Off-design performance and an optimal operation strategy for the multistage compression process in adiabatic compressed air energy storage systems

Accepted Manuscript Off-design performance and an optimal operation strategy for the multistage compression process in adiabatic compressed air energy storage systems Huan Guo, Yujie Xu, Yi Zhang, Qi Liang, Hongtao Tang, Xinjing Zhang, Zhitao Zuo, Haisheng Chen PII: DOI: Reference:

S1359-4311(18)34394-1 https://doi.org/10.1016/j.applthermaleng.2018.12.035 ATE 13042

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

17 July 2018 7 November 2018 6 December 2018

Please cite this article as: H. Guo, Y. Xu, Y. Zhang, Q. Liang, H. Tang, X. Zhang, Z. Zuo, H. Chen, Off-design performance and an optimal operation strategy for the multistage compression process in adiabatic compressed air energy storage systems, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng. 2018.12.035

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Off-design performance and an optimal operation strategy for the multistage compression process in adiabatic compressed air energy storage systems Huan Guoa, Yujie Xua,b, Yi Zhanga,b, Qi Lianga,b, Hongtao Tanga, Xinjing Zhanga, Zhitao Zuoa, Haisheng Chena,b,* a.Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China. b.University of Chinese Academy of Sciences, Beijing 100049, China. * Correspondence: [email protected]; Tel.: +86 10 82543148. Abstract: Compressed air energy storage (CAES) systems usually operate under off-design conditions due to load fluctuations, environmental factors, and performance characteristics of the system. Thus, to improve design and operation characteristics, it is important to study off-design performance of CAES systems. The compression process plays an important role in CAES systems. In this paper, we discuss the methodology for modeling off-design operation of a multistage compression process with intercooling of the most promising adiabatic CAES (A-CAES) system. The off-design performances under two proposed kinds of operating regulations are analyzed and compared. These two operating regulations are equal-power-ratio regulation (EPR) and optimizing variable inlet guide vane rotation angle (OVRA) (optimizing all stages simultaneously). Correlation between parameters such as total power consumption ratio, exergy efficiency, hot water temperature versus mass flow rate ratio, and back pressure are revealed in depth. Based on this research, the optimal operation laws, including pressure ratio distribution and efficiency distribution among all stages, are obtained, and it is found that the primary optimum principle is to enhance the isentropic efficiencies of low-pressure stages to approach design point. Finally, the optimized regulating law for the inlet guide vane rotation angles of the 4 stages is revealed. This study provides strong support for the design, operation, and control of CAES systems. Key words: Off-design performance; Multistage compression process; Exergy efficiency; Optimization

1. Introduction Compressed air energy storage (CAES) systems have the advantages such as large scale, low cost, and possess a flexible storage duration as well as a long lifespan, and two commercialized CAES plants (McIntosh and Huntorf) are in operation [1, 2]. However, conventional CAES relies on fossil fuels and bulk air storage chambers and has low efficiency and energy density[3, 4]. To solve these problems, scholars have adopted several methods to improve CAES efficiency, including alteration of working mediums or their states, optimization of thermodynamic cycles, and integration with other advanced technologies. Many novel kinds of CAES systems have been proposed and developed: adiabatic compressed air energy storage (A-CAES), liquid air energy storage (LAES), isothermal compressed air energy storage (ICAES), under water compressed air energy storage (UWCAES), and supercritical compressed air energy storage (SC-CAES). The summarization of the above CAES systems is shown in Reference [5], and where a table for it is presented. Among these novel CAES systems, A-CAES has received much attention for its independence from fossil fuels, high efficiency[6], and simple system flow[7]. CAES systems usually operate under off-design conditions. When CAES is used to store renewable energies like wind power and solar power, the input power of the compressor can

fluctuate almost constantly due to intermittency and volatility of these renewable energies [8]. When CAES is used in a micro-grid or distributed energy system, the loads of compressor and expander vary based on the overall system power balance. Similarly, the compressor works under the condition of varying back pressure (BP) if the air storage chamber is in constant volume instead of constant pressure [9]. Change in environmental conditions such as ambient temperature, pressure and humidity can also lead to off-design operation of the system. Change in environmental conditions such as ambient temperature, pressure and humidity can also lead to off-design operation of the system [10]. Currently, only a few scholars work on off-design performance in A-CAES systems. Most of these researches only include off-design models for a part of the components of their A-CAES systems rather than considering off-design models for all the components, leading to inaccurate results for the obtained off-design performance of the whole system[11, 12]. He et al. [12] investigated two cavern operational scenarios of a CAES systems (un-compensated isochoric and compensated isobaric caverns) and three types of cavern wall heat transfer conditions (isothermal, CHT (Convective Heat Transfer) and adiabatic conditions). They found that isothermal and isobaric operation of caverns can store more exergy and provide more work, as well as improving energy density. There is also research employing comprehensive component models. But the research mainly focused on the effect of packed-bed TES on system performance, resulting in the lack of attention on other components’ characteristics or studies on optimum operation strategy [13, 14]. Sciacovelli et al. [13] investigated the dynamic performance of a specific TS-CAES plant with packed bed thermal energy storage (TES). This team found that maximum round trip efficiency occurs when cycling stationary temperature profiles are established in the packed bed TES. But this research mainly focused on the effect of packed bed TES on system performance. Mazloum et al. [14] conducted a dynamic simulation on isobaric TS-CAES systems with comprehensive off-design models. The research revealed that time required to reach the steady state was about 120 s during storage periods and 382 s during production periods. But this study was lacking in its consideration of thermodynamic character. Multistage compression process with intercooling is the key operating process of A-CAES system, and investigation on its off-design performance can lay the ground work for obtaining the off-design performance of the whole system. Currently, most research focuses on the compression section in isothermal CAES and conventional CAES system rather than A-CAES system. Zhang et al. [15] investigated an isothermal compression system with liquid-piston by filling porous media into the compression cylinder. A numerical method of which RANS models are used for high Reynolds number simulations and large eddy simulation model is for the pore-scale simulations was adopted to analyze the influence of the amount of heat exchanged during compression on system performance. Yan et al. [16] conducted some experiments for an isothermal compression system to verify that all the porous mediums mentioned in the article could help to improve the compression efficiency, but efficiency differences presented were slight. Heidari et al. [17] established a model for a kind of reciprocating compressor, which can be applied in isothermal CAES systems, as well as an energy macroscopic representation that can be used in the modeling. The simulation results were well in accordance with those of the experiment. Salvini et al. [18] proposed a control strategy for a compression system with intercooling in conventional CAES or A-CAES systems, to allow the system to keep up with the variation of load. To increase the compressor’s operational flexibility, it was found that infinite step control (ISC) is preferable compared to the hydraulic variable volume clearance pocket. The above references are summarized and compared below in Table 1, which shows that isothermal CAES systems generally employ reciprocating compressors, which are not suitable for large-scale A-CAES

systems. In addition, there is no thermal exergy stored in isothermal and conventional CAES systems, so the influence of thermal exergy on the evaluation of consumed power is not taken into account, and objective function of the consumed work only takes pressure energy into consideration in these studies. Table 1 Comparison of the studies concentrated on the compression process in CAES systems With thermal System

Compressor type

exergy stored

Research content

or not The heat transfer from Reference [15]

Isothermal

Reciprocating（liquid

CAES

piston）

No

the air to inserted material and its effect on the system Experiment study for

Reference [16]

Isothermal

Reciprocating（liquid

CAES

piston）

No

selecting the porous medium in the compressor chamber

Reference [17]

Isothermal

Reciprocating（linear

CAES

reciprocating）

Introduces an modeling No

modeling of a compressor

Conventional Reference [18]

CAES or A-CAES

approach for the dynamic

Operational flexibility of Reciprocating

Yes

the compressor and its control

In technologies other than CAES, there are also many thermal systems that contain multistage compression processes, including gas turbines, refrigerating systems, and air separation systems. For example, Chen et al. [19] adopted the method of finite time thermodynamics to investigate the thermodynamic performance of an irreversible Brayton cycle with an intercooled compression process. In addition, the numerical results indicate that when the pressure ratios of all stages match each other optimally, the maximum power-output is obtained, or the maximum efficiency is obtained, but they cannot coexist. Roytta et al. [20] studied the relationship between parameters of each compressor stage and refrigeration system performance using intercooled compressors. It was found that the optimum operation point of the cycle is not at the point of optimum efficiency of either compressor stage. As with conventional CAES, research on multistage compression processes in these systems generally doesn’t take thermal exergy storage into consideration. In these researches, only mechanical exergy of high-pressure work medium is considered when assessing the compression system. However, as mentioned above, thermal exergy storage plays an important role in A-CAES systems. In CAES systems, compressor control methods of off-design operation, like the control methods of a gas turbine, can be divided into inlet/outlet valve control , rotating speed control, variable inlet guide vane control, variable diffuser vane control (in the condition with diffuser). The first control method is to change the inlet/outlet pressure and mass flow rate of the compressor, while the second one is to change the speed of the compressor by a motor. The third and the fourth methods are to change the geometry of the compressor and then the compressor’s characteristics changes [21]. For CAES systems, rotating speed control is expensive due to the use of converters. These converters are much more expensive since each stage has to be equipped one converter and

one motor, and the converters are in the MW scale. The BP / valve control can cause high pressure loss. Meanwhile the variable diffuser vane control is weak in adjusting pressure ratios, which is, however, important in CAES systems’ adjustment for the variation of pressure in the air storage chamber. Different from the above three control methods, variable inlet guide vane (VIGV) control can achieve a wide load range and high efficiency control [22]. However, owing to the specificity of compression process of CAES, no research on variable inlet guide vane control performance within the CAES system’s compression process has been found. In the past, the application of CAES was usually towards peak sheaving, which doesn’t demand high flexibility of system off-design performance. Currently, more focus has been paid to renewable energies, placing more stress on the flexibility of CAES to operate efficiently under a wide range. Thus, the

VIGV-controlled compressor for CAES system is the primary subject of this paper. In this paper, first a comprehensive off-design model of a complete A-CAES system consisting of a multi-stage compressor with intercoolers and thermal energy storage systems is developed. The further details of the system are presented under section on ‘Process description’. Then, two operating modes, including equal-power-ratio (EPR) operation and the operation of optimizing variable inlet guide vane rotation angle (OVRA) are compared. This comparison was then used to determine the comprehensive off-design characteristics and the optimization principle. Last, the optimal operating strategy is explored. This research can serve as a reference for the off-design performance study of the whole system of A-CAES, as well as for the optimal operation of its compression section.

2. Process description Figure 1 is the schematic diagram of an A-CAES system. During the charging process, ambient air is compressed to a high pressure by a multistage compressor system with intercoolers, and then stored in an air storage chamber. The power source of compressor of A-CAES can be from the grid, wind power, solar power, etc. At the same time, in the intercooler the compression heat is absorbed by a cold thermal storage medium, which is stored in a hot tank after that. During the discharging process, the high-pressure air released from air storage chamber is reheated by the hot thermal storage medium, and then expanded in the expander to generate electricity. Meanwhile, the thermal storage medium flows out of the reheater, is cooled to ambient temperature in the radiator, and then stored in a cold tank.

Fig. 1. Schematic diagram of A-CAES system To better show the thermodynamic process of the A-CAES system, a p-h diagram is shown in Fig.2. it can be seen that discharge pressure at point 11 is lower than charge pressure at point 10 because of the pressure change in air chamber. The inlet temperature of expander is lower than the

outlet temperature of compressor, which is because the intercoolers and reheaters have temperature difference at both sides. The enthalpy decrease in the expander is smaller than the enthalpy increase in the compressor, which is due to the expansion ratio being smaller than the pressure ratio, as well as compressor efficiency and expander efficiency being smaller than 1. Since outlet temperature of expander is larger than ambient temperature, the outlet enthalpy of expander is larger than the enthalpy under the ambient condition.

Fig.2. p-h diagram of the A-CAES system Multistage compression intercooled by circulating water is the main focus of this paper, and is shown in the dashed box of Fig. 1. Based on the temperature of thermal storage, as well as the characteristics of thermal storage/transfer media, a double-tank with an indirect thermal storage design was proposed for the thermal storage/transfer unit. This unit mainly consisted of intercoolers, reheaters, a hot tank, and a cold tank. Circulating water was chosen as the thermal storage/transfer medium. Thermal storage using hot tanks is widely used in many energy systems, such as solar power systems and cogeneration systems equipped with thermal storage. Choosing water as the heat storage/transfer medium can result in high performance in thermal storage capacity and heat transfer efficiency, and is economical, chemically stable, and safe. In contrast to single-tank thermal storage, double-tank thermal storage stores cold and hot media separately, which avoids thermal exergy destruction by mixing in a single tank [23, 24]. Compressors are the fundamental unit of the compression process. Piston compressors can be used in small-scale CAES systems because piston compressors have a simple structure, large pressure ratio, and are technologically mature. However, in large-scale CAES systems, centrifugal/axial compressors are usually employed because piston compressor is not suitable for the condition of large mass flow rate and piston compressor has a large vibration[25]. In this paper, 10 MW system is studied, and the compressor is centrifugal type. Table 2 Basic parameters of design point[26] Parameter Discharge power (MW) Charge power (MW)

Value 10 5.84

Discharge pressure

70

Maximum pressure during charging (bar)

100

Energy stored (kW·h) Thermal storage temperature (K)

6.11×103 440

Water temperature at exit (K)

332

Chamber volume (m3)

1748

In this paper, a 4-stage compression process is taken as a case to calculate its off-design performance. The system parameters are shown in Table 2, which comes from the optimizing parameters under the design condition [26]. Meanwhile, main detailed design parameters of the compression section are presented below in Table 3 [27]. It is worth noting that stages 1-4 of the multistage compressor are arranged from low- to high-pressure stages, and so are the serial numbers of intercoolers. Table 3 (a) Main design parameters of compression Parameters

Value

BP of compressor (bar)

100

Number of compressor stages [27]

4

Heat transfer area of each intercooler (m3)

1310 (1st stage), 1344 (2nd stage), 1785 (3rd stage), 2366 (4th stage)

Temperature difference at two sides of each intercooler (K)

5

Effectiveness of each intercooler

0.966

Pressure loss of each intercooler (bar)

0.2

Pressure of circulating water (bar)

12

Ambient temperature (K)

298

Ambient pressure (bar)

1

Table 3 (b) Main design parameters of each stage compressor 1st stage

2nd stage

3rd stage

4th stage

Mass flow rate (kg/s)

10.138

10.138

10.138

10.138

Pressure ratio

3.237

3.237

3.237

3.237

Efficiency [28]

0.84

0.84

0.84

0.84

Inlet temperature (K)

298

303

303

303

Inlet pressure (bar) Rotational speed (r/min) Power (kW)

1

3.037

9.628

30.960

15500

15500

28500

28500

1441.645

1465.538

1465.144

1468.432

3. Methodology 3.1 Component models Each device’s performance characteristics are presented as follows in order to investigate the key performance characteristics of the entire process. (1) Compressor When a compressor works under off-design condition, its pressure ratio and isentropic efficiency vary with air flow rate, rotating speed, VIGV rotation angle, and other factors. The off-design performance characteristics of a compressor mainly refers to the variation of pressure ratio and isentropic efficiency with other parameters. General performance curve for obtaining the performance characteristics of compressor is preferred in this paper. The reason is that the detailed compressor information is unknown in initial theoretical study as the condition in this paper. The general performance curve model can

also make the results be general rather than relying on the specific compression system. In reference [29, 30], one type of general formula for a compressor characteristic map is presented, which is appropriate for centrifugal and axial flow compressor s, and the compressor characteristic map is fitted on the basis of physical background and massive experimental data, shown as Eqs. (1) - (3):

 c  c1 (nc )Gc2  c2 (nc )Gc  c3 (nc )

(1)

c  [1  c4 (1  nc )2 ](nc / Gc )(2  nc / Gc )

(2)

2 c1  nc / [（ q 1- m / nc） n（ c nc - m）] 2 c2  ( q - 2mnc2 ) / [（ q 1- m / nc） n（ c nc - m）]

c3  -( qmnc - m n ) / [（ q 1- m / nc） n（ c nc - m）] 2 3 c

(3)

2

c4  0.3 where Gc  Gc / Gc 0； Gc  (Gc T1 ) / p1

(4)

nc  nc / nc 0； nc  nc / T1

(5)

 c   c /  c0 ;

(6)

c  c / c 0

where Gc is mass flow rate, nc is rotation speed, εc is pressure ratio, ηc is isentropic efficiency, and subscripts 0 and 1 represent design values and inlet parameters respectively. Eqs. (1) - (3) have been thoroughly verified by experimental data as indicated in reference [26]. So in this paper, the above general formula is adopted. In Eq. (3), the values m=1.8 and q=1.8 are used to investigate the centrifugal compressor, while values m=1.06 and q=0.36 are used to investigate the axial flow compressor (the values of m and q are obtained by fitting a lot of experiment data in References [29, 30]). In this paper, a centrifugal compressor was employed, thus, m and q are set to be 1.8 and 1.8 respectively. The value of c4 is obtained by fitting massive experiment data when constructing the general compressor model in Reference [29], and the value has been used widely in literature [30, 31]. Thus in this paper, the value is also employed. Under a certain reduced speed, a surge occurs in the compressor when the mass flow rate is below one certain value. Therefore, the mass flow rate should be higher than the mass flow rate corresponding to the surge point. Under the design reduced speed, surge margin can be calculated according to the definition [32]:

  / Gs  S .M .   s  1 100%   d / Gd 

(7)

where subscripts represent the parameters of surge point under design reduced speed, subscript d represents the parameters of design point. In this paper, surge margin under design reduced speed is set as 18%，which is derived from model experiment in Reference [27]. Under other off-design reduced speeds, the isentropic efficiency of surge point is calculated by [29]:

su,i  1  a(1  n)2

(8)

where ηsu,i is based on the isentropic efficiency of surge point under design reduced speed, n is based on reduced design speed. Through fitting the experimental data, a is set as 0.7 in this paper [33, 34]. Although the compressors in References [33, 34] are for gas turbines, the variation of the

surge point with the off-design speed in References [33, 34] is similar to that in our research. According to equations (1) and (7), the mass flow rate and pressure ratio under design reduced speed are obtained. According to equations (2), (7) and (8), the mass flow rate and pressure ratio under other reduced speed can be obtained，then the surge limit is established. In addition, the choke boundary is established at the relative efficiency of 0.85[35]. According to equations (1)-(8), general characteristics of compressor are shown in Fig.3.

（a）pressure ratio

（b）efficiency

Fig. 3. General characteristics of compressor The performance characteristics of VIGV control, calculated by Eq. (9) [36], are taken into account in present work. These equations are widely used in the general calculations of compressor characteristics under VIGV regulations. Calculations of the parameters under any VIGV rotation angle are based on their values under the designed VIGV rotation angle as shown in Eq. (9). Gc  Gc ,map (1 

b1 ) 100

 c  1  (  1)c ,map (1  c  c ,map (1 

b2  ) 100

(9)

b3 2 ) 100

where △α is the VIGV rotation angle of compressor, and subscript c,map represents design VIGV rotation angle. The values of b1, b2 and b3 are 1,1and 0.01 respectively [36]. (2) Heat storage In compression section, the designed heat storage/transfer devices consist of intercoolers, a hot tank, and a cold tank. In an intercooler, heat released by air is equal to that absorbed by the circulating water. Meanwhile, heat transfer should satisfy heat transfer equation:

Ga,c (ha,in,inte  ha,out,inte )  kint e Ainte Tinte

(10)

where G represents mass flow rate, h represents specific enthalpy, kinte represents the heat transfer coefficient of the intercooler, Ainte represents the heat transfer area of the intercooler, △Tinte is the mean temperature difference between air and water, expressed in the form of log mean temperature difference. Subscript a is for air, w for water, in and out for inlet and outlet respectively, inte for the intercooler. Heat transfer in the intercooler is the type of heat transfer between gas and liquid, and the heat transfer coefficient is not constant with varying working conditions, which is similar to the heat transfer in the economizer of a heat recovery steam generator. Thus, the empirical model of the

economizer in a heat recovery steam generator was adopted to calculate the heat transfer coefficient of an intercooler: 

m T  k / kd       md   T d 



(11)

where k is the heat transfer coefficient, m is the air flow rate, T is the mean air temperature, and subscript d represents design condition. The values of the exponents, α and β are obtained by fitting experiment data in Reference [37], and the fitting results have a good agreement with the experiment data. Thus these values are adopted in this paper. Change of the operating condition can change the effectiveness of the intercooler, and hot water temperature changes with the change of the effectiveness. Besides, the pressure loss can vary with operating condition. Pressure loss of air through intercoolers can be calculated by the model of pipeline pressure loss [38, 39]：

p / pin (m T / p)in2  2 (p / pin ) d (m T / p)in,d

(12)

where △p is the pressure loss, subscript “in” means inlet, and d means design value. There is thick insulation on the outlayer of hot tank, heat dissipation coefficients are very low. After calculation, the effect of ignoring heat dissipation on the system efficiency is less than 0.1%, and the value decreases with the improvement of the thermal insulation performance. Meanwhile, water flow out of the cooler is near ambient temperature, so in the cold tank the heat exchange between water and the environment is negligible. Therefore, heat dissipation within hot tank and cold tank is not taken into account in the present work. Since water pumps motive power consumption is a small value compared with the consumed of compressor, and the pump work covers around 0.1%-0.2% of compressor work. Thus the pump work is neglected in this paper. It is noted that this paper mainly focuses on off-design performance and operation strategy of the compression process in A-CAES systems, and the present model can’t be used to simulate the transient performance for lacking transient models. But this work of off-design study is the basis of the system dynamic study.

3.2 Calculation logic and evaluation index Two kinds of regulations for the compression section are investigated and compared to reveal the optimizing principle and the variation relationship of different parameters in depth: operating under EPR and OVRA modes. The details are presented as follows: 1) EPR operation In this operating mode, when the compressor works under variable conditions, ratios of actual input power to rated power for each stage are equivalent, as shown in Eq. (13). Since the pressure ratios are equal for each stage’s compressor under the design condition, when the compressor works under EPR, its pressure ratios for each stage are close as long as isentropic efficiencies do not change very much. Moreover, the temperatures of outlet water are approximately the same, avoiding large exergy loss from the water mixing process. Under multi-stage gear coupled configuration, EPR operation can help to improve the stability of system. So EPR operation can be used as a referential system to OVRA operation to deeply investigate the principle and law of optimization.

W1,C W1,C,d



W2,C W2,C,d



W3,C W3,C,d



W4,C

(13)

W4,C,d

In the equation, the numbers in subscript represent stage numbers, d represents design value. The evaluation indices of compression is the exergy efficiency shown in Eq.(14).

E ,C 

Eair  Ehot_water

(14)

Wtotal,c

where Wtotal,c is the total work consumed by the multi-stage compressor, Eair is the exergy of high-pressure air at the outlet of the multi-stage compressor, and Ehot_water is the exergy of hot water in the compression process. Start

EPR operation

OVRA operation G, pb,T0

Initial value of VIGV rotation angles Amend G 1st compressor

No Amend G

VIGVs have been adjusted

Yes

Amend G

Choke?

Yes

No Power，T and p at outlet

η

No

No 1st intercooler

Yes

VIGVs have been adjusted

VIGVs have been adjusted

No

△Tm, △p, Ti ,hot，T and p of air at outlet

pb matches, equation (13) satisfied?

Yes

No pb matches, maximum ηE,C satisfied?

2nd, 3rd, 4th compressors and intercoolers

Data storage

End

Yes

Yes Data storage

Caculate Thot, ηE,C, etc.

End

Fig. 4. Calculation logic of compression process (EPR and OVRA operations)

The inlet air and inlet water of compression are in ambient conditions, and their exergy value is set as 0, indicating that the inlet exergy of compression only contains the consumed shaft power. The exergy of both high-pressure air and hot water at compressor outlet is considered. Therefore Eq. (14) represents the exergy efficiency with consideration to the exergy of both air and water. 2) OVRA operation This operating mode, achieved by optimizing each stage VIGV rotation angle, aims at maximizing the exergy efficiency (Eq. (14)). The optimization method adopts sequential quadratic programming, which is widely used in the operation optimization of gas turbines, since it has the advantage of dealing easily with inequality constraints, good performance in global convergence, and rapid convergence [40-43] The two kinds of operations are both in a certain rotating speed for the constant power grid frequency, and the logical diagram of calculations for the two operations is shown in Fig. 4. The

calculation logic is: under different BP and mass flow rate ratio (MR) (MR, means the ratio of the actual mass flow rate to design mass flow rate), parameters including each stage’s VIGV rotation angle, total consumed work, and hot water temperature are obtained via iteration. Stopping condition of the iteration varies with different operations. If a compressor works under EPR, the stopping condition is that ratios of different stage’s power under off-design condition should be equal to the ratios under the design condition, as shown in Eq. (14), as well as the MR and BP both matching the set values. If a compressor works under OVRA operation, compression exergy efficiency (Eq. (14)) reaching its maximum is the stopping condition, as well as the MR and BP both matching the set values. In order to avoid large temperature differences in intercoolers (to reduce exergy loss and develop a relatively high heat storage temperature) and enable the stored heat to meet the demand of discharge process, when working under variable operating conditions, the MR of circulating water varies proportionally with the MR of air.

4. Results and discussion 4.1 Characteristics under EPR and OVRA operation Figure 5 shows the variation of total power consumption ratio with MR and BP. In EPR operation, for a given value of BP, the total power consumption ratio increases almost linearly with MR. For a given value of MR, the total power consumption ratio increases with BP. To achieve EPR operation, the total power consumption ratio and MR present variation limits which are determined by BP, and the lower limit of the total power consumption ratio and MR increases with BP. This is because when inlet pressure (affected by BP for 2-4 stages) is high and MR is low, the low reduced MR at the compressor inlet may cause a surge while VIGV has limited regulating ability. In OVRA operation, compared with that in EPR operation, the linearity between total work consumption ratio and MR for a given value of BP is stronger. Meanwhile, compared with the total power consumption ratio of EPR operation, the consumption ratio under a fixed BP and MR is lower during OVRA operation. The reduction of consumed work of OVRA becomes clearer with the increasing deviation to design point. The reason lies in that EPR operation is close to OVRA operation in design point, and the effect of the optimized VIGV rotation angle appears when it deviates from design point.

Fig. 5. Variation of total power consumption ratio with MR and BP Figure 6(a) shows the variation of the exergy efficiency with MR and BP. In EPR operation, for a given value of BP, there is a maximum value of exergy efficiency along with the

changing MR, and the optimal MR decreases with decreasing value of BP from its design value. The reason lies in that the decrease of both BP and MR can make the inlet reduced MR of compressor close to design value, which is beneficial to improve the efficiency of compressor, then the exergy efficiency improved. Moreover, the maximum exergy efficiencies of different BPs vary little due to the regulation of VIGV. The optimum MR point under design BP (100bar) is 0.95 rather than the design point, 1. The reason is that lower MR results in lower pressure loss in the intercooler, and the pressure ratio of each stage’s compressor is lower while the power consumption is also less with little variation of isentropic efficiency. Compared with that of EPR operation, the variation of the exergy efficiency with MR and BP is different to some extent in the OVRA operation. The exergy efficiency under OVRA operation is improved compared with that under EPR operation as shown in Fig. 6(b). For a given value of BP, there exists a point of MR, where the exergy efficiency in OVRA operation has almost no improvement compared with that of the EPR operation. When deviating from the point, the exergy efficiency improves quite markedly. The exergy efficiency can increase by, at most, 2.5% within the range of calculation. Meanwhile, it is found that the MR point is smaller when BP is lower. The reason is that under the MR point, the inlet reduced mass flow rate of each compressor is near to its design point, thus in that point the two operating strategies are similar. The more extensive reason are shown in Section 4.2 of this paper.

(a) Exergy efficiency (b) Increment of the exergy efficiency Fig. 6. Variation of the exergy efficiency with MR and BP Figure 7 shows the variation of the hot water temperature with MR and BP. In EPR operation, for a given value of MR, hot water temperature increases with BP, because the increase of BP results in increasing the pressure ratio and outlet temperature of the compressor. For a given value of BP, there is an MR which causes the temperature of the hot water to reach an optimally low temperature. Moreover, this MR is equal to the optimal MR of exergy efficiency.

Fig. 7. Variation of hot water temperature with MR and BP

Compared with that in the EPR operation, for a given value of BP the variation range of hot water temperature is smaller in the OVRA operation. Similer to the relationship of the exergy efficiency with hot water temperature in EPR operation, a strong relationship of the exergy efficiency and hot water temperature also exists in OVRA operation. That is, for a given value of BP, the MR corrresponding to the peak point of the exergy efficiency is also the MR corrresponding to the lowest point of hot water. Or, more simply, the value of hot water temperature for a given value of BP can reflect the exergy efficiency. The off-design performance of compression section is also affected by ambient temperature. The variation of relative parameters with ambient temperature is studied in the following segment, which is only in the case of design MR, because the variations of parameters with ambient temperature under different MRs are similar to each other. The variation of total power consumption ratio with ambient temperature is shown in Fig. 8 (a). In EPR operation, total power consumption ratio increases linearly with ambient temperature for a given value of BP. The reason is that the inlet temperature of each compressor increases linearly with the increase of ambient temperature, and power consumption of the compressor increases linearly with its inlet temperature, which can be explained by the expression of compressor power consumption under an ideal gas model. Similar to that of EPR operation, total power consumption ratio linearly increases with ambient temperature for a given value of BP in OVRA operation. The reason is the same as that in EPR operation. For a given value of BP, the increase of total power consumption ratio with ambient temperature is slightly larger than that of EPR.

(a) Total power consumption ratio

(b) Hot water temperature

Fig. 8. Variation of work consumption ratio and hot water temperature with ambient temperature The variation of hot water temperature with ambient temperature is shown in Fig. 8 (b). It can be seen that hot water temperature increases linearly and strongly with ambient temperature, which can be explained by the linear relationship between temperatures at the inlet and outlet of the compressor. Meanwhile, the change rate of hot water temperature to ambient temperature under different BPs shows little difference. In OVRA operation, the variation is consistent with that of EPR operation, and it is also the reason for such variation. It can be seen that the thermal storage temperature increases by 1.4K approximately when the ambient temperature increases by 1K. This can be remarkable in practical operation since the pressure of water should be taken into account to avoid vaporization.

4.2 Comparison of parameters in different stages To reveal the optimizing principle, the distribution of pressure ratio and efficiency for the 4 stages under EPR operation as well as OVRA operation is illustrated in this section. The following comparison is based on ambient temperature conditions. 1）Pressure ratio distribution The variation of pressure ratio with MR under different BP for 4 stages is shown in Fig. 9. In EPR operation, for a given BP the difference of pressure ratios for 4 stages is small, and the variation of the pressure ratio with MR is small since the pressure loss in the intercooler changes only a little. In OVRA operation, there exists a near point of intersection of the curves (the pressure ratio of each stage varying with MR). At this point, the pressure ratio of each stage in the OVRA operation is the same as that of the EPR operation. The MR of the point is larger as the BP is larger. When MR deviates from the point, the difference of the pressure ratios for the 4 stages increases, where the pressure ratios of the 1st and 4th stages vary largely in opposite directions, the pressure ratios of the 2nd and 3rd stages vary slightly, and the pressure ratios of the two stages in the OVRA operation are close to that of the EPR operation. The reason for the existence of the near point of intersection is that at this point the reduced mass flow rate is near the design value, and the pressure ratios of each stage are near to each other, resulting in higher exergy efficiency. This is shown in Fig. 10. In the two sides of the point of intersection, the relative values of the pressure ratios for all 4 stages are different because the reduced mass flow rates lie on the different sides of the design value for the two sides of the intersection.

（a）pb=70bar

（b）pb=100bar

Fig. 9. Variation of pressure ratio with MR for 4 stages Meanwhile, under a certain BP, the pressure ratio of each stage varies almost linearly with MR. In OVRA operation, the pressure ratios vary with BP more significantly for the latter stages. In addition, the variations of the pressure ratios for the first-three stages are small, and the change of BP is mainly due to the change of the pressure ratio of the 4th stage. 2）Efficiency distribution The variation of compressor isentropic efficiency with MR under different BPs for the 4 stages is shown in Fig. 10. In EPR operation, for a given value of BP, a peak point exists in the curve of isentropic efficiency with MR for each stage (the peak point under low BP is not shown in the figure since the border is limited), and the MR of the peak point is smaller for the latter stage. As the BP increases, the difference of MR of the peak point between different stages decreases. In OVRA operation, in the zone of low MR and high BP, the isentropic efficiencies of the 1st, 2nd, and 3rd compressors vary with MR to some extent. However, in the zone of high MR, the isentropic efficiencies of the 1st, 2nd, and 3rd compressors almost do not change with MR, staying around the design point, 0.84. Meanwhile the isentropic efficiency of the 4th compressor in the OVRA operation is always lower than that in the EPR operation. Thus, the high efficiencies of the 1st, 2nd, and 3rd are at the cost of the low efficiency of the 4th compressor. The reason of the above phenomena can be obtained from the analysis of the pressure ratio distribution (mentioned above), and the reason stated in brief is: in OVRA operation, the VIGV adjustment of the first three stages make the three stages working in high-efficiency point (close to design point), while the 4th stage works to maintain the value of BP (the 2nd stage and 3rd stage also make contribution to maintain the value of BP, but the 4th stage plays a major role). Hence, it can be concluded that the optimization principle is to improve the isentropic efficiencies of the first several stages, making them closer to their respective design points.

(a) pb=70bar

(b) pb=100bar

Fig. 10. Variation of compressor isentropic efficiency with MR for 4 stages (For a given value of BP, the difference of the optimal MRs between EPR operation and OVRA operation is small, as is the difference of exergy efficiencies between the two optimal MRs. Thus, the difference of optimal MRs is not considered)

Meanwhile, it also can be seen that for a given value of BP, the MR achieving maximum exergy efficiency is not the MR where efficiencies of 4 stages are all larger. Also, the first MR is smaller than the second one. The reason is that the smaller MR results in less pressure loss in the intercooler. To better show the relationship of the thermodynamic processes between component scale and the system scale, and to clearly show the parameter changes under off-design conditions, the p-h and T-s diagrams under part-load condition (pb=70, MR=1) are drawn in Fig.11, where the points are corresponding to the points in Fig.1. In the p-h diagram, the ordinate adopts log10 type. It can be seen that, the air pressure increases with the compression proceeds, and the four compression processes in the four compressor stages have a near equal ordinate length under EPR operation, which is because the four stages almost have an equal pressure ratio in EPR operation. It also clearly shows that the changes of pressure ratios in the first and last stages are large between the EPR operation and the OVRA operation. The pressure ratio changes result in the larger enthalpy change in the first stage, and smaller enthalpy change in last stage compared with that in other stages. The T-s diagram for the compression process is shown in Fig.11 (b). It can be seen that the entropy change is consistent with the change of the logarithmic pressure. The reason is that the entropy change is in direct proportion to the change of the logarithmic pressure for a given temperature and under the assumption of ideal gas. It also can be seen that the temperature change is consistent with the enthalpy change. The reason lies that gas enthalpy is mainly affected by temperature rather than pressure.

(a) p-h diagram (b) T-s diagram Fig.11. Themodynamic diagrams for the compression process under two operation at part load To summarize the two operations briefly, a comparison of EPR and OVRA operations is shown in Table 4. The objectives, exergy efficiencies, pressure ratio distributions and efficiency distributions under the two operations are compared. It can be seen that OVRA operation has a very different parameter distributions among the stages because of the different objective, which make the OVRA operation has a better off-design performance. Table 4 Comparison of EPR and OVRA regulations

Objective

EPR operation

OVRA operation

To make power ratio among all

To make exergy efficiency highest

stages constant, and set as referent

operation Exergy efficiency

Near 82%-87% with BP varying in 70-100bar and MR varying in

Improved as large as 2.5% compared with EPR

0.7-1.3 Pressure ratio distribution Efficiency distribution

Concentrated among all stages

Decentralized among all stages

Efficiency varies with stage

First three stages almost have the

number and operating condition

design efficiency

4.3 Optimal operation strategy The variation of the 1st VIGV optimal rotation angle with MR and BP is shown in Fig. 12(a). It can be seen that the 1st VIGV rotation angle varies little with BP in OVRA operation. When MR is large, BP almost has no effect on the 1st VIGV rotation angle. The 1st VIGV rotation angle linearly increases with MR. The variations of the 2nd, 3rd, 4th VIGV rotation angles with the 1st VIGV rotation angle are shown in Fig.12 (b), (c), and (d), respectively. It can be seen that the VIGV rotation angle is affected more with BP in latter stages. When the MR increases, the effect of pressure ratio on each stage’s VIGV rotation angle decreases. Compared with that of the 1st stage, the variation ranges of the last three stages’ VIGV rotation angles with MR are small, with the 3rd VIGV rotation angle being the smallest. Moreover, when it gets close to design point, the 3rd VIGV rotation angle doesn’t change with the 1st VIGV rotation angle. The first three stages’ VIGV rotation angles increase with MR while the increment ratios decrease from the 1st to the 3rd stage. For the last stage, the stator even decreases with MR. The regulation principles of VIGV rotation angle mentioned above can have reference value to practical operation.

(a) The 1st VIGV rotation angle

(b) The 2nd VIGV rotation angle

(c) The 3rd VIGV rotation angle (d) The 4th VIGV rotation angle Fig. 12. Regulation of VIGV in different stages under OVRA operation

5. Conclusions The off-design performance and optimal operation strategy of the multistage compression process in a promising adiabatic CAES (A-CAES) system are studied in this paper. A comprehensive general off-design model is established for the process. Two operational modes, optimizing VIGV rotational angle (OVRA) (optimizing all stages simultaneously) and equal pressure ratio (EPR), are proposed and compared for exploring the optimization principle. The detailed conclusions are as follows: (1) The relationships between key operation parameters, which include power consumption, exergy efficiencies, hot water temperature, back pressure (BP) and mass flow rate ratio (MR) under the two operation modes are revealed in depth. For a given value of BP, a point of MR exists where the exergy efficiency has no obvious improvement in the OVRA operation compared to that of the EPR operation. When MR is changed from this point, the exergy efficiency improves markedly for the OVRA operation. The efficiency can increase by, at most, 2.5% within the range of calculation. (2) It is found that from low to high mass flow rate areas, the relative magnitude among each stage’s pressure ratio is changed in OVRA operation. Meanwhile, the region at which the values of pressure ratio/MR for different compressor stages intersect increases with increasing BP. Under the BP of 70bar, 80bar, 90bar and 100bar, the area of this intersection lies in the MR of 0.71, 0.8, 0.9 and 0.95, respectively. The farther MR deviates from this region, the greater the differences between the 4 stages’ pressure ratios. The main regulation principle for OVRA operation is to improve the isentropic efficiencies of low-pressure stages, to make them closer to design point. (3) The optimal regulating law of VIGV is obtained. Under OVRA operation, BP shows almost no influence on the VIGV rotation angle of the first stage. When the MR increases, the effect of pressure ratio on each stage’s VIGV rotation angle decreases. Compared with that of the 1st stage, the variation ranges of the last three stages’ VIGV rotation angles with MR are small. The first three stages’ VIGV rotation angles increase with MR while the increment ratios decrease from the 1st to the 3rd stage. For the last stage, the VIGV rotation angle even decreases with the increase of MR. Acknowledgements: The authors acknowledge the support provided by National Key R&D Program of China (2017YFB0903602), National Natural Science Foundation of China (51676181， 51522605), The frontier science research project of CAS (QYZDB-SSW-JSC023), Beijing Science and Technology Plan (D161100004616001; D161100004616002). Nomenclature

A= surface area of compressed air reservoir, m2 cp= constant pressure specific heat, J/kg/K E= exergy, J G= mass flow rate, kg/s G = relative reduced mass flow rate G = reduced mass flow rate h= specific enthalpy, J/kg k= heat transfer coefficient, W/(m2·k) m=constant

M= mass, kg n= rotating speed, rpm n = relative reduced rotating speed n = reduced rotating speed p= pressure, Pa T= temperature, K u= specific internal energy, J/kg W= power, W q=constant

Greek symbols α= VIGV angle ε= pressure ratio  = relative pressure ratio

η= efficiency  = relative isentropic efficiency κ= isentropic exponent

Subscripts a= air c= compressor d= design point i (1,2,3,4)= i-th stage in= inlet

inte=intercooler out= outlet s= surge point w= water 0= design value

Acronyms A-CAES=adiabatic compressed air energy storage BP= back pressure CAES= compressed air energy storage EPR= equal-power-ratio

MR= mass flow rate ratio VIGV= variable inlet guide vane OVRA= optimizing VIGV rotation angle DP= design point

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Highlights

Comprehensive model of a multistage compression process is established for the first time. Off-design performance of the compression process using VIGV regulation is revealed in depth. An optimization principle is obtained after a multi-parameter synchronous optimization. The optimal regulation law for VIGVs is given for the operation of the compression process.