OPTIMAL CONTROL EXPERIMENTATION OF COMPRESSION TRAJECTORIES FORA LIQUID PISTON AIR COMPRESSOR

Farzad A. Shirazi, Mohsen Saadat, Bo Yan, Perry Y. Li, and Terry W. SimonDepartment of Mechanical Engineering

University of MinnesotaMinneapolis, Minnesota 55455

Email: fshirazi@umn.edu

ABSTRACT

Air compressor is the critical part of a Compressed Air En-ergy Storage (CAES) system. Efficient and fast compression ofair from ambient to a pressure ratio of 200-300 is a challengingproblem due to the trade-off between efficiency and power den-sity. Compression efficiency is mainly affected by the amountof heat transfer between the air and its surrounding during thecompression. One way to increase heat transfer is to implementan optimal compression trajectory, i.e., a unique trajectory max-imizing the compression efficiency for a given compression timeand compression ratio. The main part of the heat transfer modelis the convective heat transfer coefficient (h) which in generalis a function of local air velocity, air density and air tempera-ture. Depending on the model used for heat transfer, differentoptimal compression profiles can be achieved. Hence, a goodunderstanding of real heat transfer between air and its surround-ing wall inside the compression chamber is essential in order tocalculate the correct optimal profile. A numerical optimizationapproach has been proposed in previous works to calculate theoptimal compression profile for a general heat transfer model.While the results show a good improvement both in the lumpedair model as well as Fluent CFD analysis, they have never beenexperimentally proved. In this work, we have implemented theseoptimal compression profiles in an experimental setup that con-tains a compression chamber with a liquid piston driven by a wa-ter pump through a flow control valve. The optimal trajectoriesare found and experimented for different compression times. Theactual value of heat transfer coefficient is unknown in the exper-iment. Therefore, an iterative procedure is employed to obtain h

corresponding to each compression time. The resulted efficiencyversus power density of optimal profiles is then compared withad-hoc constant flow rate profiles showing up to %4 higher effi-ciency in a same power density or %30 higher power density ina same efficiency in the experiment.

1 IntroductionGas compression and expansion has many applications in

pneumatic and hydraulic systems, including in the CompressedAir Energy Storage (CAES) system for offshore wind turbinethat has recently been proposed in [1, 2]. In the proposed CAESsystem, high pressure (∼20-30MPa) compressed air is stored in adual chamber storage vessel with both liquid and compressed air.Since the air compressor/expander is responsible for the majorityof the storage energy conversion, it is critical that it is efficientand sufficiently powerful. This is challenging because compress-ing/expanding air 200-300 times heats/cools the air greatly, re-sulting in poor efficiency, unless the process is sufficiently slowwhich reduces power [3, 4]. There is therefore a trade-off be-tween efficiency and power.

Most attempts to improve the efficiency or power of the aircompressor/expander aim at improving the heat transfer betweenthe air and its environment. One approach is to use multi-stageprocesses with inter-cooling [5]. Efficiency increases as the num-ber of stages increase. To improve the efficiency of the compres-sor/expander with few stages, it is necessary to enhance the heattransfer during the compression/expansion process. A liquid pis-ton compression/expansion chamber with porous material inserts

Proceedings of the ASME 2013 Heat Transfer Summer Conference HT2013

July 14-19, 2013, Minneapolis, MN, USA

HT2013-17613

1 Copyright © 2013 by ASME

has been studied in [3]. The porous material greatly increases theheat transfer area and the liquid piston prevents air leakage [6].Numerical simulation studies of fluid flow and enhanced heattransfer in round tubes filled with rolled copper mesh are studiedin [7]. Application of porous inserts for improving heat transferduring air compression has also been investigated [8].

In addition, the compression/expansion trajectory can beoptimized and controlled to increase the efficiency for a givenpower or to increase power for a given efficiency. For high com-pression/expansions ratios 200:1-350:1, such Pareto optimal tra-jectories have been shown, theoretically, to increase the powerof the compressor/expander by 200-500% at the same efficiency,over ad-hoc trajectories such as linear and sinusoidal trajecto-ries [3, 9, 10]. In [3, 9], the optimal trajectories were derivedanalytically based on simple heat transfer models and by con-sidering thermodynamic losses alone. For example in [3], theproduct of the heat transfer coefficient and heat transfer surfacearea hA is assumed to be constant; and in [9], hA is allowed tovary with air volume to take into account the decrease in sur-face area as the porous material is submerged in the liquid pis-ton. In both cases, the optimal trajectories consist of fast adi-abatic portions at the beginning and at the end. For the con-stant hA case [3], the middle portion is isothermal resulting inan Adiabatic-Isothermal-Adiabatic (AIA) trajectory, whereas forthe volume dependent hA(V ) case [9], the temperature differencefrom the ambient is inversely proportional to

√hA(V ) leading to

an Adiabatic-Pseudo-Isothermal-Adiabatic (APIA) trajectory.Optimal trajectories have also been numerically obtained

for cases with varying heat transfer coefficient, and consider-ing the effect of liquid friction and flow rate constraints of thesystem [10]. Application of such optimal trajectories has beenverified for a liquid piston compressor/expander using CFD andthe results have been compared with AIA and APIA trajectories.

While the theoretical improvement in power/efficiency withthe optimal trajectories for air compression over conventionallinear and sinusoidal trajectories has been validated analyticallyand numerically, this paper provides the experimental validationof the usefulness of such optimal trajectories. One issue withimplementing the optimal trajectories experimentally is the un-certainty of the heat transfer model. The transient heat transfercoefficient depends in reality on many factors. Even if a con-stant heat transfer coefficient h is assumed, its value is often dif-ficult to access a-priori. If the estimate of h is inaccurate, thereare two consequences: 1) the pressure-volume or temperature-volume trajectories of the air being compressed, which determinethe thermodynamic performance cannot be tracked properly; 2)the “optimal” trajectory itself cannot be defined correctly.

To overcome these issues, a temperature-volume trajectorytracking controller that adaptively estimates the unknown heattransfer coefficient h was developed and presented in our previ-ous paper [11]. This controller is used to track an estimated op-timal temperature-volume profile. The achieved thermodynamic

profile is then used to compute an average heat transfer coef-ficient. This value is then used to compute the next estimatedoptimal temperature-volume profile. With this adaptive-iterativeoptimal control strategy, the compression efficiency successivelyimproves for a given power-density.

The rest of the paper is organized as follows. Experimen-tal setup capable of 10:1 compression ratio is discussed in thenext section. A theoretical background is introduced in Section 3on optimal trajectories used in experiments. Section 4 discussesthe experimental tracking results. The iterative procedure to ob-tain the optimal compression trajectories is discussed in Section5. The experimental results of constant flow rate and optimaltrajectories are presented and discussed in Section 6. Section 7concludes the paper.

2 Experimental SetupIn order to investigate the performance of the optimal tra-

jectories for air compression an experimental setup was builtat power fluid control lab at University of Minnesota shown inFigure 1. A water pump circulates water inside the circuit andprovides the required flow rate to compress air inside the com-pressor chamber. Two pressure transducers are mounted in thesetup to measure the upstream water pressure and downstreamair pressure. An Omega FTB-1412 turbine flow-meter and an En-dress+Hauser Promass 80 Coriolis flow-meter are used to mea-sure the water volume and flow rate. In one hand, the Coriolismeter measurement has a time lag of ∆t = 71 ms and on the otherhand, the turbine meter is not accurate at small flow rates. There-fore, we combine the volume information of two meters to obtaina more accurate measurement of the water volume flowing insidethe chamber and consequently the volume of the air under com-pression. The water and air volume at each instance of time aredetermined as follows.

Vw(t) = VCor(t −∆t)+VT M(t)−VT M(t −∆t) (1)V (t) = V0 −Vw(t)

where V0 is the initial volume of the air. We add the turbinemeter volume difference between time t and t −∆t to the vol-ume measured from Coriolis meter to compensate for the exist-ing time-lag. A relief valve is mounted in the circuit to keepthe pressure of the circuit below 160 psi. Filters are also in-cluded to remove unwanted particles from flowing water. A 16-bit PCI-DAS1602/16 multifunction analog and digital I/O boardis used to acquire the system data. This board is linked withMATLAB/Simulink via xPC Target to send and receive the con-trol and measured signals. The sampling frequency is chosen tobe 4000 Hz. The main reason for the high sampling frequency isto correctly capture the output signal of Coriolis and turbine me-ters. A Burkert 6223 Servo-assisted proportional control valve

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FIGURE 1. EXPERIMENTAL SETUP

is employed to manipulate the flow rate inside the chamber. Thetemperature of the air inside the compressor is obtained from theideal gas law using the pressure and volume measurements asfollows.

T =PV

P0V0T0 (2)

where P and V are the measured air pressure and volume, andP0 and T0 are initial pressure and temperature of the air, respec-tively. In each experiment, the nut on the cap of the compressoris opened to make the initial air pressure equal to the ambient.Water can be added to the column to set the initial air volume atthe desired value. The maximum available flow-rate in the setupis 302cc/s. It is noted that we only use the data up to the pressureratio of 10.

3 Theoretical Background on Optimal TrajectoriesIn this section, a brief background on optimal compression

trajectories is presented. The interested reader is referred to [10]for a detailed mathematical modeling. Calculating the optimalcompression profile can be addressed as a functional optimiza-tion problem (or optimal control problem) for which the costfunction is the input work defined as follows.

J =∫ tc

0

(mRT(t)

V(t)−P0

)V(t) dt︸ ︷︷ ︸

Compression Work

+∫ tc

0Γ(V(t),V(t))

V(t) dt︸ ︷︷ ︸Friction Work

+ (rP0 −P0)Vf︸ ︷︷ ︸Isobaric Cooling Work

(3)

where m, T and V are the mass, temperature and volume of airunder compression, R is the specific gas constant of air, Γ is theviscous friction power (between liquid piston and its surround-ing solid wall), r is the final pressure ratio, and tc and Vf are the

compression time and air volume at which the desired pressureratio is obtained. While the viscous friction energy loss is rela-tively large for small diameter tubes and large flow rates, such aloss can be neglected due to the large diameter of the compres-sion tube used in this experiment (d = 5.04cm). In Eq. (3), termV(t) is the control input while T (air temperature) is a dynamicstate for which, the dynamic equation comes from the first lawof Thermodynamics as

T(t) =h(t)A(t)

mCv(TW −T(t))+(1− γ)

T(t)

V(t)V(t) (4)

where Cv and γ are the heat capacity of air at constant volumeand the heat capacity ratio of air, respectively. h is the heat trans-fer coefficient between air and solid boundary which, in gen-eral, varies with time during compression based on air proper-ties (h(t) = h(T(t),V(t),V(t))). Tw is the wall temperature which isassumed to be constant (at ambient temperature) during the com-pression process (due to large heat capacity of the solid boundarycompared to air). Finally, A(t) is the active heat transfer area be-tween the air and cylinder wall, tube’s cap and liquid piston.

A(t) =4V(t)

d+

πd2

2(5)

While the air is initially at T0, V0 and P0 (at t = 0), it is desired tocompress it to a final compression ratio of r in a final compres-sion time of tc. Therefore, there exists a final manifold for therelated optimal control problem that relates the final volume andtemperature as follows.

T(tc)

V(tc)− r

T0

V0= 0 (6)

In summary, finding optimal compression profile which resultsthe maximum compression efficiency is equivalent to minimiz-ing the cost function defined by Eq. (3), subject to dynamic con-straint of Eq. (4) and final manifold of Eq. (6). Additional in-equality constraints due to hardware limitations can also be im-posed. For example, here we have considered the liquid pumplimitation so that the liquid piston flow rate is limited.

|V(t)| 6 Vmax (7)

The continuous optimal control problem is then parameterizedas a finite dimensional problem and solved numerically by stan-dard algorithms for constrained parameter optimization. Oncethe control input is defined over the time range, air temperature

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0 0.5 1 1.5 20

50

100

150

200

250

300

Time (s)

Flo

w R

ate

(cc/

s)

50 100 150 200 250 300

300

350

400

450

Volume (cc)T

empe

ratu

re (

K)

50 100 150 200 250 3000

2

4

6

8

10x 10

5

Volume (cc)

Pre

ssur

e (p

si)

0 0.5 1 1.5 2

300

350

400

450

Time (s)

Tem

pera

ture

(K

)

FIGURE 2. SAMPLE OPTIMAL COMPRESSION PROFILE(FLOW RATE-TIME, TEMPERATURE-VOLUME, PRESSURE-VOLUME AND TEMPERATURE-TIME) FOR r = 10, tc = 2s ANDh = 16 W

m2.K

can be found by solving Eq. (4). Figure 2 illustrates sample flowrate, temperature-volume and pressure-volume optimal trajecto-ries obtained from the optimization problem for r = 10, tc = 2.5s,h = 10 W

m2.K , V0 = 300cc and Vmax = 280cc/s. The efficiencyof trajectories is obtained from dividing the isothermal processwork over the input compression work for the same pressure ra-tio of r as follows.

Wiso = P0V0 ln(r) (8)

Win = −∫ V f

V0

(P−P0)dV +(rP0 −P0)Vf (9)

η =Wiso

Win(10)

In calculation of both compression works the isobaric ejectionwork is also included. The improvement of compression effi-ciency for a fixed compression power (or vice versa) resulted byoptimal compression trajectories is shown in Figure 3 comparedto constant flow rate profiles. As a rule of thumb, this improve-ment is more noticeable at lower power densities and higher pres-sure ratios. As mentioned earlier, the heat transfer coefficient his a complex function of air properties, flow regime and the com-pression chamber geometry. As the result, it is usually difficultto have an accurate model of h during the compression process.In this work, we use an iterative procedure to estimate a constantaverage value for h to design the optimal compression profiles.

101

102

70

75

80

85

90

95

100

Power Density (kW/m3)

Effi

cien

cy %

Optimal ProfileConstant FlowAdiabatic

FIGURE 3. COMPRESSION EFFICIENCY VS. POWER DENSITYFOR r = 10 AND h = 10 W

m2.K

100 150 200 250 300 350 400 450 500 550 600

300

320

340

360

380

400

420

440

460

Air Volume (cc)

Tem

pera

ture

(K

)

tc=3s

tc=5s

tc=7s

FIGURE 4. EXPERIMENTAL T-V TRACKING CURVES OF OP-TIMAL TRAJECTORIES FOR COMPRESSION TIMES OF 3s, 5sAND 7s

4 Optimal Trajectory Tracking ResultsFigure 4 illustrates the tracking of the temperature-volume

optimal trajectory for compression times of 3s, 5s and 7s. Thereader is referred to [11] for more details on design and imple-mentation of the controller employed to track the optimal trajec-tories. The experimental temperature profiles for compressiontimes of 1.5s and 2s are also shown for optimal and constantflow-rate trajectories in Figure 5. It can be seen that constantflow-rate trajectories result in higher temperatures at the secondhalf of the process that makes an important effect on decreasingthe compression efficiency compared to optimal trajectories.

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0 0.5 1 1.5 2250

300

350

400

450

500

t(s)

T(K

)

Optimal for tc=1.5s

Constant flow−rate for tc=1.5s

Optimal for tc=2s

Constant flow−rate for tc=2s

FIGURE 5. EXPERIMENTAL TEMPERATURE PROFILES OFOPTIMAL AND CONSTANT FLOW-RATE COMPRESSION TRA-JECTORIES FOR COMPRESSION TIMES OF 1.5S AND 2S

5 Iterative Calculation of Optimal TrajectoriesThe optimal trajectories are designed for constant values of

h. Here, we consider an iterative algorithm that takes the ad-vantage of both experimental data and theory to obtain optimalcompression trajectories for the experimental setup. In this algo-rithm, for a given compression time tc and a maximum flow rateVmax, an initial value of heat transfer coefficient h0 is assumedand the optimal compression trajectory, e.i., temperature versusvolume Ti(V ) is determined using the optimization code. Thistrajectory is used in the experiment for tracking. The experi-mental data is acquired and analyzed to obtain the correspondingefficiency ηi, compression time tci and an average value of actualhi. This calculated hi is again inserted in the optimization code toobtain the next optimal trajectory. This iterative procedure is car-ried on until the efficiency is not increased anymore and a finalvalue of hN is obtained, where N is number of iterations. Figure 6shows a sample convergence curve of h for tc = 1.5s. Followingthe procedure explained above, an initial guess of h0 = 8 W

m2K con-verges to h4 = 12 W

m2K after 4 iterations that results in the same ex-perimental compression time. Figure 7 shows the h profiles ver-sus volume for optimal trajectories experimented in successiveiterations of the iterative procedure for tc = 3s and V0 = 608cc.It is observed from this figure that the constant h assumption isvalid for a major part of the compression process. The main dif-ference appears at the end of the compression where h increasessharply resulting in lower final temperatures in the actual experi-ment (Figure 4) than the designed optimal trajectories. The finalresults of the iterative procedure are listed in Table 1 for differ-ent compression times. The efficiencies are obtained for optimal

1 2 3 4

8

8.5

9

9.5

10

10.5

11

11.5

12

Iteration number

h i (W

/m2 .K

)

hguess

hactual

FIGURE 6. CONVERGENCE OF GUESS AND ACTUAL VALUESOF hi FOR tc = 1.5s AND V0 = 300cc AFTER 4 ITERATIONS

100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

70

80

90

Air Volume(cc)

h (W

/m2 K

)

h1=12 W/m2K

h2=14 W/m2K

h3=12.3 W/m2K

h4=13 W/m2K

FIGURE 7. COMPARISON OF HEAT TRANSFER COEFFICIENTPROFILES DURING COMPRESSION FOR SUCCESSIVE ITERA-TIONS OF AVERAGE h value FOR tc = 3s AND V0 = 608cc

trajectories corresponding to the converged h values.

6 Experimental ResultsAs explained before, the trade-off between efficiency and

power density of air compression plays an important role inCAES systems. In order to obtain a benchmark for power-efficiency characteristics of our experimental setup we first con-ducted a series of experiments with different constant flow-ratelinear volume trajectories. Table 2 summarizes the compres-sion time and efficiency values for the tested constant flow rates.Figure 8 shows the comparison of pressure-volume curves for

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TABLE 1. SUMMARY OF RESULTS FOR OPTIMAL TRAJECTO-RIES

tc(s) V0(cc) h( Wm2K ) η(%)

10 608 16 91.56

7 608 15.75 88.43

5 608 15 85.70

3 608 13 80.94

3 300 18 82.90

2 300 16 79.57

1.5 300 12 77.28

TABLE 2. COMPRESSION TIME AND EFFICIENCY VALUESFOR THE TESTED CONSTANT FLOW RATES

tc(s) V0(cc) V (cc/s) η(%)

9.80 715 65.0 86.86

6.01 715 106.0 84.24

4.43 715 144.1 81.99

3.54 715 179.8 80.14

3.03 715 216.2 79.63

2.62 715 250.0 77.99

2.36 715 280.7 77.79

2.19 715 301.9 77.18

1.21 300 201.9 76.20

optimal, iso-thermal, adiabatic and constant flow-rate trajecto-ries for a compression ratio of 10, tc = 3s and V0 = 300cc. Theisothermal curve corresponds to an infinitely slow, 100% effi-cient process while the lowest efficiency of 70.4% is obtainedfrom the adiabatic infinitely fast compression. It is observedthat the optimal trajectory P-V curve is found to be shifted to-wards the isothermal curve compared to the constant flow-rateprocess that results in up to 4% higher efficiency for a samepower density. Figure 9 also illustrates the P-V curves for 3s,5s and 7s optimal trajectories and their comparison with isother-mal and adiabatic compressions. A summary of efficiency ver-sus power density of optimal and constant flow rate trajectoriesis shown in Figure 10. It is observed that for a same value ofpower density the efficiency of optimal trajectories can be higherup to 4%. This difference is more significant at higher efficien-cies that is consistent with the simulation results discussed inSection 3. On the other hand, for a same efficiency the powerdensity can be increased up to 30% for a pressure ratio of 10.The actual temperature-volume tracks the desired T-V optimaltrajectory closely. Therefore, the efficiency of each process is

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

150PV curves

Volume (cc)

Pre

ssur

e (p

si)

Optimal for tc=3s

IsothermalAdiabaticConstant flow rate for t

c=3s

FIGURE 8. COMPARISON OF PRESSURE-VOLUME CURVESFOR OPTIMAL, ISO-THERMAL, ADIABATIC AND CONSTANTFLOW-RATE TRAJECTORIES FOR tc = 3s AND V0 = 300cc

100 200 300 400 500 600

20

40

60

80

100

120

140

Air Volume (cc)

Pre

ssur

e (p

si)

Adiabatictc=3s

tc=5s

tc=7s

Isothermal

FIGURE 9. EXPERIMENTAL P-V CURVES OF OPTIMAL TRA-JECTORIES FOR COMPRESSION TIMES OF 3s, 5s and 7s

obtained to be approximately the same as its theoretical value asshown in Figure 11. The uncertainty in volume measurement canaffect the efficiency calculation. The percentage of error in vol-ume measurement is about ±1% based on the specification of theflow-meters used in the experiment. Numerical sensitivity anal-ysis shows ±1% error in efficiency due to ∓1% error in volumemeasurement throughout the experiment.

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0 20 40 60 80 100 120 140 160 18076

78

80

82

84

86

88

90

92

Power Density (kW/m3)

Effi

cien

cy (

%)

Optimal TrajectoryLinear trajectory

FIGURE 10. COMPARISON OF EFFICIENCY VS POWER DEN-SITY CURVES OF OPTIMAL AND CONSTANT FLOW RATECOMPRESSION TRAJECTORIES FOR r = 10

20 40 60 80 100 120 140 16076

78

80

82

84

86

88

90

92

Power Density (kW/m3)

Effi

cien

cy (

%)

Optimal Trajectory ExperimentOptimal Trajectory Theory

FIGURE 11. COMPARISON OF THEORETICAL AND EXPERI-MENTAL EFFICIENCY VS POWER DENSITY CURVES OF OPTI-MAL TRAJECTORIES FOR r = 10

7 ConclusionsThis paper presented the experimental investigation of op-

timal trajectories for air compression. The experimental setupand procedure were discussed in detail. An adaptive nonlinearcontroller designed in [11] is employed to command the desiredflow-rate to the compressor chamber and track the temperature-volume optimal trajectories. Since the actual value of h is notknown, an iterative procedure was introduced to obtain the opti-mal trajectory using the experimental data. For a given compres-sion time and compression ratio, the final optimal trajectory wasdetermined based on the converged average value of h. The ex-

perimental trajectory tracking results and efficiency-power den-sity calculations were presented. It was shown that for a samepower density and a relatively low pressure ratio of 10 the opti-mal trajectories can improve the efficiency up to 4% comparedto linear constant flow rate trajectories.

REFERENCES[1] P. Y. Li, E. Loth, T. W. Simon, J. D. Van de Ven and

S. E. Crane, “Compressed Air Energy Storage for OffshoreWind Turbines,” in Proc. International Fluid Power Exhibi-tion (IFPE), Las Vegas, USA, 2011.

[2] M. Saadat and P. Y. Li, “Modeling and Control of a NovelCompressed Air Energy Storage System for Offshore WindTurbine,” in Proc. American Control Conference, pp. 3032-3037, Montreal, Canada, 2012.

[3] C. J. Sancken and P. Y. Li,“Optimal Efficiency-Power Rela-tionship for an Air Motor/Compressor in an Energy Storageand Regeneration System,” in Proc. ASME Dynamic Systemsand Control Conference, pp. 1315-1322, Hollywood, USA,2009.

[4] M. Nakhamkin, E. Swensen, R. Schainker and R. Pollak,“Compressed Air Energy Storage: Survey of Advanced CAESDevelopment,” in Proc. ASME International Power Genera-tion Conference, pp. 1-8, San Diego, USA, 1991.

[5] J. D. Lewins, “Optimising and intercooled compressor for anideal gas model,” Int. J. Mech. Engr. Educ., vol. 31, Issue 3,pp. 189-200, 2003.

[6] J. D. Van de Ven and P. Y. Li, “Liquid Piston Gas Compres-sion,” Applied Energy, vol. 86, Issue 10, pp. 2183-2191, 2009.

[7] M. Sozen and T. M. Kuzay, “Enhanced Heat Transfer inRound Tubes with Porous Inserts,” Int. J. Heat and FluidFlow, vol. 17, Issue 2, pp. 124-129, 1996.

[8] A. Rice, Heat Transfer Enhancement in a Cylindrical Com-pression Chamber by Way of Porous Inserts and the Optimiza-tion of Compression and Expansion Trajectories for VaryingHeat Transfer Capabilities, Master thesis, University of Min-nesota, 2011.

[9] A. T. Rice and P. Y. Li, “Optimal Efficiency-Power Trade-off for an Air Motor/Compressor with Volume Varying HeatTransfer Capability,” in Proc. ASME Dynamic Systems andControl Conference, Arlington, USA, 2011.

[10] M. Saadat, P. Y. Li and T. W. Simon, “Optimal Trajectoriesfor a Liquid Piston Compressor/Expander in a CompressedAir Energy Storage System with Consideration of Heat Trans-fer and Friction,” in Proc. American Control Conference, pp.1800-1805, Montreal, Canada, 2012.

[11] F. A. Shirazi, M. Saadat, B. Yan, P. Y. Li and T. W. Si-mon,“Iterative Optimal Control of a Near Isothermal LiquidPiston Air Compressor in a Compressed Air Energy StorageSystem,” to appear in American Control Conference, Wash-ington, DC, 2013.

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