An efficient and power dense high pressure air compressor/expander (C/E) is critical for the success of a compressed air energy storage (CAES) system. There is a tradeoff between efficiency and power density that is mediated by heat transfer within the compression/expansion chamber. This paper considers the optimal control for the compression and expansion processes that provides the optimal tradeoff between efficiency and power. Analytical Pareto optimal solutions are developed for the cases in which hA, the product of the heat transfer coefficient and heat transfer surface area, is either a constant or is a function of the air volume. It is found that the optimal trajectories take the form “fast-slow-fast” where the fast stages are adiabatic and the slow stage is either isothermal for the constant-hA assumption, or a pseudo-isothermal (where the temperature depends on the instantaneous hA) for the volume-varying-hA assumption. A case study shows that at 90% compression efficiency, power gains are in the range of $500−1500%$ over ad hoc linear and sinusoidal profiles.

## Introduction

Compressed air energy storage (CAES) is widely accepted as an economic grid scaled energy storage approach needed to meet the challenge of integrating more and more intermittent and variable renewable energy resources such as wind or solar into the electrical grid. A novel CAES system for off-shore wind turbines is proposed in Ref. [1]. For this and other CAES systems, a critical component is the high pressure air compressor/expander (C/E) responsible for transforming energies between mechanical and pneumatic domains. To be effective, the C/E needs to be capable of high pressure ($∼200$ bar), efficient, powerful enough to handle the power requirements, and compact enough so as to minimize capital cost and physical footprint. A C/E with these characteristics are also useful in other applications such as making liquefied gas or fueling vehicles powered by natural gas.

There is an inherent tradeoff between efficiency and power density (power for a given size) that is a function of heat transfer. For example, as will be explained in Sec. 2, an isothermal process is the most efficient but requires a long cycle time for heat transfer to take place; an adiabatic process can be infinitely fast, but is inefficient. A common hardware approach to improving this tradeoff is to enhance heat transfer by adding extra thermal capacitances and heat transfer surface areas. Examples include elastomeric foams, [2], thin metal strands [3], and sprays of tiny water droplets [4,5]. Our group's approach is to use a liquid piston compressor/expander in conjunction with porous media inserts to increase the specific surface area for heat transfer while allowing free movement of the piston which is the liquid/air interface [610].

In this paper, a software approach is proposed in which the compression and expansion trajectories are optimized and controlled to further improve performance. In the case of the liquid piston compressor/expander, arbitrary trajectories can be implemented simply by varying the liquid flow rates. Pareto optimal solutions are sought that optimize powers for given efficiencies or optimize efficiencies for given powers. In the literature, similar works are found in Refs. [1115] where optimal control was derived using calculus of variations to maximize work output of heat engines. Here, we focus on air compressor/expanders and derive analytical solutions. Besides an ideal gas assumption, it is assumed that the temperature of the heat source/sink is constant, and hA, the product of the heat transfer coefficient (h) and the heat transfer area A, is either a constant or a function of the air volume V. Although these assumptions are an idealization, these analytical solutions offer important insights into the problem of high pressure, efficient, and powerful compressor/expanders. For a specific problem with more general assumptions, numerical techniques such as parametric optimization and dynamic programming can be applied. See, e.g., our more recent works [16,17].

It is shown that the Pareto optimal compression/expansion trajectories take the form of consecutive fast-slow-fast segments. Specifically, in the constant hA product case, the optimal trajectories consist of adiabatic–isothermal–adiabatic segments. In the volume dependent hA(V) case, the isothermal segment is replaced by a pseudo-isothermal segment whose temperature depends on the instantaneous hA. The improvements can be quite significant. For example, in a CAES case study, the system is $5−15$ times more powerful with optimized trajectories than with ad hoc sinusoidal and linear trajectories. Preliminary results in this paper were presented in Refs. [18] and [19] where only sketches of the proofs were provided.

The remainder of the paper is organized as follows: Section 2 discusses the system model, assumptions, and definitions. Section 3 derives the optimal compression trajectory solution for the constant hA case. Section 4 extends the result to the volume varying hA(V) case. Section 5 applies the solution to an energy storage scenario. Sections 6 and 7 discuss the results and offer some remarks.

## System Model and Problem Statement

For the purposes of this paper, the air compressor/expander is assumed to be used in the open accumulator energy storage architecture as shown in Fig. 1, although much of the analysis is generally applicable. The distinguishing feature of this configuration is that the compressed air pressure in the storage vessel can be maintained constant regardless of the amount of compressed air present. This is accomplished by adjusting the liquid volume in the vessel which contains both liquid and compressed air.

To store energy, for every compression cycle, ambient air (assumed to be an ideal gas) at temperature and pressure, T0 and P0, is compressed to a temperature and pressure Tc and Pc inside the compression/expansion chamber according to a time-dependent pressure–volume trajectory: $ζc(t)=(P(t),V(t))$. The compressed air is then ejected isobarically to the accumulator to be stored. Inside the accumulator, the compressed air is allowed to cool back to ambient temperature.

For energy regeneration, the compressed air is first injected from the accumulator into the compression/expansion chamber isobarically. The air then expands from Pc and T0 to the ambient pressure P0 and the final temperature Te according to a pressure–volume trajectory $ζe(t)=(P(t),V(t))$.

Work by the ambient pressure P0 is subtracted out because it is assumed to be the case pressure of the compressor/expander. In accordance with the open accumulator concept, when compressed air is ejected into or injected from the storage vessel, an equivalent volume of liquid from the storage vessel is removed or added. Since the input/output work of the gas compressor/expander is retrieved/provided by the hydraulic pump/motor, the net work associated with the ejection/injection process is zero. However, additional recompression work is needed to maintain the compressed air pressure inside the accumulator as the compressed air is cooled.

The assumption that the air has time to return to equilibrium with the environment before expansion is compatible with applications that require storage on the order of hours or longer and is a conservative assumption in other cases.

The dynamics of the compression/expansion process are determined from the first law of thermodynamics and the equation of state for an ideal gas applied to the air within the compression/expansion chamber
$ncvṪ=−PV̇−q=−nRTVV̇−qP=nRTV$
(1)
where $cv=R/(γ−1)$ is the constant volume specific heat of air on a molar basis, γ is the ratio of the specific heats, (P, V, T) are the air pressure, volume, and temperature, n is the air quantity in moles, R is the universal gas constant, q is the heat transfer rate (out of the gas) described by
$q=hA(V)(T−T0)$
(2)

where h is the heat transfer coefficient, A is the heat transfer surface area, and the heat sink/source temperature is assumed to be ambient T0. $V̇$, which determines the compression/expansion trajectory, can be considered the control input. In this paper, the hA product is assumed to be either a constant or a function of air volume only.

The assumption that air in the compressor/expander is an ideal gas is an idealization that is less accurate at higher pressures. In the case of the liquid piston compressor/expander, vaporization and condensation of the water also have thermodynamic effects (although the effect is only appreciable at low efficiency operating regimes to increase power density but its effect is insignificant at high efficiency regimes [21]). However, in light of the simplified heat transfer model, the ideal gas model is reasonable in order to obtain analytical near closed-form solutions. As mentioned earlier, more complex models can be investigated numerically such as for detailed designs.

The cycle compression work input $Win$ and expansion work output Wout are given by the shaded areas as shown in Fig. 2.4 These areas exclude the ejection and injection work, but the compression work includes the isobaric cooling work in accordance with the open accumulator concept. The compression work input is

$Win(ζc):=−∫0tc(P(t)−P0)V̇(t) dt+nR(1−1/r)(Tc−T0)$
(3)

$Wout(ζe):=∫0te(P(t)−P0)V̇(t) dt$
(4)
Note that $Win=Wout$ if ζc and ζe are isothermal processes at T = T0. Hence, the work for isothermal compression or expansion is the maximum work available and is defined to be the stored energy for compressing this amount of gas to Pc. It is given by
$E:=nRT0[ln(r)−1+1r]$
(5)

where $r:=Pc/P0>1$ is the pressure ratio.

Compression and expansion efficiencies are defined as
$ηc(ζc):=EWin(ζc) and ηe(ζe):=Wout(ζe)E$
(6)
The rate at which energy is stored or regenerated is the power. Therefore, compression and expansion powers are defined as
(7)
where tc and te are the compression and expansion process times to execute trajectories ζc and ζe. From the first law (1) and Newton's law of cooling (2), tc and te are
$tc/e:=∫dt=∫q dthA(V)(T−T0)=∫ncvdT+PdVhA(V)(T−T0)$
(8)

where $dT=(PdV+VdP)/nR$ and the integration is executed over the trajectories $ζc(·)$ and $ζe(·)$. Ejection to and injection from the storage vessel, fresh air intake from and expanded air exhaust to the low pressure buffer (or atmosphere) are assumed to take no time as they are not limited by heat transfer. Therefore the power defined in Eq. (7) can be consider the thermodynamic limited power. The actual power will be lower due to the finite time needed for these nonthermodynamic limited processes.

There is an inherent conflict between efficiency and power. For example, isothermal compression/expansion at T = T0 are 100% efficient, but from Eq. (8), the process times are infinite, resulting in zero power. On the other hand, adiabatic processes take zero times (hence high power) but, as illustrated in Fig. 2, the energy loss is large so that efficiencies are low. Note from Eq. (8) that the numerator determines the P–V curve and hence efficiency, whereas increasing the denominator decreases process time and increases power. Thus, increasing the heat transfer capability given by hA(V) can increase power without sacrificing efficiency. In this paper, we go further by utilizing optimal control to make the best use of the available heat transfer capability.

Problem statement. To determine Pareto optimal trajectories $ζc*(t)$ for compression from (P0, T0) to Pc with subsequent isobaric cooling, and $ζe*(t)$ for expansion from $(Pc,T0)$ to P0, such that: for a given efficiency, no other trajectory provides more input power $Win$ or output power $Wout$ than the Pareto optimal trajectory or for a given power, no other trajectory operates at higher efficiency than the Pareto optimal trajectory. The optimization problem is expressed as
(9)
where $η*$ is some prescribed efficiency. Or equivalently
(10)

where $Pow*$ is some prescribed power. In Eqs. (9) and (10), $ηc/e$ and $Powc/e$ are defined in Eqs. (6) and (7), respectively.

Remark 1. In the optimization procedures below, we solve the problem of minimizing process times tc or te while constraining input work $Win$ or output work $Wout$ (or optimizing $Win$ or $Wout$ while prescribing process times). It will be shown in Remark 2 that it is equivalent to maximizing power while prescribing efficiency (or maximizing efficiency while prescribing power).

## Constant hA Optimal Trajectories

In this section, we derive the Pareto optimal compression/expansion trajectories for the case when the hA product in Eq. (2) is a positive constant and the heat sink/source is at temperature T0. Instead of optimizing with respect to efficiency and power, we optimize with respect to input/output work and process time.

The derivation proceeds in two steps:

Step 1. A physically feasible process ζ between two arbitrary endpoints (P0, V0) and (Pf, Vf) can be improved with an A-I-A process $ζ*$ between the same endpoints consisting of an adiabatic (A) process, an isothermal (I) process, and a final adiabatic (A) process. The work associated with the adiabatic–isothermal–adiabatic (AIA) process is the same but the process time will be the same or reduced.

Step 2. For the given initial (P0, V0), final pressure Pf, and for each prescribed efficiency, the final volume Vf and the isothermal temperature $T*$ of the AIA process are optimized to minimize the process time while matching the prescribed efficiency (or to optimize work while matching the prescribed process time).

Step 1 is further divided into two substeps. Let ζ be a process between two arbitrary endpoints (P0, V0) and (Pf, Vf) that is physically feasible with heat transfer with the heat source/sink at the temperature T0. Since any such process can be uniformly approximated by a sequence of isothermal (I) and adiabatic (A) processes, without loss of generality, ζ is assumed to consist of such alternate I and A process steps as illustrated in Fig. 3. In step 1A, we show that an isothermal–adiabatic–isothermal (IAI) subsequence can be replaced by an AIA subsequence with reduced process time but equal work. Then, in step 1B, this procedure is propagated through the entire original process ζ to form a global AIA replacement sequence with equal work and reduced process time.

### Step 1A: Improving Isothermal–Adiabatic–Isothermal Process Time With Adiabatic–Isothermal–Adiabatic Process.

Consider an IAI subsequence of the original process curve ζ that has been approximated by alternate A and I steps (Fig. 3). Suppose that the IAI subsequence traverses through states A-B-C-D, where A-B and C-D are isothermal steps at temperatures TA and TD, and B-C is an adiabatic step.

The boundary work done ($W=∫PdV$) on the gas is
$WIAI=WAB+WBC+WCD=nR[TAln(PB/PA)+(TD−TA)γ−1+TDln(PD/PC)]$
(11)
Since the adiabatic step takes no time, the process time is
$tIAI=nRhA[TAln(PB/PA)TA−T0+TDln(PD/PC)TD−T0]$
(12)

To be physically feasible, each term should be positive.

Consider now an alternate AIA sequence that traverses the states A-E-F-D where A-E and F-D are adiabatic steps and E-F is isothermal at temperature TE. The work on this alternate sequence is
$WAIA=WAE+WEF+WFD=nR[(TD−TA)γ−1+TEln(PF/PE)]$
(13)
Note that the total adiabatic works are the same for the two sequences since the temperature changes are the same. The process time becomes
$tAIA=nRhA[TEln(PF/PE)TE−T0]$
(14)

which must be positive to be physically feasible.

It can be shown that the products of the isothermal pressure ratios are the same for the two sequences
$PBPA·PDPC=PFPE:=rI$
(15)
Thus, we can write for some $x∈ℜ$
$PBPA=rIx, PDPC=rI1−x$
If the temperature TE is chosen such that the work WAIA by A-E-F-D (AIA) is the same as the work WIAI by A-B-C-D (IAI), then utilizing Eq. (15) and comparing Eqs. (11) and (13), TE must be
$TE=xTA+(1−x)TD$
(16)
Let $Δs$ be the time difference normalized by the AIA time
$Δs:=tIAI−tAIAtAIA$
(17)
Thus, $Δs>0$ would signify a decrease in process time by the AIA sequence. Utilizing Eqs. (15) and (16) and some algebra, the normalized time difference can be written as
$Δs=T0TE[x(TE−TA)(TD−TA)(TA−T0)(TD−T0)]$
(18)

To evaluate the sign of $Δs$, we consider the following cases to determine the sign of the terms inside $[·]$ in Eq. (18).

Case 1. $rI>1, 0. In this case, $TA>T0$ and $TD>T0$ (in order for each term in Eq. (12) to be positive). From Eq. (16), TE is between TA and TD.

Case 2. $rI>1$, x < 0. In this case, $TA (for each term in Eq. (12) to be positive). Combining this with Eq. (16), we have $TA.

Case 3. $rI>1, 1. In this case, $TD (for each term in Eq. (12) to be positive). Combining this with (16), we have $TD.

Case 4. $rI<1, 0. In this case, $TA and $TD (in order for each term in Eq. (12) to be positive). From Eq. (16), TE is between TA and TD.

Case 5. $rI<1$, x < 0. In this case, $TD (for each term in Eq. (12) to be positive). Combining this with Eq. (16), we have $TE.

Case 6.$rI<1, 1. In this case, $TA (for each term in Eq. (12) to be positive). Combined this with Eq. (16), we have $TE.

The effects on the signs of each term inside $[·]$ are summarized in Table 1.

For the boundary case of rI = 1, AIA collapses to a single adiabatic step so that tAIA = 0. It would be smaller than the finite tIAI except when $PB/PA=PD/PC=1$ where the IAI also collapses to a single adiabatic step. If x = 0 or x = 1, one of the I's in the original IAI collapses to a point. They can already be considered an AIA with a trivial A. Thus, the process time is not affected.

The overall effect is that in all cases, $Δs≥0$ while only when $PB/PA=1$ or $PD/PC=1, Δs=0$.

### Step 1B: Improving Process Time of Whole Process by a Global Adiabatic–Isothermal–Adiabatic Process.

Step 1A shows that a physically feasible IAI process can be replaced by an AIA process with the same endpoints and the same boundary work but a shorter process time. The procedure can be propagated to attain a global AIA process that improves upon an arbitrary physically feasible process with the same endpoints and the same work but a shorter process time.

To see this, suppose that the original process ζ is adequately approximated by the sequence of isothermal and adiabatic segments
$I1A1I2︸A1′I1′A2′A2I3…AkIk…$
Replacing $I1A1I2$ by $A1′I′1A2′$ with the same work but reduced time, we have
$A1′I1′A2′A2︸A2″I3…AkIk…$
Since $A2′A2=A2″$ is together a single adiabatic, we have a sequence with fewer segments
$A1′I1′A2″I3…AkIk…$

The process can be continued by iteratively replacing an IAI sequence by an AIA, and combining the resulting consecutive A's into a single A segment. As the number of segments reduces, eventually, the replacement sequence becomes a three-segment AIA. The two end points and the boundary work for the final AIA are unchanged, but the process time is reduced. The result for step 1 is summarized as follows:

Proposition 1.Given the initial and final points (P0, V0) and (Pf, Vf), the process with the least process time t that provides the prescribed feasible boundary work$W=∫0t−PV̇dt$consists of a sequence of AIA segments.

Proof. The proof is simply that any feasible candidate process that provides the prescribed work can be subjected to the improvement process above unless it itself is an AIA process. □

### Step 2: Optimizing the Adiabatic–Isothermal–Adiabatic Trajectory.

In this step, the AIA trajectory is optimized to minimize the process time. We first focus on the compression process. It is assumed that the initial $(P0,V0,T0)$ and the final $Pc=rP0$ are given and the work input is prescribed. Note the molarity $n=(P0V0)/(RT0)$.

Let the AIA process goes through points $(Pi,Vi,Ti)$ for $i=0,1,2,3$ with $P3=Pc$ and T1 = T2. Steps $0−1$ and $2−3$ are adiabatic steps and steps $1−2$ is isothermal. The process can be uniquely parametrized by temperatures $(T1,Tc)$. Therefore, using the property of the adiabatic process
so that the isothermal pressure ratio is
$rI:=P2P1=r(TcT0)−γγ−1$
The prescribed work in terms of $T1,Tc$ is therefore
$Win(T1,Tc)=nR{T1[ln(r)−γγ−1ln(TcT0)]+γγ−1(Tc−T0)−T0(r−1)/r}$
(19)
The process time is
$tc(T1,Tc)=nRhA(T1T1−T0)[ln(r)−γγ−1ln(TcT0)]$
(20)
The constrained minimization problem with prescribed work input $W†$ can be formulated as an unconstrained optimization using Lagrange multiplier
$minT1,Tc,λtc(T1,Tc)+λ(Win(T1,Tc)−W†)$
so that the first-order extremum conditions are
$∂tc∂T1+λ∂Win∂T1=0∂tc∂T3+λ∂Win∂Tf=0$
(21)
Solving for and equating λ from each expression yields
$λ=T0T1−T0=T1(T1−T0)(Tc−T1)⇒T1=T0·Tc$
(22)

This establishes the relationship between the isothermal temperature of the AIA trajectory and the initial and final temperatures. Different choices of T1 (or Tc) result in different work input.

A similar procedure can be applied to minimize the work input while satisfying a prescribed time. This results in the same relation (with λ replaced by $1/λ$ in Eq. (21)) between T0, T1, and Tc in Eq. (22).

For the expansion case, we specify the initial $(rP0,Vs,T0)$ and the final pressure P0. The AIA is parameterized by the isothermal temperature T1 and final temperature Te. The output work and expansion time are given instead by
$Wout(T1,Te)=nR{T1[ln(r)−γγ−1ln(T0Te)]+γγ−1(T0−Te)−T0(1−1/r)}$
(23)

$te(T1,Te)=nRhA(T1T0−T1)[ln(r)−γγ−1ln(T0Te)]$
(24)
The optimal trajectory can be found by optimizing the time while matching the prescribed work output, or optimizing work output while matching the prescribed time. Both result in the same expression
$T1=T0·Te$
(25)

Note that since optimizing time while prescribing work, and optimizing time while prescribing time give the same trajectories, the optimal trajectories are the Pareto optimal.

We summarize the results for the constant hA case in the following theorem.

Theorem 1. Let P0and T0be the ambient pressure and temperature. Assume an ideal gas model, and in Eq.(2), the heat transfer coefficient-area product hA is constant and the heat sink/source is at T0.

Compression. The trajectory$ζc*$that compresses the gas from (P0, T0) to a pressure of rP0with r > 1 consisting of an instantaneous adiabatic compression from (P0, T0), followed by an isothermal compression at Tiso, and finally followed by another instantaneous adiabatic compression to the desired pressure rP0and the final temperature of Tc such that
$Tiso=T0⋅Tc$

is a Pareto optimal trajectory with respect to minimizing input work$Win$given by Eq.(3)and compression time tc given by Eq.(8).

Expansion. The trajectory,$ζe*$that expands the gas from$(rP0,T0)$to P0with r > 1 consisting of initial and final instantaneous adiabatic stages and an intermediate isothermal stage where the isothermal stage temperature Tisoand final temperature Te are related by
$Tiso=T0⋅Te$

is a Pareto optimal with respect to maximizing output work (4) and minimizing process time (8).

For both compression and expansion, the Pareto frontier is generated by choosing different Tiso(or final temperatures Tc or Te).

Remark 2. Although Theorem 1 is stated in terms of input/output work (3) and (4) and process time (8), the same trajectories are optimal with respect to efficiencies (6) and powers (7). It is so because prescribing efficiency is equivalent to prescribing input/output work. Also, for compression, maximizing power is equivalent to minimizing process time. For expansion, the first-order optimality condition for maximizing power with the prescribed work output using the Lagrange multiplier is, in place of Eq. (21), given by

for $θ=te$ and T1. Thus, solving for and equating $t(tλ+1)/Wout$ for each θ gives the same condition as for the case of minimizing expansion time with a constraint in output work.

## Volume Dependent hA Optimal Trajectory

In this section, we extend the result in Sec. 3 to the case when the heat transfer coefficient-area product hA(V) in Eq. (2) is a differentiable function of the gas volume V. This will be especially important with a liquid piston compressor/expander with the chamber being filled with porous media so that the heat transfer area will decrease as the chamber volume decreases. The solution is derived in two steps:

Step 1. We will first show that Pareto optimal monotonic compression or expansion trajectories must consist of segments of two adiabatic (A) segments sandwiching a pseduo-isothermal (pI) segment (i.e., ApIA). Instead of being at a constant temperature as in the constant hA case, the temperature within the pseudo-isothermal segment varies with the hA for the instantaneous volume.

Step 2. The parameters for the ApIA trajectory will be optimized to minimize the process time for the prescribed work (or equivalently, to optimize work for the prescribed process time).

### Step 1: Optimal Trajectories Consist of Adiabatic (A), Pseudo-Isothermal (pI), and Adiabatic (A) Segments.

We discretize the differentiable hA(V) function into N (which will be taken to be $∞$) constant hA segments such that: for $i=1,2,…,N$
(26)

Here, $VN represent the transition volumes at which the hA product changes to a new value. From Proposition 1, over each constant hA volume interval $(Vi,Vi−1)$ and for given initial and final pressures and volumes, the optimal trajectory with respect to work and time takes the form of an AIA trajectory.

Consider consecutive constant hA intervals $(Vi,Vi−1)$ for $i=1,…,N$. Let the isothermal temperatures be $Ti−1iso$ for the interval $(Vi,Vi−1)$ and the temperature at Vi be Ti for each i (see Fig. 4).

The boundary work ($∫PdV$) over these N segments is
$W=nR(TN−T0)γ−1+∑j=1NnRTjisoγ−1ln[Tj−1Vj−1γ−1TjVjγ−1]$
(27)
and the process time over these N segments is
$t=∑j=1NnRTjiso(γ−1)hAi(Tjiso−T0)ln[Tj−1Vj−1γ−1TjVjγ−1]$
(28)
Using the Lagrange multiplier method to optimize t (or W) while constraining W (or t), the first-order optimality condition gives for $θ=Tiiso,Ti$ for each $i=1,…,N$
$λ∂W∂θ=−∂t∂θ$
(29)
where λ is the Lagrange multiplier common for all θ. Derivatives with respect to $Tiiso$ for different i give
$[λ−T0hAi(Tiiso−T0)2]ln[Ti−1Vi−1γ−1TiViγ−1]=0$
(30)

This implies each segment is either an entire adiabatic ($Ti−1Vi−1γ−1=TiViγ−1$) or $T0/λ=hAi(Tiiso−T0)2$. Derivative with respect to Ti gives information in regard to the transition temperature Ti (see Ref. [10] for details). However, this information is not useful as we take $N→∞$.

Since hA(V) is a differentiable function, as $N→∞,|hAi−hAi+1|→0$. Hence, two consecutive isothermal temperatures, $Tiiso$ and $Ti+1iso$, become infinitesimally close and the adiabatic portion (of the AIA for each constant hA interval) that transitions between the two isotherms also vanishes. This means that as $N→∞$, contiguous nonadiabatic intervals form intervals where the temperature is given by
$T0λ=hA(V)(T(V)−T0)2$
(31)

with $λ>0$ being the parametrization. The segments in Eq. (31) are referred to as pI segment since it reduces to an isotherm when hA(V) is a constant.

Note that for each $λ>0$ in Eq. (31), there are two possible solution branches: $T(V)−T0>0$ or $T(V)−T0<0$. The appropriate branch must be consistent with the direction of heat transfer according to Eqs. (1) and (2) by checking the sign consistency in
$nR(TpI′γ−1+TpIV)V̇=−hA(V)(TpI(V)−T0)$
where the superscript $′$ denotes derivative with respect to V and
$TpI′=∓12T0λhAhA′hA$

with the $−/+$ signs corresponding to $T>T0$ and $T, respectively. For the compression process, $(T−T0)>0$ is the nominal branch with decreasing volume. For the expansion process, $(T−T0)<0$ is the nominal branch with increasing volume. Also when hA(V) increases monotonically with V (as is expected in practical applications), only the nominal branch is feasible.

Beside the pseudo-isothermal segments, from Eq. (30), adiabatic segments can also exist. However, they can only exist at the beginning or end of the trajectories, or as transitions between the two feasible pseudo-isothermal branches of Eq. (31). It is because two points on the same branch of a feasible pI (31) cannot be connected by an adiabatic as transversing the pI involves continuous heat transfer in one direction whereas traversing the adiabatic involves no heat transfer.

For the practical case where $∂hA/∂V≥0$, it is easy to see from Fig. 5 that a pI-A-pI trajectory where the two pI's correspond to two branches (and different volume directions) is less efficient and takes more time than a direct pI-A or A-pI. Thus, trajectories with interior A segments are not optimal in this case. For more generic hA(V) functions, by restricting volume trajectory V(t) to be nonincreasing for compression; and nondecreasing for expansion, then only the nominal branch of the pI is possible.5

The above discussion leads to the following result:

Proposition 2.If hA(V) is is a non-decreasing differentiable function of V or the volume trajectory is restricted to be monotonic, the Pareto optimal trajectory must be an ApIA trajectory consisting of two adiabatic (A) segments sandwiching a pI segment.

### Step 2: Optimizing the ApIA Trajectory.

We assume that either $hA′(V)≥0$ or the volume trajectory is restricted to be monotonic so that Proposition 2 holds. In this step, the ApIA trajectory is optimized to minimize the process time $tc/e$ in Eq. (8) for a prescribed Win in Eq. (3) or Wout in Eq. (4). For compression, the initial $(P0,V0,T0)$ and the final $Pc=rP0$ are given. For expansion, the initial $(rP0,Vs,T0)$ and the final pressure P0 are given.

Each ApIA trajectory is uniquely specified by the volume Va at which the first A transitions to the pI, and the volume Vb at which the pI transitions to the final A (Fig. 6). It can also be specified by the Lagrange multiplier λ in Eq. (31) (implicitly defined by Va), and the final temperature Tc (for compression) or Te (for expansion). Let Ta and Tb be the temperatures at Va and Vb.

For compression, the pI segment is characterized by
$TpI(V)=T0+T0λ·hA(V)$
(32)
The input work Win is
$WinnR=(Ta−T0)+(Tc−Tb)γ−1−∫VaVbTpI(V)VdV+rP0nR(Vc−V0r)−P0V0nR(1−1r)=γ(Tc−T0)γ−1−Tb−Taγ−1+T0ln(VaVb)−T0λ∫VaVbdVVhA(V)−T0(1−1r)$
(33)
From Eq. (1), along a pseudo-isothermal
$−nR(TpI′γ−1+TpIV)︸G(λ,V)dV=hA(V)(TpI(V)−T0)dt$
where $TpI′=∂TpI/∂V$ so that the process time tc is
$tcnR=−∫VaVbG(λ,V)hA(V)(TpI(V)−T0)dV=−∫VaVbdVVhA(V)−T0λ∫VaVbdVVhA(V)+12(γ−1)[1hA(Va)−1hA(Vb)]$
(34)
As before, the adiabatic stages are instantaneous and do not contribute to the compression time. In the earlier equations $λ,Ta,Tb,Tc$ are related to Va and Vb by
$Ta=(V0Va)1γT0$
(35)

$λ=T0hA(Va)(Ta−T0)2$
(36)

$Tb=T0+hA(Va)hA(Vb)(Ta−T0)$
(37)

$Tc=(VbV0rP0)11+γTbγ1+γ$
(38)
Using the same technique as in the constant hA case to equate the Lagrange multiplier while constraining Win (or tc) and optimizing tc (or Win), the optimality condition is obtained as (using Va and Vb as independent variables)
$∂tc∂Va∂Win∂Vb=∂tc∂Vb∂Win∂Va$
(39)

Equation (39) and the constraint $tc=tc*$ (or $Win=Win*$), together with the dependent variables in Eqs. (35)(38), form a system of two equations and two unknowns which may be solved for the optimal Va and Vb. A combination of symbolic and numerical analysis can be used (Fig. 6). The Pareto frontier can be generated by evaluating Win and t for all solutions (Va, Vb) that satisfy Eq. (39).

The expansion case is similar, with the pseudo-isothermal given by
$TpI(V)=T0−T0λ·hA(V)$
(40)
The output work Wout(4) and process time te given by
$Wout=γ(T0−Te)γ−1−Ta−Tbγ−1+T0ln(VbVa)−T0λ∫VaVbdVVhA(V)−T0(1−1r)$
(41)

$tenR=−∫VaVbdVVhA(V)+T0λ∫VaVbdVVhA(V)+12(γ−1)[1hA(Va)−1hA(Vb)]$
(42)
The optimality condition is
$∂te∂Va∂Wout∂Vb=∂te∂Vb∂Wout∂Va$
(43)

Theorem 2. Let the product of the heat transfer coefficient and heat transfer area in Eq.(2)be a differentiable function hA(V) of V. Suppose that either hA(V) is monotonically increasing or the volume trajectory is restricted to be monotonic.

Compression.The Pareto optimal volume trajectory that compresses a gas from (P0, T0) to rP0 with respect to the input work Winin Eq.(3)and process time tc in Eq.(8)consists of an initial adiabatic portion, followed by a pseudo-isothermal portion given by Eq.(31), and ending with a final adiabatic portion. The transition points and the choice of the pseudo-isothermal curve satisfy Eq.(39).

Expansion.The Pareto optimal volume trajectory that expands a gas from$(rP,T0)$to P0 with respect to the output work Woutin Eq.(4)and process time te in Eq.(8)consists of an initial adiabatic portion, followed by a pseudo-isothermal portion given by Eq.(40), and ending with a final adiabatic portion. The transition points and the choice of the pseudo-isothermal curve satisfy Eq.(43).

Following the same argument as in Remark 2, the trajectories given in Theorem 2 are also optimal with respect to efficiencies (6) and powers (7).

## Case Study

In this case study, we consider a CAES application for a wind turbine. A simplified system is shown in Fig. 7 in which a liquid piston C/E is connected to the mechanical shaft of the wind turbine-electric generator.

The chamber of the C/E is a 12 m3 cylindrical drum, with an aspect ratio of unity. To increase the heat transfer area, it is filled uniformly with a perfectly conducting metallic wire mesh bonded to the isothermal chamber walls. Excess wind energy is used to power the water pump/motor that pumps water to fill the chamber, compressing the air above the water piston. To regenerate energy, compressed air is expanded causing the water piston to retreat and to motor the pump/motor to power the generator. As the water fills the chamber, the porous mesh becomes submerged and the surface area in contact with the air is reduced. The heat transfer coefficient h is assumed to be 100 W/m2 K (a constant). Therefore, the hA product increases with air volume V affinely as
$hA(V)=h[4V(1−ϵ)dϵ+4VϵD+πD22]$
(44)

where $ϵ=99.5%$ is the mesh porosity, m is the diameter of a strand of mesh, D = 2.48 m is the chamber diameter, and V is the instantaneous air volume. The temperature of the wire mesh is assumed to be constant at $T0=298 K$. The pressure compression ratio is r = 350, and the nominal power of the CAES system is .

The optimal tradeoff between compression efficiency (6) and storage power (7) using optimized ApIA trajectories is shown in Fig. 8. As comparisons, tradeoffs using suboptimal (since hA is not a constant) AIA trajectories, sinusoidal and linear trajectories are also plotted. Sinusoidal and linear profiles are included as they are commonly generated using a reciprocating crank-slider and a constant speed piston, respectively. For each type of trajectory, a tradeoff exists in that efficiency decreases as power increases. At any efficiency, the optimal ApIA trajectory has higher power than any other trajectories. For example, at 90% efficiency, the optimal ApIA trajectory is 60% more powerful than the AIA solution, 500% more powerful than sinusoidal compression, and more than 1500% more powerful than linear compression. Correspondingly, for a given power requirement, the chamber of the compressor that uses the optimal trajectory can be five times more compact than the one that uses a sinusoidal profile. At the nominal power of 1 MW, the ApIA trajectory achieves an efficiency of 80.3% whereas the efficiencies of all other trajectories are in the range of 60–65%.

Figure 9 compares the various types of volume trajectories normalized by the total process time for two different efficiencies: 90% and 60%. As expected, the suboptimal AIA and the optimal ApIA both have an instantaneous stage, a slow stage, and a final instantaneous stage. The ApIA's have shorter adiabatic portions than AIAs. Also, higher efficiency trajectories have shorter adiabatic compression stages.

The expansion efficiency versus power output tradeoff is shown in Fig. 10. Similar to compression, the optimal ApIA has the highest efficiency for a given power and the highest power for a given efficiency.

As an example, the transition temperatures and metrics for the optimal ApIA trajectory at the nominal 1 MW power are shown in Table 2.

## Discussion

It has been assumed that the adiabatic portion of the compression/expansion process takes zero time. In real situations, there are physical limitations to the compression and expansion rates so truly adiabatic processes are not possible. This effect is investigated for the case when hA is a constant, i.e., the optimal trajectories are AIA.

Let the compression rate be limited by $V̇max$ and define p to be the ratio of $V̇max$ to the maximum compression rate during the isothermal sections. Since $V̇max$ is finite, finite time is needed to traverse volume change during the adiabatic sections, decreasing power. Efficiency is however conservatively assumed to be unchanged.

The effect of p on the efficiency-power relationship for the constant-hA case is shown in Fig. 11. As expected, with finite p, the power is reduced from the case when the adiabatic take no time, with greater effect at low efficiency, high power situation. However, even at p = 4, the difference is hardly noticeable whenever efficiency is greater than 80%.

Another key assumption made in this paper is that hA is a constant or only a function of gas volume. While the heat transfer area being volume dependent is accurate, the heat transfer coefficient is a function of other factors such as speed, density, temperature, viscosity, and conductivity. Some correlations for different porous media as heat exchangers can be found in Refs. [10], [22], and [23]. To accommodate such general cases as well as to take into consideration the physical limitations in compression/expansion speeds, the optimal efficiency-power tradeoff problem can be solved numerically either using generic nonlinear optimization algorithms (after parameterizing the trajectories) [14,16]; or using dynamic programming [17]. Interestingly, even with complex heat transfer correlation and limits in compression/expansion rates, the optimal solutions also consist of maximum rates at the beginning and the end, and relatively slow rate in between. These are similar to the AIA or ApIA trajectories that are derived for the simplified case in this paper.

To implement the optimized trajectories using the liquid piston compressor/expander concept, only the displacement of liquid pump/motor needs to be adjusted in real-time to control flow rate. This operation is not energy intensive. One issue, however, is that optimal trajectories typically require large flow rates and hence larger variable displacement pump/motors. This drawback can be mitigated with a combined solid piston and liquid piston approach [24].

The benefits of applying optimized compression trajectories have been validated experimentally in Refs. [25] and [26] for the low pressure (10 bar) and the high pressure (200 bar) settings, respectively. Optimal trajectories were shown to double the power densities for the same efficiency or to increase efficiency by 4–5% for the same power. These gains are consistent with the expectations based on the limitations in maximum flow rates and the heat transfer capabilities under the test conditions.

The results for the air volume dependent hA(V), $hA′≥0$ case in Theorem 2 is a direct extension to the constant hA case in Theorem 1. However, for the generic case where $hA′$ can be negative, the result is weaker, since the volume trajectories are restricted to be monotonic to ensure that ApIA is optimal. The restriction is needed because of the possibility that $(T−T0)>0$ and $(T−T0)<0$ can both be feasible solutions when $∂hA(V)/∂V≠0$ and we have not ruled out that the optimal trajectories can involve switching between these two branches. Our conjecture is that ApIA are indeed optimal even without the restriction that trajectory be monotonic. This, however, has not been proved. In any case, it is expected that in practical situations, $hA′≥0$ is a reasonable assumption.

## Conclusions

In this paper, Pareto optimal trajectories for compressing or expanding gas that are Pareto optimal with respect to efficiency and power are obtained for the cases that hA, the product of heat transfer coefficient and heat transfer area, is a constant or only a function of volume. The optimal solutions consist of adiabatic steps sandwiching either an isothermal or a pseudo-isothermal step. Analytic solutions for these solutions are obtained. A case study motivated by compressed air energy storage application for wind turbine shows that the optimal solutions can increase power density 5–15 times over ad hoc trajectories without sacrificing efficiency. These solutions can be extended for more complex heat transfer situation with numerical based solutions, but the analytical solutions for these idealized cases offer important insights.

## Funding Data

• Institute on the Environment, University of Minnesota (Project No. RS-0027-11).

• National Science Foundation (Grant No. EFRI-1038294).

• Center for Compact and Efficient Fluid Power, a National Science Foundation Engineering Research Center (Grant No. EEC-0540834).

4

For a standalone compressor/expander not in an open accumulator, $Win$ and $Wout$ would include additional term $(Pc−P0)Vs=nRT0(1−1/r)$. This has the effect of increasing the efficiency and power values. However, since the extra term is a constant, it does not affect the optimization process.

5

As adiabatic transition to the non-nominal branch must involve a reversal of volume change direction. For example, in a compression process to reach the branch with $(T−T0)<0$ either from an initial temperature of T0 or from a branch with $(T−T0)>0$, the transition adiabatic segment must increase volume to reduce temperature. Similar argument applies for compression.

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