air preheater Turbine subsystem Turbine stage Condenser Regenerative heaters and deaerator Solar field subsystem Solar collector unit HTF/water heat exchanger
Qcoal b, ori c, t , ori Qsolar , input
(Qsolar , input + Qcoal b ) c, t Qcoal b, ori c, t , ori Qsolar , input c, t
+
Qcoal Qsolar , input
(
b c, t
b, ori c, t , ori )
(8)
Scenarios 1: the surplus solar energy is dumped. The SAPG unit would be operated under the design condition when all the highpressure regenerative extracted steam is displaced. Scenarios 2, i.e. measure 1: the surplus solar energy is charged into a TES, which will be used to supplement the preheating of feedwater when solar energy is insufficient. Scenarios 3, i.e. measure 2: the surplus solar energy is used to preheat feedwater continuously, which would make the temperature of feedwater entering the boiler higher than the designed value.
where Qb,input is the heat exchanged inside the boiler (MWth), Qloss,exhaust is the heat loss due to exhaust gas, which is the main heat loss in the boiler (MWth). The main influential factors of Qloss,exhaust are the exit flue gas temperature and the volumetric flow rate. Qloss,other is the other heat loss, including the heat lost due to unburned gaseous combustibles, unburned solid combustibles, radiation and convection (MWth). The excess air coefficient is approximate constant of 1.25 in this case. Therefore, the boiler efficiency can be reflected effectively by the temperature of exhaust gases when the other heat lost keep almost unchanged. Combined with Eqs. (1) to (4), the ESTE ηeste can be expressed as follows: este
r (r + 1) D (r + 1) D 1
There are three possible ways/scenarios to deal with the surplus solar energy/heat:
Qcoal = Qb, input + Qloss, exhaust + Qloss, other =
(7)
Psolar , annual
4. Scenarios set
where PSAPG is the overall power generated in the SAPG plant (MWe), Qcoal is the calorific value of coal combustion (MWth), ηc,t and ηb are the thermal cycle efficiency and the boiler efficiency. The subscript ori stands for the original plant, i.e. without solar heat input. Due to the solar input, the changes in steams flows make the turbine and boiler work under off-design conditions. Therefore, the efficiencies would change correspondingly. The thermal cycle efficiency ηc,t and the boiler efficiency ηb can be expressed as follows: c, t
(Ccapital·AF ) + CO & M
where r is discount rate and D is lifetime of the power station (year). In the study, r and D are set 6% and 30 years respectively.
(2)
b, ori c , t , ori
(6)
where Psolar,annual is the annual electricity generated from the solar energy (kWh); Ccapital is the increased total capital cost after the solar heat is introduced into the SAPG system ($); CO&M is annual operating and maintenance expenditure of solar field ($); AF is annuity factor defined as
where IDNI is the DNI (W m−2), Asf is the aperture area of the solar field (m2), Qsolar,input is the solar heat input (MWth), ηs is the efficiency of solar field, Psolar is the power generated from solar heat (MWe), which can be calculated as follows:
Psolar = PSAPG
este · S
In order to investigate the economic effect of an SAPG plant, the levelized cost of energy (LCOE) ($ kWh−1) can be calculated through follows [28].
To evaluate the technical performance of an SAPG, the effective solar-to-electricity efficiency (ESTE) ηeste in an SAPG plant is defined as follows [22]:
=
Psolar = IDNI Asf /106
3.2. Levelized cost of energy
3.1. Effective solar-to-electricity efficiency
este
=
(5) 4
Module order B.1 B.2 B.E1 B.P1, B.S1 to B.S6, B.R1 to B.R3 B.A T.1 T.3 T.4, T.5, T.6 S.1 S.2
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water spray) could restore the steam temperature to a certain degree. When the surplus solar heat is used to preheat feedwater continuously (i.e. the measure 2), it would intensified the redistribution. Namely, the drop of the steam temperature at boiler’s outlets might exceed the acceptable level for safety. Therefore, how much the surplus solar energy could be integrated to the system must be determined. The maximum should meet the following requirements:
remains almost unchanged, the power generated in both modes are kept generally unchanged also. 5.1.2. Impacts on the boiler, fuel consumption and boiler efficiency Fig. 7 shows the temperature of feedwater entering the boiler. The feedwater temperature keeps constant at designed 249 °C in both modes until 100% displacement. It can be seen that when the solar input is less than the given maximum value (99.9 MWth in FS mode and 129.1 MWth in PB mode), the feedwater temperature out of the economizer is lower than 353 °C (i.e. 10 °C below the saturating temperature of 363 °C at 19.56 MPa), thus meeting the Requirement a). Fig. 8 shows the coal consumption vs. the solar input in both modes. In FS mode, the coal consumption declines as the solar input increases and the primary steam flow decreases (shown in Fig. 4). When surplus solar heat is used to continue preheating the feedwater, as suggested by measure 2, the temperature of the feedwater entering the boiler increases, thus reducing the heat required in the boiler, i.e. the fuel consumption. While in PB mode, the coal consumption almost keeps unchanged until surplus solar heat is introduced. Fig. 9 shows the material flows diagram of the boiler subsystem. Based on the parameters analyzed above, Table. 2 summarizes the changes trends of the steam flow rate and temperature at the boiler inlets with the increase of solar input, before and after (as measure 2 suggested) the solar heat demand is reached. In both modes, the increase of reheating steam flow due to the increase of solar input (before the maximum solar input value reaches) requires more heat to be exchanged in the reheating stage to avoid temperature drop of reheated steam. Swings up the burner tilt and reduces the TWS attemperations are two ways adopted to achieve this [20]. When the surplus solar heat is used to heat feedwater, i.e. measure 2, the temperature of feedwater entering boiler (water walls) increases. The steam generated in the water walls (evaporation stage) increases. Thus, to avoid temperature drop in primary steam, more heat to be exchanged in superheating stage is required. Swinging up the burner tilt and reducing the TWS attemperations can also help to avoid this problem, as shown in Fig. 10. However, there are limits in these two ways. If the flow rates of steam attemperations are down to zero and the burners’ tilt angles are adjusted to the maximum (i.e. 30°), the temperatures of the steams are still not fully restored, the (surplus) solar energy input would have to be reduced to ensure the operation safe/stable. Therefore, there is a maximum solar energy input to meet the safety Requirement b), i.e. maintaining temperatures of primary/ reheated steam within ± 5 °C, as shown in Fig. 11. It can be seen from Figs. 10 and 11, the burner tilt has been swung
a) To ensure the economizer safe, the temperature of the feedwater in the economizer outlet must be at least 10 °C lower than the saturation temperature, according to the manufacturer. b) At the outlets of the boiler, the temperature of steam must not differ from the designed temperature by ± 5 °C [29]. c) The temperature of the exhaust flue gas (out of the air heater) should not be lower than the dew temperature to ensure the air heater safe. By a safety assessment, the maximum solar energy is determined to be 99.9 MWth with surplus solar heat of 28.0 MWth in FS mode and 129.1 MWth with surplus solar heat of 50.8 MWth in PB mode. The detail safety assessment will be discussed in Section 5.1.2. 5.1. Impacts of inputting solar energy on an SAPG plant In an SAPG plant, as solar input increases, more and more the 1st and 2nd extraction steam will be displaced until 100% displacement. Depending on the operation mode (PB mode or FS mode), the mass flow rates of primary/reheated steam and feedwater are adjusted accordingly. Therefore, the changes of solar input will affect the operations of boiler and turbines, and then the whole plant. The detailed impacts of the solar input change on the key parameters or the performance of the plant are discussed below. 5.1.1. Impacts on the turbine and thermal cycle efficiency Fig. 3 shows the flow rates of 1st and 2nd extracted steam varies with the changes of the solar input. It can be seen in Fig. 3, the 2nd and 1st extracted steam is gradually reduced and eventually dropped to zero (100% displacement) as solar input increases to 71.9 MWth in FS mode and 78.3 MWth in PB mode. When the amount of solar heat exceeds these values, it becomes surplus. Fig. 4 shows the flow rates of primary and reheated steam varies with solar input. Since the 2nd extraction is located after the reheating stage, thus the flow rate of the reheated steam can be expressed by Eq. (9).
mr = m p
mex,1 + mTWS, r
(9)
where m stands for the mass flow rate, and the subscript r, p, ex,1 and TWS,r represent the reheated steam, primary steam, 1st extracted steam and reheating TWS. As more 2nd and 1st extraction steam is saved with the increase of solar heat input, the quantity of the exhaust steam entering the condenser increases, which increases the heat discharged in the condenser, resulting in a decrease in the thermal cycle efficiency ηc,t, defined in Eq. (3), as shown in Fig. 5. Furthermore, due to the reduction of the steam flow through the turbine (Fig. 4) in FS mode, the isentropic efficiency of the turbine decreases, resulting in a larger reduction in ηc,t than that in PB mode [30]. It is noted that the slight drops at the end of the ηc,t curves in both modes are caused by the reduction in the temperature of reheating steam (detail see Section 5.1.2). Fig. 6 shows the SAPG plant’s power generated in FS mode and PB mode. In FS mode, with the increase of solar heat input, the primary steam flow decreases to keep power generated roughly constant. In PB mode, the primary steam flow remains unchanged, which results in an increasing power generated. If surplus solar heat is used to continue heat the feedwater as the measure 2 suggests, the turbine subsystem
Fig. 3. Flow rates of 1st and 2nd extracted steam in FS mode and PB mode. 71.9 MWth and 78.3 MWth are the heat demands in FS mode and PB mode respectively. 5
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Fig. 7. The temperature of feedwater. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 4. Flow rate of primary/reheated steam in FS mode and PB mode. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 8. The coal consumptions in FS mode and PB mode. Fig. 5. The thermal cycle efficiency ηc,t varies with solar input in FS mode and PB mode.
Fig. 9. The heat transfer fluid flows diagram of the boiler subsystem: TWS is the tempering water spray.
up to the maximum (30°) and the reheated TWS has been reduced to the minimum (zero tons·h−1) when the inputting solar heat reaches 86.1 MWth in FS mode and 121.1MWth in PB mode. Then, if continuously increasing surplus solar heat input, the temperature of the reheated steam starts to deviate from design temperature of 543 °C and eventually drop beyond 5 °C with surplus solar heat input of 99.9 MWth
Fig. 6. Power generated PSAPG varies with solar input in FS mode and PB mode.
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Table 2 The states of the input parameters of boiler subsystem before and after the heat demand is reached: ↑, ↓ and — stand for increase, decrease, unchanged/slightly change respectively; Temp. stands for Temperature. FS mode
PB mode
Before
Feedwater Reheating steam Flue gas
After
Before
After
Flow rate
Temp.
Flow rate
Temp.
Flow rate
Temp.
Flow rate
Temp.
↓ ↑ ↓
— — —
— — ↓
↑ — —
— ↑ —
— — —
— — ↓
↑ — —
In summary, 99.9 MWth with the surplus solar heat of 28.0 MWth in FS mode and 129.1 MWth with the surplus solar heat of 50.8 MWth in PB mode are the maximum surplus solar inputs that meet the Requirement a), b) and c) to ensure the plant safe. Fig. 14 shows the differences of the flue gas temperature between that of the original boiler (i.e. without solar input) and that integrated with various solar input. It reveals the exhaust flue gas temperatures decrease in both modes with various solar inputs. Commonly, the excess air coefficient is approximate constant (of 1.25). The adiabatic flame temperature varies with the temperature of the air out of the air preheater. Owing to the upward swing of the burner tilt would reduce the heat to be transferred in the evaporation stage, thus the temperature of the flue gas out of combustion section is increased. From Fig. 14 it can be seen when solar input is just enough to displace all extraction steam, the temperature of the flue gas leaving the water walls rises with swung up of burner tilt in PB mode, while decreasing in FS mode. The reason is that, in PB mode, the flue gas flow almost remains unchanged, while in FS mode, the flue gas flow reduced due to less coal consumption, as shown in Fig. 8. When the surplus solar heat is used to continue preheating the feedwater (i.e. the measure 2), the temperature of feedwater entering to the boiler increases. Therefore, less heat is required for the evaporation stage, which causes the temperature rising of flue gas out of the water walls in both modes. In the preheating stage, i.e. the economizer, if surplus is used, the temperature of feedwater increases. The temperature difference between flue gas and feedwater decreases. Thus, the heat transfer declines, resulting in a higher temperature of flue gas out of economize increases, as the pentagrams shown. Additionally, the reduction of flue gas flow through the air preheater is also the main factor that contributes to the decrease in the
Fig. 10. The adjustment of burner tilt and TWS in FS mode and PB mode.
Fig. 11. The temperatures of primary/reheated steam in FS mode and PB mode.
in FS mode, and of 129.1 MWth in PB mode. Hence, the maximum solar input to maintain temperatures of primary/reheated steam is determined to meet Requirement b). The introduction of solar heat make the boiler work under off-design conditions, thus changing the boiler efficiency ηb. As shown in Fig. 12, the detailed simulation shows that as solar input increases, the boiler efficiencies increase in both modes. The reason is that, as solar input increases, exhaust gas temperature decreases, as shown in Fig. 13. According to Eq. (4), the definition of boiler efficiency ηb increases. However, for safety purpose, the minimum exhaust temperature should higher than the dew point (about 100 °C) [31], i.e. to meet the Requirement c).
Fig. 12. The boiler efficiency ηb varies with solar input in FS mode and PB mode. 7
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Fig. 15. Power generated from solar Psolar and effective solar-to-electricity efficiency ηeste in FS mode and PB mode. The black dash and the red dash stand for the heat demands in FS mode and PB mode respectively.
Fig. 13. The temperature of exhaust flue gas (out of air heater) in FS mode and PB mode.
temperature of exhaust flue gas.
with a TES with an enough capacity, the solar heat dumped may be avoided. However, due to the high cost of TES, it may not be an economic choice. If the surplus solar heat is used to continue preheating feedwater (measure 2), the more solar input, the higher benefits would be, as shown in Fig. 15. However, since there is a maximum solar input to ensure operation safe, when the solar heat collected excesses the maximum value, the exceeded solar heat should be dumped, thus affecting the technical performance. To make a comparison among these three ways to deal with the surplus solar heat, i.e. being dumped, being charged and preheating continuously, it is necessary to analyze and optimize the annual economic performances.
5.1.3. Impacts on effective solar-to-electricity efficiency Fig. 15 shows the correlation between solar input Qsolar,input and the ESTE ηeste calculated by Eq. (5). From Fig. 15 it can be seen that generally when the solar input Qsolar,input is in the safety interval, the more Qsolar,input, the more power generated by solar energy Psolar and the higher ESTE ηeste in both modes. In FS mode, when the plant runs under the design condition (with 71.9 MWth solar input) and the maximum solar input condition (with 99.9 MWth solar input), the ESTEs ηeste are 36.9% and 39.5% respectively; while in PB mode, they are 40.0% (with 78.3 MWth solar input) and 42.4% (with 129.1 MWth solar input) respectively. 5.1.4. Summary and discussion Summarily, according to the impact analysis, the conclusion can be drawn that using surplus solar heat to continue preheating feedwater is acceptable, if the adjusting mechanism is adopted correspondingly. By the safe assessment, the maximum (surplus) solar input can be determined, which is 99.9 MWth (SM ≈ 1.4) in FS mode and 129.1 MWth (SM ≈ 1.7) in PB mode to ensure a safe and stable operation. Moreover from Fig. 15 indicates above, as the solar input increases to the maximum value, the effective solar-to-electricity efficiency keeps increasing. If TES is equipped with the SAPG plant, as measure 1 suggests, the solar heat stored in the TES should be used/discharged as soon as the solar input drops below the heat demand level, to bring the solar displacement back to 100% and meet a maximum efficiency. Equipped
6. Economic analysis of the three ways to deal with the surplus solar heat In order to compare the economic performances of the three ways to deal with the surplus solar heat, the SAPG plant described in Section 2 is assumed to be located in the middle of the Tibetan Plateau, Lhasa, one of the most solar resourceful areas in China. The combination of low solar zenith angles with low atmospheric attenuation and good weather contributes an annual radiation of 2600 kWh m−2 y−1 in a typical year. In the solar field subsystem, the mean temperature difference between the heat transfer fluid and feedwater is assumed to be 20 °C [32]. The aperture area of each loop is 1413 m2 (47.1 m × 5 m × 6). The highest DNI value is chosen near the solar noon (1144 W m−2 on 25th
Fig. 14. The differences of the flue gas temperature between that of the original boiler and that integrated with various solar input. 8
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Table 3 Design point parameters for the solar field of an SAPG plant. Design point parameters DNI (W·m−2) Altitude (m) Longitude (°) Latitude (°) Ambient temperature (°C) Incidence angle (°) Solar filed size (m2) Solar heat collected (MWth) Efficiency of solar field (%)
Value 1144 3658 91.1 29.7 26.1 6.4
FS mode 91,845 72.0 68.5
PB mode 100,323 78.6 68.5
Table 4 Main initial investment of an SAPG plant [33]. Items
Unit
Value
Direct cost (DC) Solar field Thermal energy storage
$·m−2 $·kWh−1
242.0 31.4
Indirect cost Contingency Engineer, procure, construct Project, land, management Operating and maintenance expenditure
% % % %
10.0 15.0 3.5 1.7
of of of of
DC DC DC DC
Fig. 17. Levelized cost of energy and solar heat wasted of Scenarios.2, i.e. measure 1, varies with solar field size and TES capacity in FS mode.
Fig. 18. Levelized cost of energy and solar heat wasted of Scenarios.2, i.e. measure 1, varies with solar field size and TES capacity in PB mode.
Fig. 16. Levelized cost of energy and solar heat wasted of Scenarios.1 with solar field size in FS mode and PB mode.
June P.M.14:00) as the design point. It requires 65 loops (91,845 m2) in FS mode and 71 loops (100,323 m2) in PB mode to meet the heat demand of 71.9 MWth and 78.3 MWth respectively. Table. 3 summarizes the parameter values at the design point. The costs of the main components in the study are given in Table 4, based on which the LCOE, defined in Eq. (6), is calculated. Fig. 16 shows the LCOE with different solar field areas if the surplus solar heat is dumped. Figs. 17 and 18 show the LCOE with different solar field areas when the plant is run with measure 1 in FS mode and PB mode, respectively. Fig. 19 shows the LCOE with different solar field areas when the plant is run with measure 2. As Figs. 16–19 show, with solar field size increasing, the LCOE curves drop firstly, then rise. There is a minimum LCOE value in each scenario to obtain the best economic performance. Table. 5 summaries the best performances of three ways to deal with the surplus solar energy. It can be seen by comparing Figs. 16–19, preheating continuously,
Fig. 19. Levelized cost of energy and solar heat wasted of Scenarios.3, i.e. measure 2, varies with solar field size in FS mode and PB mode.
measure 2, is the best way to reduce the solar heat dumped, in terms of LCOE and annual ηste, which can meet 5.2 cents/kWh and 18.2% in FS mode and 4.7 cents/kWh and 20.2% in PB mode, respectively. In this 9
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input by using which to continue preheating feedwater. However, there is a maximum amount surplus solar heat input to make sure the plant, especially the boiler runs safely. In this case, the maximum solar energy is determinedto be 99.9 MWth (solar multiple value ≈ 1.4) in fuel-saving mode and 129.1 MWth (solar multiple value ≈ 1.7) in power-boosting mode. 2) In general, when the solar input is less than the maximum value, the more solar heat input Qsolar,input, the higher the effective solar-toelectricity efficiency ηeste will be. Therefore, measure 2 would have better technical performance than measure 1, and both measures would be superior to dumping surplus solar heat. 3) Due to the high cost of a thermal energy storage system, measure 2 is also more favorable than measure 1, in terms of levelized cost of energy. 4) Compared to the fuel-saving mode, the power-boosting mode is preferable technically and economically. The optimal levelized costs of energy by using measure 2 are 5.2 cents/kWh in fuel-saving mode and 4.7 cents/kWh in power-boosting mode, respectively.
Table 5 The best performances of three ways to deal with the surplus solar energy: Asf is the solar field size and annual ηste is the annual solar-to-electricity efficiency. Three ways
Being dumped Being charged Preheating continuously
FS mode
PB mode
Asf (×103 m2)
LCOE (cents/ kWh)
Annual ηste (%)
Asf (×103 m2)
LCOE (cents/ kWh)
Annual ηste (%)
111.0 124.0 137.0
5.3 5.5 5.2
17.8 17.9 18.2
120.0 133.0 192.0
4.9 5.0 4.7
19.3 19.4 20.2
case, after equipping the plant with a TES, i.e. measure 1, although the technical performance is improved, the economic performance is decreased in either PB or FS mode due to the high cost of TES. 7. Conclusions
Declaration of Competing Interest
This study investigates, thoroughly, the impacts of three ways to deal with the surplus solar heat. The two measures, i.e. being charged and preheating continuously, are compared with being dumped by a desktop case study of a 330 MWe solar aided power generation plant. As a solar aided power generation plant can be operated in either powerboosting or fuel-saving mode, the technical (in terms of effective solarto-electricity efficiency ηeste) and economic (in terms of levelized cost of energy) performances of the three ways in each mode are also studied. The following conclusions are therefore drawn:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Funding: This work was supported by Guangzhou Power Supply Co., Ltd. Science and Technology Program (ZHKJXM20180152), National Natural Science Foundation of China No. 51876057 and the China Scholarship Council No. 201806730031.
1) A solar aided power generation plant is able to accept surplus solar Appendix A Air preheater
A tri-sector air preheater is configured to use the flue gas preheating the primary air and secondary air, as Fig. A.1 shown. The lumped parameter method is used in modeling [34]. It is assumed that ignoring axial heat exchange of the air heater [35]. Due to air leak, the mass flow of flue gas, primary air and secondary air out of the air heater can be expressed as follows.
mG, out = mG, in + mPG, leak + mSG, leak mP, out = mP, in mPG, leak mPS, leak mS, out = mS, in mSG, leak + mPS, leak
(A.1)
where m stands for the mass flow rate (kg s−1), the subscript G, P, S, in, out represent flue gas, primary air, secondary air, inlet and outlet respectively. The subscript type of AB, leak stands for the leak from A to B. In the air preheater, the heat transfer Q (W) can be expressed as follows
Fig. A1. Scheme of an air preheater. 10
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QGM = kG AG (Tavg, G
TM , G ) = mG, in cG, in TG, in
mG, out cG, out TG, out + QPG, leak + QSG, leak
QMP = kP AP (TM , P
Tavg , P ) = mP, in cP, in TP, in
mP, out cP, out TP, out
QPG, leak
QMS = kS AS (TM , S
Tavg, S ) = mP, in cP, in TP , in
mP , out cP, out TP, out
QPG, leak + QP S, leak
QPS, leak (A.2) −2
−1
2
where the subscript M stands for the metal body; k, A, T and c represent heat transfer coefficient (W m K ), heat exchange area (m ), temperature (K) and heat capacity (J kg−1 K−1), Tavg are the average temperatures (K), which can be calculated as follows.
Tavg, G = Tavg , P = Tavg , S =
TG, in + TG, out 2 TP , in + TP , out 2 TS , in + TS , out 2
(A.3)
In Eq. (A.2), TM are the average temperatures of the metal body (K), which can be expressed as follows. G G+ P+ S P G+ P+ S S G+ P+ S
mM cM
dTM , G
mM cM
dTM , P
mM cM
dTM , S
d d d
=
(TM , P
=
(TM , S
=
(TM , G
TM , G )
G+ P+ S
TM , P )
G+ P+ S
TM , S )
G+ P+ S
mM cM + QGM mM cM
QMP
mM cM
QMS
(A.4)
where mM and cM are the mass of the whole metal body (kg) and heat capacity (J kg−1 K−1); θ are the angle of the flow section (°), shown in Fig. A.1; ω is the rotational speed of the metal body in the air preheater (°·s−1), τ is the time step, 1 s. In Eq. (A.2), the heat transfer coefficient k can be calculated as follows.
kG = kP = kS =
G
de P
de S
de
( ) ( ) ( )
VG d e 0.8 0.4 Pr , G CT , G G
VP de 0.8 0.4 Pr , P CT , P P VS de 0.8 0.4 Pr , S CT , S S
(A.5) −1
−1
2 −1
where λ, Pr and v are thermal conductivity (W m k ), Prandtl number and kinematic viscosity (m s ), respectively; de is equivalent diameter (m); V is the flow velocity (m s−1). CT is the temperature correction factor, which can be calculated as follows.
CT , G = 1
( ) =( )
CT , P = CT , S
Tavg , P 0.5 TM
Tavg, S 0.5
(A.6)
TM
where TM is the average temperatures of the whole metal body, which can be expressed as follows.
TM =
TM , G
G
+ TM , P P + TM , S G + P + S
S
(A.7)
Then, the temperature of exhaust gas, primary air and secondary air (out of air preheater) can be calculated as follows.
TG, out = TP, out = TS , out =
mG, in cG, in TG, in + kG AG TM , G
TG, in + QPG, leak + QSG, leak 2
kG AG + mG, out cG, out 2 TP , in QPG, leak 2 kP AP + mP , out cP, out 2
mP , in cP, in TP, in + kP AP TM , P
mS , in cS , in TS, in + kS AS TM , S
TS, in 2
QPS, leak
QSG, leak + QPS, leak
kS A S + mS, out cS, out 2
(A.8)
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