DRAFT: NONLINEAR CONTROLLER DESIGN WITH BANDWIDTH CONSIDERATIONFOR A NOVEL COMPRESSED AIR ENERGY STORAGE SYSTEM

Mohsen SaadatDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: saad0021@umn.edu

Farzad A. ShiraziDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: fshirazi@umn.edu

Perry Y. LiDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: perry-li@umn.edu

ABSTRACTTo achieve both accumulator pressure regulation and gen-

erator power tracking for a Compressed Air Energy Storage(CAES) system, a nonlinear controller designed base on an en-ergy based Lyapunov function. The control inputs for the storagesystem are the pump/motor displacements inside the hydraulictransformer and the liquid piston air compressor/expander.While the pump/motor inside the liquid piston has a low band-width, the other pump/motor inside the hydraulic transformerhas a relatively higher bandwidth. On the other hand, the pneu-matic energy storage path of open accumulator has high en-ergy density, whereas the hydraulic path is more power dense.The nonlinear controller is then modified based on these proper-ties. In the proposed approach, the control effort is distributedbetween the two pump/motors based on their bandwidths: Hy-draulic transformer reacts to high frequency events, while theliquid piston air compressor/expander perform a steady stor-age/regeneration task. As the result, liquid piston air compres-sor/expander will loosely maintain the accumulator pressure ra-tio, while the pump/motor in hydraulic transformer preciselytracks the desired generator power. This control scheme alsoallows the accumulator to function as a damper for the storagesystem by absorbing power disturbances from the hydraulic pathgenerated due to wind gusts.

INTRODUCTIONExtracting energy from wind is perhaps one of the most at-

tractive industries in renewable energy generation. However, themain disadvantage and constraint in providing a good perfor-mance and capacity for the wind turbines is the intermittency andmismatch between available wind power and electrical power de-

mand. Therefore, large scale energy storage systems can be sig-nificantly useful to improve the capacity factor of wind farms byproviding the steady and predictable power for grid as well ascapturing maximum available wind power in normal situations.Storing energy in high pressure compressed air is attractive sinceincreasing the pressure ratio of the compressed air can result anappreciable energy density. For example, at a pressure ratio of350 (35 MPa), 170MJ of energy can be stored in 1m3 of vol-ume. Other major benefits of CAES systems are their low costand long operation life. A novel CAES system has been pro-posed and modeled in [1, 2] (Fig. 1). The excess energy fromthe wind turbine is stored in the storage vessel prior to electric-ity generation, while the generator power is maintained at thedesired value (demand power from electrical grid). This allowsto downsize the electrical components and reduce the involvedpower electronics. In particular, downsizing the generator willconsequently improve the capacity factor of the system definedbased on the generator size.

Two main challenges in the proposed CAES system are i) thelow efficiency and power density of the air compressor/expander,ii) efficiency and power reduction of the compressor/expanderwhen the pressure inside the storage vessel reduces as com-pressed air depletes. The first concern can be solved by de-ploying liquid piston air compressor/expander [3] with a cham-ber filled with porous materials, beside using optimal compres-sion/expansion trajectory [4] and water spray cooling/heatingmethod [5]. The open accumulator concept is a solution for thesecond issue [6]. Energy can be stored or extracted by pump-ing or releasing i) pressurized liquid similar to a conventionalhydraulic accumulator or ii) compressed air similar to a conven-tional air receiver. In both cases, energy is stored in the com-pressed air. By coordinating the hydraulic and the pneumatic

paths, the pressure can be maintained regardless of energy con-tent. However, this coordination for the purpose of pressure reg-ulation can affect the generator power due to tandem shaft con-nection between the pump/motors connected to the pneumaticand hydraulic paths of the open accumulator as well as inductiongenerator. Therefore, a good control algorithm is essential for si-multaneous achievement of the pressure regulation and generatordemand power tracking. The efficient performance of the CAESsystem is significantly dependent on this controller design. Notethat both objectives should be satisfied in presence of supply ordemand power variations.

In this paper, an energy-based controller is first developedto exponentially stabilize the system states and meet the above-mentioned control objectives. However, this controller doesnot take advantage of the frequency characteristics of the sys-tem as well as meeting the bandwidth requirement for the liq-uid piston air compressor/expander unit. Therefore, a filter isintroduced in the controller implementation to channelize thehigh-frequency and low-frequency commands to the hydraulicpump/motor and the liquid piston compressor/expander, respec-tively. In this scheme, the pump/motor responds to the fastchanges in the input power due to wind gusts and the liquid pis-ton compressor acts against long-term variations in either windspeed or electrical grid demand power. Simulation results havebeen presented for the combined wind turbine and CAES sys-tem while the turbine torque is controlled by the standard torquecontroller.

ModelingA short summary of the system model is followed that de-

scribes different subsystems and their function in the combinedwind turbine and storage system:A variable displacement pump (B) is directly (no gearbox) at-tached to the wind turbine (A) in nacelle which converts windpower to hydraulic power. Such a direct coupling requires a com-paratively large displacement pump to transmit a large power(i.e. order of MW) since the wind turbine angular speed islow ( 20rpm). At the ground level, there is a tandem con-nection of a variable displacement hydraulic pump/motor (C),a near-isothermal liquid piston air compressor/expander (F) anda fixed speed induction generator (G), all driven by the pump(B). The liquid piston air compressor/expander, the main unit forstoring/regenerating energy, consists of a compression/expansionchamber filled with some porous material (F1) and a liquid pis-ton pump/motor (F2). The porous material is used in addition tooptimal compression/expansion trajectory and water spray to en-hance the heat transfer inside the chamber to improve its thermalefficiency which has a significant effect on the overall efficiencyof the storage system. The open accumulator (E) which is in factthe storage vessel contains both air and liquid. Hydraulic pathcan be utilized to accommodate high power transient events suchas wind gust or sudden power demand (from grid), whereas thepneumatic path can be reserved for steady storage/regeneration

function.

Figure 1. CAES SYSTEM ARCHITECTURE

In summary, the overall dynamic of the combined wind turbineand storage system can be found as:

Jrwr =�Dp

2pP0(r�1)�Gp(wr)+

12

r0pR2rCP(b,l)

V 3w

wr(1)

Jgwg =�Dpm

2pP0(r�1)�

Dl p

2pP0ln(r)hsgn(Dl p)

trm �Gpm(wg)

�Gl p(wg)�Tg(wg) (2)

V r =Dl p

2pwg +

Dp

2pwrr+

Dpm

2pwgr�Ll p(Pw)

� rLp(r)� rLpm(r) (3)

V =�✓

Dp

2pwr +

Dpm

2pwg

◆+Lp(r)+Lpm(r) (4)

where r, g, p, pm and l p are subscripts standing for turbine rotor,generator shaft, pump in nacelle, pump/motor in hydraulic trans-former and the pump/motor connected to the liquid piston aircompressor/expander chamber, respectively. In these equations,D is used to show the displacement for hydraulic actuators, G isthe mechanical loss and L is the volumetric loss. Moreover, ris the pressure ratio of the air inside the accumulator and V isthe volume of the compressed air. Additionally, Rr and Vw arethe radius of the wind turbine rotor and the wind speed whileCp is the turbine power factor as a function of blade’s pitch an-gle (b) and tip speed ratio (l). It should be noted that while thedisplacement of the actual liquid piston pump/motor in a cyclevaries rapidly to achieve the desired compression/expansion pro-file [4], a cycle mean value is used here shown by Dl p. In theother words, cycle-by-cycle behavior is approximated by a timeaverage model. htrm is then the thermodynamic efficiency of the

compression/expansion chamber over one cycle. Finally, P0 andr0 are the ambient pressure and density of air and J is the angularmoment of inertia.

Controller DesignThe overall control objectives of the combined wind tur-

bine and the storage system can be summarized as: i) Capturingmaximum available power from wind by controlling the windturbine angular speed; ii) Maintaining the accumulator pressureratio over the storage or regeneration mode; and iii) Providingthe required power demanded by the electricity grid. Since thewind turbine shaft and the generator shaft are not coupled inthe proposed architecture, it would be possible to achieve allthese goals at the same time. Wind turbine control is performedby utilizing standard torque control approach through the hy-draulic pump located in the nacelle. The accumulator pressureand generator power will be controlled by the variable displace-ment pump/motors inside the hydraulic transformer as well asthe liquid piston air compressor/expander unit.

Storage System and GeneratorIn conventional CAES systems, the pressure in the storage

vessel reduces as compressed air in the storage vessel depletes,making it difficult for the air compressor/expander to maintaineither its efficiency or power at all energy levels. However, in theopen accumulator design, it is possible to maintain the pressureno matter how much compressed air is inside the vessel. So, it isimportant to design a controller such that it maintain the desiredpower on the generator shaft as well as the desired pressure ratioin the accumulator.

Design of an appropriate nonlinear controller begins bychoosing a suitable Lyapunov function. Here, the idea is to usean energy based Lyapunov function relying on the energy of gen-erator shaft as well as compressed air inside the accumulator.Note that because the storage vessel is assumed to have enoughspace for the compressed air, the air volume dynamics given byEqn. 4 will not be controlled at this level (i.e. a high levelsupervisory controller will control air volume). The Lyapunovfunction considering the generator shaft speed and accumulatorpressure tracking errors is defined as:

E =12

Jgw2g +

Z V1

VP0 (r� rd)dv (5)

where the errors are defined as:

r = r� rd (6)

wg = wg �wdg (7)

Note that the pressure dependent part of this Lyapunov functionis in fact the required energy to compress/expand a fixed mass

of air at pressure rP0 and volume V to the desired pressure ofrdP0. In this definition, the energy level of air at desired pres-sure is zero. If such a compression/expansion takes place as anisothermal process, the energy level at current pressure ratio rand volume V is:

E =12

Jgw2g +P0V

✓rln(

rrd)� r

◆(8)

by taking time derivation of Eqn. 8 and substituting equivalentterms from Eqns. 2, 3 and 4, we will get:

E =�wgP0F �P0�Gr+Hg(r)r2� (9)

where F , G and H are:

F =Dpm

2p(r�1)+

Dl p

2pln(r)hsgn(Dl p)

trm

+Gpm(wg)+Gl p(wg)+Tg(wg)

P0(10)

G =� 12p

✓Dl pwg

rd+Dpwr +Dpmwg

◆

+Ll p(Pw)

rd+Lp(r)+Lpm(r) (11)

H =Dl p

2pwg �Ll p (12)

Here, ln( rrd) is divided into two parts based on its Taylor series

expansion around rd as:

ln(rrd) =

rrd

� r2✓

12r2

d� r

3r3d+ ...

◆

| {z }g(r)

(13)

Now, if Dpm and Dl p are controlled such that F = K1wg and G =K2r where K1 and K2 are two positive gains, then E becomes:

E =�P0K1w2g �P0 (K2 +Hg(r)) r2 (14)

Therefore, if K2 is chosen large enough, K2 +Hg(r) is alwayspositive and E is negative definite. Note that such an analogy isvalid over a finite range of Hg(r) on which the lower bound isknown before hand. Since E is positive definite and radially un-bounded, based on Lyapunov stability criteria, system is asymp-totically stable at the desired pressure ratio and generator shaftspeed [7]. According to Eqns. 10 and 11, Dpm and Dl p can be

obtained as:

2

64Dpm

Dl p

3

75= 2p

2

64(r�1) ln(r)hsgn(Dl p)

trm

�wg �wgrd

3

75

�1

⇥

0

B@

2

64K1 0

0 K2

3

75

2

64wg

r

3

75

�

2

64Gpm+Gl p+Tg

P0

Ll prd

+Lp +Lpm � Dp2p wr

3

75

1

CA (15)

Although the convergence of states is guaranteed by this con-troller, there is a major practical drawback for this controller:High frequency terms appeared in the control command ofpump/motor displacement in the liquid piston air compres-sor/expander unit (Dl p). Such a high frequency signal generatedby the wind gust has bad effects on the pump/motor of the liq-uid piston air compressor/expander and may reduce its operationlife significantly (in general, this subsystem has a relatively lowbandwidth). In addition to wind gust, a fast demand power track-ing can also be a source for creating high frequency command forthe liquid piston pump/motor.

Control Effort Distribution (based on subsystems’bandwidth)

The frequency concern can be solved by using a unique fea-ture of the storage system. Investigating the pressure dynamicand generator shaft dynamic from Eqns. 2 and 3 reveals thatwhile the generator shaft speed has a fast dynamic in response tothe actuator’s displacements, the accumulator pressure dynamicis relatively slow (except for the case when the air volume in-side the storage vessel is too small). Such a property is used hereto modify the nonlinear control commands that relaxes the highfrequency concern. The idea is to use a low pass filter to removethe high frequency components of the liquid piston pump/motordisplacement command signal:

˙Dl p =1t�Dl p � Dl p

�(16)

where t depends on available bandwidth of the pump/motorin liquid piston air compressor/expander. The effect of filter-ing (on generator shaft speed) will be compensated then by thepump/motor displacement command (inside the hydraulic trans-former):

Dpm =� ln(r)r�1

hsgn(Dl p)trm Dl p�

2pP0(r�1)

�Gpm +Gl p +Tg �K3wg

�

(17)Note that the convergence of both states (pressure and speed) canbe proved even by such a modification [8]. In this way, the liquidpiston pump/motor will loosely control the accumulator pressure

ratio around its desired value during the storage or regenerationphase. On the other hand, the hydraulic pump/motor will pre-cisely control the generator shaft speed to maintain the desiredgeneration power (Fig. 2). Note that a precise control of gener-ator shaft speed is important since generator power is a sensitivefunction of its speed. The control effort (to achieve both pres-sure regulation and precise shaft speed tracking) will be sharedbetween two pump/motors inside the hydraulic transformer andliquid piston air compressor/expander based on their availablebandwidths.

Non$Linear*

Controller*

Low$Pass*

Filter*

Generator*Sha5*

Dynamic*

(compensator)*

!gDlp

rDlp

Dpm

Figure 2. NONLINEAR CONTROLLER AND THE COMPENSATOR AR-CHITECTURE

Such a control scheme has an other attractive outcome for thestorage system: the open accumulator will play the role of a bigdamper for the system by absorbing all the high frequency fluc-tuations in the flow coming from the pump in nacelle. These highfrequency power flows will be stored in the accumulator throughthe hydraulic path which has a high power density and is suit-able for this task. As mentioned earlier, these high frequencycomponents are mainly generated by the gusts in wind speed.

Wind Turbine Torque ControlNacelle pump is used to control the turbine speed via the

standard torque controller to maximize power capture [9]:

Tp =�Ksw2r (18)

Ks =12

r0pR5rCmax

p

l⇤3 (19)

Pump displacement Dp(t) is then set according to Eqn. (1).When the measurements and turbine model are perfect, the ro-tor speed converges to the optimal value (Fig. 3 and Eqn. 20). Inregion 2 of wind speed, b = 0 in order to capture maximum windpower. In this case, l⇤ = 8.1 gives the maximum power factor

(Cmaxp = 0.48).

wr =1

2Jrr0pR5

r w2r

✓Cp

l3 �Cmax

p

l⇤3

◆)

8>>>>>><

>>>>>>:

Cp <Cmax

p

l⇤3 l3 ) wr < 0

Cp >Cmax

p

l⇤3 l3 ) wr > 0

(20)Speed regulation of rotor at higher wind speeds (region 3 of windspeed) typically achieved using PID controller:

b(t) = Kpwr(t)+KI

Z t

0wr(t)dt+KD

dwr(t)dt

(21)

where wr = wr �wdr is the rotor speed error, the difference be-

tween the desired rotor speed and the measured rotor speed. Inthis region, the primary objective is to limit the turbine powerso that safe mechanical loads are not exceeded. Power limita-tion can be achieved by pitching the blades or by yawing the tur-bine out of the wind (not considered here), both will reduce theaerodynamic torque below what is theoretically available froman increase in wind speed.

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

CP< (CP−max/�*3).�3CP> (CP−max/�*3

).�3

Tip Speed Ratio (� )

Pow

er F

acto

r (C P)

CP(�=0,� )

(CP−max/�*3).�3

Figure 3. POWER FACTOR VERSUS TIP SPEED RATIO FOR ZEROPITCH ANGLE

Simulation ResultsA 65-hour simulation has been run to show how the com-

bined wind turbine and storage/regeneration system works usingthe proposed nonlinear controller. The wind profile is generatedby superimposing a 10-minute average wind speed profile andthe corresponding turbulent wind (Fig. 4). The mean wind speedis a recorded series of data at 50m elevation (provided by Renew-able Energy Research Laboratory (RERL)) while the turbulentwind speed generated by Turbsim software based on the meanwind speed profile [10]. It is assumed that the storage systemis designed for a 3MW wind turbine while the demand power is

set at 700kW and the electricity line frequency is 60Hz (over thewhole time period). The total storage size is 500m3. The rest ofthe constant parameters used in this simulation are given in Table1. Note that W is the bandwidth considered for each actuator inthe combined system. More details regarding these parameterscan be found in [2].

Table 1. Constant parameters used for simulation

Property Value Unit Property Value Unit

Rr 45 m Jr 2e6 Kg.m2

bmax 40 degree Wb 0.03 Hz

wratedr 20.5 rpm rd 200 �

fs 60 Hz p f 4 �

Vs 415 volt hA 5e4 Watt/K

Dratedp 630 lit/rev Wp 2 Hz

Dratedpm 6 lit/rev Wpm 2 Hz

Dratedl p 120 lit/rev Wl p 0.002 Hz

Jg 20 Kg.m2 Vacc 500 m3

0 6 12 18 24 30 36 42 48 54 60 650

5

10

15

Time (hr)

Win

d S

peed

(m/s

)

1−second Sample Data 10−minute Mean

Figure 4. MEAN AND TURBULENT WIND SPEED USED FOR SIMU-LATION

Fig. 5 shows the captured wind power through the pumplocated at nacelle as well as accumulator energy. The accumu-lator energy increases when the captured wind power is morethan the constant electric demand power and vise versa (consid-ering the inefficiencies on the path between pump in nacelle andgenerator). An important consequence of such a constant powergeneration is the enhancement of capacity factor defined basedon generator size.

Fig. 6 (top) shows the accumulator pressure ratio that is al-ways close to its desired value (200) no matter how much energyis stored. Since the accumulator plays the role of a damper forthe overall storage system, the pressure deviation amplitude isdirectly related to the fluctuations in the wind turbine speed. The

0 6 12 18 24 30 36 42 48 54 600

0.5

1

1.5

2

2.5

3

Pow

er (M

W)

Time (hr)

0 6 12 18 24 30 36 42 48 54 60 0

3

6

9

12

15

18

Ener

gy (M

Whr

)

Generator Power (Captured) Wind Power Accumulator Energy

Figure 5. TURBINE SHAFT POWER, GENERATOR POWER AND AC-CUMULATOR ENERGY (BOTTOM)

turbine rotor speed is controlled such that the optimal tip speedratio is maintained in the second region of wind speed. Note thatthe hydraulic pump in Nacelle has a relatively large displacement(rated at 630 lit/rpm) since the wind turbine speed is small and nogearbox is used. Fig. 6 (middle) shows the air mass flow rate andliquid volume flow rate to/from the accumulator. While liquidand air flows are in opposite directions, they have more or lessthe same ratio all the time. This shows how the controller triesto maintain the pressure inside the vessel by adding/subtractingenough volume of liquid through the hydraulic path. Fig. 6 (bot-tom) shows the displacements corresponding to the pump/motorlocated inside the hydraulic transformer and the pump/motor lo-cated in the liquid piston air compressor/expander unit. Whilethe latter is responsible to control the energy storage/regenerationin longer time scales (i.e. 10-minutes), the former is utilized tocompensate the low-pass effect and maintain the generator power(or shaft speed). Particularly, it is important to notice that thepump/motor in the hydraulic transformer works as a motor mostof the time while in the pumping mode it just uses a small amountof its displacement. In order to solve possible practical issues re-garding this fact, one can divide this pump/motor into a large mo-tor and a relatively smaller pump/motor (with probably a higherbandwidth). In this way, the small pump/motor can be utilized athigher displacements which will increases its overall efficiency.

The performance of the standard torque controller appliedon the wind turbine shaft through the hydraulic pump in na-celle is shown in Fig. 7. As shown, the controller tracks thedesired wind turbine power versus wind speed curve in region2 of wind speed (Vwind 12m/s) while the blade’s pitch anglesis deviated from zero to cut the extra wind power in region 3of wind speed (Vwind .12m/s). The overall efficiency for the en-tire system (wind turbine and storage/regeneration system) overthe 65-hour simulation is found 72%. The loss stream regardingeach pump/motor as well as the thermal loss inside the compres-sion/expansion chamber is shown in Fig. 8.

0 6 12 18 24 30 36 42 48 54 60199.8

199.9

200

200.1

200.2

200.3

200.4

Acc

umul

ator

Pre

ssur

e R

atio

Time (hr)0 6 12 18 24 30 36 42 48 54 60 0

4

8

12

16

20

24

Win

d Tu

rbin

e A

ngul

ar S

peed

(rpm

)

0 6 12 18 24 30 36 42 48 54 60−25

−12.5

0

12.5

25

Liqu

id V

olum

e Fl

ow R

ate

(lit/s

)Time (hr)

0 6 12 18 24 30 36 42 48 54 60−5

−2.5

0

2.5

5

Air

Mas

s Fl

ow R

ate

(Kg/

s)

0 6 12 18 24 30 36 42 48 54 60−6

−3

0

3

6

Dis

plac

emen

t (lit

/rev)

Time (hr)

0 6 12 18 24 30 36 42 48 54 60 −120

−60

0

60

120

Dis

plac

emen

t (lit

/rev)

Pump/Motor in Hydraulic Transformer

Pump/Motor in Liquid Piston Air C/E

Figure 6. PRESSURE RATIO OF THE ACCUMULATOR AND THEWIND TURBINE ANGULAR SPEED (TOP); AIR MASS AND LIQUIDVOLUME FLOW RATES TO/FROM THE ACCUMULATOR (MIDDLE);DISPLACEMENT OF PUMP/MOTOR IN HYDRAULIC TRANSFORMERAND THE LIQUID PISTON AIR COMPRESSOR/EXPANDER (BOTTOM)

ConclusionA nonlinear controller designed for a CAES system that has

been proposed and modeled in earlier works. An energy basedLyapunov function is derived for the purpose of controller de-sign. A modification is then performed by utilizing a low-passfilter in one channel while compensating the filter effect by othercontrol channel to solve the frequency issues for the actuatorsand meet their bandwidth constraints. Therefore, the liquid pis-ton air compressor/expander which is naturally slow (low band-width) is used to store/regenerate energy in long time scales(steady). The pump/motor inside the hydraulic transformer thathas a relatively higher bandwidth will then compensate the fil-tering effect to maintain the generator output power. In this way,

both short term and long term control objectives are satisfied bythe same controller. An important consequence of such a controlarchitecture is the useful role of the storage vessel (open accumu-lator) for the combined system: absorbing all the high frequencypower fluctuations through the hydraulic path which has a highpower density. The controller design methodology used in thiswork is highly matched with both the physical constraints andbenefits of different subsystems in the proposed CEAS system.

0 3 6 9 12 150

1

2

3

4

Wind Speed (m/s)

Cap

ture

d Po

wer

by

Pum

p (M

W)

Simulation Result Reference Curve

Figure 7. WIND TURBINE CAPTURED POWER VS. WIND SPEED

0

4

8

12

16

20

24

28

32

Overall Energy Loss for the Storage System (efficiency : 72%)

Loss

Stre

am (P

erce

nt o

f Inp

ut)

Pump in Nacelle Pump/Motor in Hydraulic Transformer

Pump/Motor in Liquid Piston Air C/E

Liquid Piston C/E Chamber (thermal)

Figure 8. LOSS STREAM FOR THE COMBINED SYSTEM BASED ONPERCENT OF INPUT ENERGY

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] Li, P., Van de Ven, J., and Sancken, C., Open Accumula-tor Concept for Compact Fluid Power Energy Storage, Pro-

ceedings of the ASME Int. Mechanical Engineering Congress,Seattle, WA, 2007, pp 42580

[4] 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.

[5] C. Qin and E. Loth, “Liquid Piston Compression withDroplet Heat Transfer,” in Proc. 51st AIAA Aerospace Sci-ences Meeting, Grapevine, TX, 2013.

[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] J. E. Slotin and W. Li, Applied Nonlinear Control, Prentice-Hall, 1991

[8] M. Wang and P. Y. Li, ”Displacement Control of Hy-draulic Actuators Using a Passivity Based Nonlinear Con-troller”, ASME DSC Conference, [DSCC2012-MOVIC2012-8784], Ft. Lauderdale, Nov. 2012.

[9] L. Y. Pao, K. E. Johnson., A Tutorial on the Dynamics andControl of Wind Turbines and Wind Farms, American ControlConference, ACC, St. Louis, MO, June 2009

[10] TurbSim Users Guide for Version 1.40., Technical ReportNREL/TPxxx, September 2008