Subsea Long-Duration Energy Storage for Integration with Offshore Wind Farms

Abstract

Long-duration energy storage systems are becoming a vital means for decarbonizing the global economy. However, with floating wind farms being commissioned farther offshore, the need to co-locate energy storage with the energy harnessing units is becoming more essential. This paper presents a transient thermal analysis of the charging process of a subsea open-cycle hydro-pneumatic energy storage system. The proposed system is designed for integration with floating wind turbines in deep water sites. Situating the system subsea presents unique opportunities for integration with offshore wind plants through the exploitation of well-known subsea pipeline technology and the surrounding seawater environment, which acts as a natural heat sink/source. The results obtained from numerical modeling in Python© Version 3.7.4 present the variation in various operating parameters with time. The outcomes reveal that the proposed system is able to achieve a work ratio and an energy storage capacity ratio of up to 0.80 and 0.95, respectively. Furthermore, the proposed open-cycle system is predicted to boost the energy storage density by a factor ranging between 2.00 and 8.10 when compared to the energy storage density of conventional closed-cycle units. Namely, the energy storage density of the long-duration energy storage can reach up to 16.20 kWh/m³ when operated in an open-cycle configuration.

1. Introduction

The unprecedentedly growing share of renewables in the electricity supply calls for augmented flexibility in the energy systems. Indeed, as more floating offshore wind turbine (FOWT) plants [1,2,3] go online, the need for long-duration energy storage (LDES) technologies that sustain durations longer than typically 6 h is becoming more urgent.

LDES can be achieved via mechanical [4,5,6], thermal [7], and electro-chemical means [8]. Hydro-pneumatic energy storage (HPES) solutions have been considered as a preferred form of compressed air energy storage (CAES) for offshore deployment due to the ability of achieving quasi-isothermal conditions [4,5]. Isothermal-based approaches for CAES avoid the need for thermal energy storage and hence simplify the deployment of such technologies in the tough marine environment. However, the limited volumetric energy storage density (ESD) of conventional HPES units has been a major limiting factor in the development and uptake of such systems.

Conventional HPES systems operate in a closed-cycle (C-HPES) configuration whereby excess renewable energy (RE) drives a water pump to compress a gas via a liquid-piston (LP) mechanism. The gas (i.e., air) and liquid are contained within the same chamber, and thus, the compression and storage of the air during charging also occur within the same vessel. Therefore, with the liquid occupying a substantial volumetric capacity of the storage vessel, the amount of energy that can be stored is limited, typically to 2 kWh/m³ for a peak working pressure of 200 bar [4,9,10].

In an effort to overcome the limitations of C-HPES units and increase the ESD, research and development is investigating alternative operating methods. One effective way to increase the ESD is to operate in an open-cycle configuration (O-HPES). The alternative O-HPES technique addresses the issues of C-HPES by being designed to start operation from atmospheric pressure, thus increasing the operating pressure ratios. Furthermore, instead of one vessel containing both compressed air and a liquid, the O-HPES functions via a series of vessels acting as liquid-piston air compressors (LPACs) with the compressed air then stored in a dedicated air receiver (AR). The concept of operation is detailed further in Section 2.

With the acknowledgement that open-cycle configurations can significantly advance the storage capacities of C-HPES systems, a number of studies [10,11,12] have indeed focused on modeling the performance of O-HPES systems. In terms of augmentation factors, Chen et al. [10] reported that the ESD of the HPES system operating at a peak pressure of 100 bar was doubled when changing from the closed-cycle to the open-cycle configuration. The literature available, however, proves to be very limited in providing sufficient insight on utility-scale systems required to sustain the intermittent load of multi-Megawatt FOWTs. The limited amount of published literature [10,11,12] has only considered the operation of O-HPES systems with LPACs in the form of cylinders having lengths and diameters in the range of mm and exposed to the open atmosphere, with pressure ratios limited to a maximum of 100 [10]. Indeed, despite the existence of various concept designs of offshore CAES [13,14,15] and C-HPES [16] solutions, at the time of writing, no studies investigating subsea, large-scale O-HPES systems could be found in the open literature. There presently exists a clear knowledge gap related to the transient thermal behavior of large-scale LPACs operating in a subsea environment.

This paper aims to address the present-day knowledge gap pertaining to the thermal performance of large-scale O-HPES systems operating in a subsea environment. The focus of the paper is only on the charging cycle of the storage system. Deploying the O-HPES unit subsea provides multiple logistical benefits, including reduced spatial demands for land, efficient utilization of the offshore space, and better behind-the meter flexibility for wind farm operators in dispatching energy to the grid. By investigating the transient thermal performance of an underwater O-HPES unit during the charging process, key performance indicators (KPIs) can be quantified and compared to values quoted in the literature for conventional CAES and/or C-HPES systems [4,9,10]. Consequently, any added benefits or challenges pertaining to the functionality of the system in an oceanic setting can be clearly identified.

The proposed underwater O-HPES system is described in Section 2, followed by the background theory related to thermal behavior in Section 3. Section 4 describes the numerical model developed and the validation procedure conducted to ensure the reliability of the outcomes. The results for time series simulations are presented in Section 5. The salient findings from this study, also including the outcomes from a parametric investigation, are discussed in Section 6, together with recommendations for further research.

2. Conceptual Model

Figure 1 shows the conceptual design of the proposed subsea O-HPES system being integrated within a FOWT semi-submersible platform. A pair of LPACs is assumed to be suspended from the platform at a vertical distance below mean seawater level (MSL), $Z_{lp}$. Apart from reducing the seabed preparation and footprint that would be required for installation, the LPACs shall further provide added stability to the FOWT by lowering the center of gravity of the overall system, as already proven by similar existing concepts [17].

Initially, only valve 1 is opened while the rest of the valves are closed. To start charging the system, excess RE drives a pump-turbine (P-T) in pump mode. The latter can be either situated subsea as represented in Figure 1 or else integrated within the platform itself [4]. The cycle initiates with the suction of atmospheric air into LPAC A through the atmospheric air breather and valve 1 while pumping water from LPAC A into LPAC B. The process causes LPAC B to first enter a compression phase (CP) and eventually inject pressurized air through check-valve 4 into an AR situated $Z_{ar}$ meters below MSL. When LPAC B is full of water, the operation is reversed by closing valve 1 and opening valve 2. The P-T empties LPAC B, filling LPAC A with water, allowing the latter to compress the air, initially at atmospheric pressure, and inject it through check valve 3 into the AR. The injection of the compressed air from the LPAC to the AR is known as the release and storage phase (RSP). Thus, both LPACs operate at π radians out of phase from one another. The whole cycle repeats as many times as required, achieving continuous energy storage until the pressure in the AR reaches the maximum allowable working pressure. As the pressure in the AR increases gradually with time during the charging process, the pressure at which the LPACs have to inject air into the AR also increases. For the discharge cycle, in which the O-HPES system generates energy, the operation is reversed by controlling valves 3 and 4 to release air from the AR into the LPACs and create two oscillating LPs with a phase shift of π radians. This, in turn, provides a continuous supply of pressurized water to drive the P-T in turbine mode while returning the expanding air back into the atmosphere. However, the discharging process is not modeled in the present study.

As depicted in Figure 1, both the LPACs and the AR can take the form of steel (grade X70) pipelines. Steel is recyclable, and pipeline sections are the cheapest form of steel structure available that can be materialized by adopting the approach of repurposing offshore pipelines for energy storage (ROPES) [18]. The ROPES technique allows for existing offshore pipelines approaching the end of life to be repurposed for energy storage applications. For the scope of the current research, it is assumed that a rigid umbilical line (UL) connects each LPAC to the AR. Air flows from the former to the latter upon the opening of the check- (or release) valves exhibited in Figure 1. Note that both the LPACs and the AR are assumed to be hemispherical ended vessels (refer to Figure 2 and Figure 3).

3. Proposed Mathematical Model for Simulating Thermal Performance

The transient thermodynamic problem for the subsea O-HPES system was solved through the application of the mass and energy conservation principles to the relevant vessels and applying the pertinent boundary conditions at different stages of operation. All parameter units in the forthcoming equations are given in SI units. Furthermore, all pressure values are considered to be in absolute (not gauge) value. Moreover, all parameters with the subscript lp are attributed to the LPAC(s), whereas the subscript ar denotes the properties pertaining to the air in the AR. Similarly, the subscript ul signifies the air conditions along the UL connecting the active LPAC and the AR.

3.1. Mass Conservation

The mass conservation law was applied across the LPACs and the AR. The generic rate of change of the mass of air m in terms of the fluid pressure p, volume V, and temperature T change rates was solved via Equation (1), where R is the specific gas constant and t is time:

$${\displaystyle \frac{d}{dt}}\left(m\right)={\displaystyle \frac{pV}{RT}}\left[\left({\displaystyle \frac{1}{p}}\right){\displaystyle \frac{dp}{dt}}+\left({\displaystyle \frac{1}{V}}\right){\displaystyle \frac{dV}{dt}}-\left({\displaystyle \frac{1}{T}}\right){\displaystyle \frac{dT}{dt}}\right]$$

(1)

Until the active LPAC reaches the pressure of the AR, the former operates in a closed configuration, with the LPAC and AR treated as two separate control volumes (CVs). With a closed check-valve (3 or 4) during the CP, no mass of air flows across any of the CV boundaries. Therefore, Equation (1) equates to zero for both the active LPAC and the AR. Note also that the second term on the right-hand side (RHS) of Equation (1) drops for the AR since the geometry of said vessel is fixed and the air will always occupy the same amount of volume.

For the RSP, as the check-valve (3 or 4) opens, the process of operation changes from a non-flow to a flow process. Consequently, Equation (1) was solved with ${\displaystyle \frac{d}{dt}}\left(m\right)$ as an unknown for both the active LPAC and AR. However, by the principle of mass conservation itself, the mass of air leaving the active LPAC must equate to the mass of air entering the AR, which is also equivalent to the air mass flowrate through the UL. Therefore, Equation (2) was formulated linking all the structural bodies (i.e., active LPAC, AR, and UL), where the negative sign in front of the term on the left-hand side (LHS) implies air mass flow out of the CV (i.e., LPAC).

$$-{\displaystyle \frac{d}{dt}}{\left(m\right)}_{lp}={\displaystyle \frac{d}{dt}}{\left(m\right)}_{ar}={\displaystyle \frac{d}{dt}}{\left(m\right)}_{ul}$$

(2)

3.2. Energy Conservation

The conservation of energy principle expressed in a form that is applicable to any CV undergoing any process is given by the following:

$$\left({\displaystyle \frac{d{Q}_{in}}{dt}}+{\displaystyle \frac{d{W}_{in}}{dt}}+\sum {\displaystyle \frac{d{E}_{m,in}}{dt}}\right)-\left({\displaystyle \frac{d{Q}_{out}}{dt}}+{\displaystyle \frac{d{W}_{out}}{dt}}+\sum {\displaystyle \frac{d{E}_{m,out}}{dt}}\right)=\Delta {\displaystyle \frac{d{E}_{sys}}{dt}}$$

(3)

In Equation (3), Q represents the thermal energy, W denotes the work done, E_m denotes the energy transport, and $E_{sys}$ is the total net energy within the system. The subscripts in and out represent the energy into and out of the system, respectively. Thus, Equation (3) gives the rate of change of net energy of the system as the difference between the total energy inputs and outputs. Similar to the mass conservation in Equation (1), Equation (3) was applied for the LPACs and AR in both the CP and RSP. For the former phase, the third and sixth terms on the LHS of Equation (3) are always nullified due to the non-flow process. Furthermore, the fifth term always drops for the AR due to a constant volume process.

The net heat transfer (i.e., ${\displaystyle \frac{d{Q}_{out}}{dt}}-{\displaystyle \frac{d{Q}_{in}}{dt}}$) between the air undergoing compression and the surrounding water was computed by considering all the heat transfer interfaces (I1, I2, and I3 for the LPACs and I4, I5, and I6 for the AR) in contact with air in both the LPACs and the AR, as portrayed in Figure 2 and Figure 3. The corresponding thermal circuits were solved via the following equations:

$$\Delta {\left({\displaystyle \frac{dQ}{dt}}\right)}_{lp}={\left({\displaystyle \frac{d{Q}_{out}}{dt}}-{\displaystyle \frac{d{Q}_{in}}{dt}}\right)}_{lp}=q_1+q_2+q_3$$

(4)

$$\Delta {\left({\displaystyle \frac{dQ}{dt}}\right)}_{ar}={\left({\displaystyle \frac{d{Q}_{out}}{dt}}-{\displaystyle \frac{d{Q}_{in}}{dt}}\right)}_{ar}=q_4+q_5+q_6$$

(5)

In Equations (4) and (5), $q_1$, $q_2$, $q_4,$ $q_5$, and $q_6$ represent the heat transfer rate in Watts, accounting for (i) an internal convective resistance between the compressed air inside the vessel (LPAC or AR) and the internal steel surface of the same vessel; (ii) a conductive resistance across the LPAC/AR wall, accounting also for the thermal capacitance of steel; and (iii) an external convective resistance between the external steel surface of the LPAC/AR and the surrounding seawater environment. Term $q_3$ represents a heat transfer rate across the internal water–air interface in the LPAC. Note that, for the LPACs, $q_1$ was treated separately from $q_2$ since the heat transfer rate from a cylindrical interface is different from the heat transfer rate from a hemispherical shell. The same statement applies for $q_5$ and $q_6$ for the AR.

The external and internal heat transfer coefficients (HTCs) $h$, as marked in Figure 2 and Figure 3, were derived from existing empirical correlations for the Nusselt ($Nu$) number listed in

Table 1. Correlations utilized for the derivation of the different HTCs for the LPACs and AR.

Location	Type of Convection	HTC	Correlation/Fixed Value	Source
External	Natural	$h_{o,1}$$\mathrm{and} h_{o,4}$	$Nu={\left\{0.60+{\displaystyle \frac{0.387R{a}^{\frac{1}{6}}}{{\left[1+{\left(\frac{0.559}{Pr}\right)}^{\frac{9}{16}}\right]}^{\frac{8}{27}}}}\right\}}^2$	Churchill-Chu [22]
External	Forced	$h_{o,1}$$\mathrm{and} h_{o,4}$	$Nu=0.30+\left\{{\displaystyle \frac{0.62R{e}^{\frac{1}{2}}P{r}^{\frac{1}{3}}}{{\left[1+{\left(\frac{0.4}{Pr}\right)}^{\frac{2}{3}}\right]}^{\frac{1}{4}}}}\right\}{\left[1+{\left({\displaystyle \frac{Re}{282,000}}\right)}^{{\displaystyle \frac{5}{8}}}\right]}^{{\displaystyle \frac{4}{5}}}$	Churchill-Bernstein [23]
External	Natural	$h_{o,2}, h_{o,5}$$\mathrm{and} h_{o,6}$	$Nu=0.533R{a}^{0.25}$	Lewandoski et al. [24]
External	Forced	$h_{o,2}, h_{o,5}$$\mathrm{and} h_{o,6}$	$Nu=2+\left(0.4R{e}^{{\displaystyle \frac{1}{2}}}+0.06R{e}^{{\displaystyle \frac{2}{3}}}\right)P{r}^{0.4}{\left({\displaystyle \frac{\mu }{{\mu }_w}}\right)}^{{\displaystyle \frac{1}{4}}}$	Whitaker [19]
Internal	Natural	$h_{i,1}$$\mathrm{and} h_{i,4}$	$Nu=1.15R{a}^{0.22}$	Ludovisi-Garza [25]
Internal	Forced	$h_{i,1}$$\mathrm{and} h_{i,4}$	$Nu={\displaystyle \frac{\left(\frac{f}{8}\right)\left(Re-1000\right)Pr}{1+12.7{\left(\frac{f}{8}\right)}^{\frac{1}{2}}\left(P{r}^{\frac{2}{3}}-1\right)}}$	Gnielinski [20]
Internal	Natural	$h_{i,2}, h_{i,5}$$\mathrm{and} h_{i,6}$	$Nu=0.2357R{a}^{0.242}$	Shiina et al. [26]
Internal	Forced	$h_{i,2}, h_{i,5}$$\mathrm{and} h_{i,6}$	100 W/m2 K	[27]
Internal	Natural/Forced	$h_3$	200 W/m2 K	[11]

. The shortlisted equations in the aforementioned table were chosen following an extensive literature review for existing models for internal/external forced and free convective heat transfer. In Table 1, the parameters $Ra$, $Re$, and $Fr$ denote the Rayleigh, Reynolds, and Froude numbers of the fluid in question. The parameters $\mu$ and ${\mu }_w$ in the Whitaker formula [19] imply the dynamic viscosities of the fluid at the free-stream and solid wall temperatures. Similarly, the parameter $f$ in the Gnielinski [20] correlations symbolizes the friction factor, which can be found explicitly from the Haaland formula [21]. Further detail on every correlation may be reviewed in the relevant sources [18,19,20,21,22,23,24,25,26,27].

3.3. The Bernoulli Equation Along the UL

Delving in depth into the modeling of the UL was beyond the scope of the current work. However, a simplified representation was critical to account for the minor pressure discrepancy between the LPACs and the AR. The pressure in the AR was expected to always be slightly less than the pressure in the active LPAC due to the loss across the UL, which results from the summation of the head loss, frictional loss, and secondary losses. Consequently, the connecting pipeline was mathematically represented by the Bernoulli equation:

$$p_{lp}+{\displaystyle \frac{1}{2}}{\rho }_{lp}{U}_{lp}^2+{\rho }_{lp}g{Z}_{lp}=p_{ar}+{\displaystyle \frac{1}{2}}{\rho }_{ar}{U}_{ar}^2+{\rho }_{ar}g{Z}_{ar}+losses$$

(6)

In Equation (6), $\rho$ is the air density in the respective vessels, $U$ is the air flow velocity, $g$ is the acceleration due to gravity, and $Z$ is the vertical head with the MSL taken as the datum. The term losses is representative of frictional and secondary losses. In order to apply Equation (6), it was assumed that the air flow across the UL was steady, inviscid, and incompressible and that the UL was a rigid body with a fixed cross-sectional area.

3.4. Key Performance Indicators (KPIs)

As a key metric for the performance of the offshore O-HPES system during charging, the polytropic index n during the CP was evaluated using the following formula:

$$n={\displaystyle \frac{\log \left(\frac{p_{f,lp}}{p_{i, lp}}\right)}{\log \left(\frac{V_{i, lp}}{V_{f, lp}}\right)}}={\displaystyle \frac{\log r_p}{\log r_v}}$$

(7)

Equation (7) was applied to every LP stroke individually, where $p_{i,lp}$ (equivalent to atmospheric pressure) and $p_{f,lp}$ denote the initial and final pressure of air in the active LPAC for the given stroke, respectively. Similarly, $V_{i, lp}$ and $V_{f, lp}$ signify the air volume in the active LPAC at the initial and final instances of the given stroke. Additionally, $r_p$ and $r_v$ give the pressure ratio and volumetric ratio of every compression, respectively. An $n$ value of unity indicates ideal isothermal conditions. Any deviations thereof thus give an indication of how far the compression stroke is from the ideal scenario. To account for the worst conditions in the parametric study presented in Section 5.2, the value of $n_{max}$ was monitored. The parameter $n_{max}$ gives the maximum value of $n$ across all strokes during a charging period.

In conjunction with the value of $n_{max}$, the maximum air temperature $T_{max}$ reached at the end of the CP in the LPAC, prior to the RSP, is indicative of the system thermal management. The closer $T_{max}$ remains to the initial working temperature value, the more isothermal the process is. The value of $T_{max}$ was derived from the iterative, numerical solution when solving for Equations (1) and (2) and subsequently from the plots generated as presented in Section 5.1 (refer to Figure 10).

A second KPI considered in the present study was the work ratio WR given in Equation (8). The WR can be considered as representing the thermal efficiency of the CP and RSP. In other words, it is defined as the ratio of ideal work done for the isothermal process to the actual work needed to increase the pressure of the AR from the pre-charge value to the maximum pressure value.

$$WR={\displaystyle \frac{W_{ideal}}{W_{real}}}$$

(8)

where

$$W_{ideal}=\left[p_{f,ar}{V}_{ar}ln\left({\displaystyle \frac{p_{f,ar}}{p_{atm}}}\right)-p_h{V}_{ar}\left({\displaystyle \frac{p_{f,ar}}{p_{atm}}}-1\right)\right]-\left[p_{p,ar}{V}_{ar}ln\left({\displaystyle \frac{p_{p,ar}}{p_{atm}}}\right)-p_h{V}_{ar}\left({\displaystyle \frac{p_{p,ar}}{p_{atm}}}-1\right)\right]$$

(9)

and

$$W_{real}={{\displaystyle \int }}_{V_{0,lp}}^{V_{f,lp}}pdV-p_h{V}_{f,lp}\left(r_p-1\right)$$

(10)

In Equation (9), $p_{f,ar}$ and $p_{p,ar}$ denote the final and pre-charge pressures of the AR. The term $p_{atm}$ signifies the atmospheric pressure that sets the initial working conditions for the LPACs. Moreover, the term $p_h$ symbolizes the hydrostatic pressure that increases in direct proportion with the sea depth (i.e., $p_h={\rho }_{sw}g{Z}_{lp}$. For $Z_{lp}$, refer to Figure 1). Indeed, the second term in every square bracket on the RHS of Equation (9) denotes the positive work done by the external hydrostatic pressure, which needs to be considered when dealing with subsea O-HPES systems, and which, in effect, reduces the exploitable energy storage capacity (ESC). The term $V_{ar}$ implies the volumetric capacity of the AR, which is constant throughout. The actual work done $W_{real}$ equates to the integral of the real work done during all compression strokes, where $p$ in Equation (10) is the instantaneous pressure.

Similar to the WR, the ESC ratio (ESCR) is indicative of how much energy is actually stored as compared to the ideal ESC rating (in kWh) of the system, with the latter attained under theoretical isothermal conditions:

$$\mathrm{ESCR}={\displaystyle \frac{ES{C}_{real}}{ES{C}_{ideal}}}$$

(11)

where

$$\begin{gathered} ES{C}_{real}=m_{i,ar}R{T}_{i,ar}ln\left({\displaystyle \frac{p{\prime }_{f,ar}}{p_{i, ar}}}\right)+\left(m_{f,ar}-m_{i,ar}\right)R{T}_{i,ar}ln\left({\displaystyle \frac{p{\prime }_{f,ar}}{p_{atm}}}\right) \\ -\left(p_h{V}_{ar}\left({\displaystyle \frac{p{\prime }_{f,ar}}{p_{atm}}}-1\right)-p_h{V}_{ar}\left({\displaystyle \frac{p_{p,ar}}{p_{atm}}}-1\right)\right) \end{gathered}$$

(12)

and

$$ES{C}_{ideal}={\displaystyle \frac{W_{ideal}}{3600\times 1000}}$$

(13)

In Equation (12), $m_{i,ar}$ is the initial mass of air entrapped in the AR at the pre-charge pressure $p_{i, ar}$ and $m_{f,ar}$ is the total mass in the AR at the end of the cyclic charging process. The parameter $R$ is the specific gas constant, and $T_{i,ar}$ is the initial air temperature in the AR assumed to be in thermal equilibrium with the surrounding seawater temperature, which, in turn, is assumed constant. In essence, the first term in Equation (12) is the work required to pre-charge the AR with $m_{i,ar}$ at $p_{i, ar}$. The second term denotes the work done to charge the AR to $p_{f,ar}$ with a mass of air that has been compressed from atmospheric pressure $p_{atm}$. The third term accounts for the work done by the hydrostatic pressure throughout the whole process. Note, however, that in Equation (12), $p{\prime }_{f,ar}$ rather than $p_{f,ar}$ is considered, accounting for the energy lost as the air in the AR cools down from the final state at $p_{f,ar}$ and $T_{f,ar}$ to reaching thermal equilibrium again with the surrounding seawater at $p{\prime }_{f,ar}$ and $T{\prime }_{f,ar}=T_{i,ar}$. The value of $p{\prime }_{f,ar}$ is obtained assuming a constant volume process and applying Gay-Lussac’s law.

The ideal and real ESDs of the system in kWh/m³ could be further computed by dividing the $ES{C}_{ideal}$ and $ES{C}_{real}$ with the volumetric capacity of (i) the AR alone or (ii) the AR and the LPACs, as given in Equations (14) through (17).

$$ES{D}_{ideal,1}={\displaystyle \frac{ES{C}_{ideal}}{V_{ar}}}$$

(14)

$$ES{D}_{real,1}={\displaystyle \frac{ES{C}_{real}}{V_{ar}}}$$

(15)

$$ES{D}_{ideal,2}={\displaystyle \frac{ES{C}_{ideal}}{V_{ar}+2V_{lp} }}$$

(16)

$$ES{D}_{real,2}={\displaystyle \frac{ES{C}_{real}}{V_{ar}+2V_{lp} }}$$

(17)

Consequently, the ESD ratio (ESDR) was obtained from Equation (18), which essentially equates to the ESCR in Equation (11).

$$\mathrm{ESDR}={\displaystyle \frac{ES{D}_{real,1}}{ES{D}_{ideal,1}}}={\displaystyle \frac{ES{D}_{real,2}}{ES{D}_{ideal,2}}}$$

(18)

4. Numerical Model

Section 4 describes how the proposed mathematical model of Section 3 was implemented into a numerical solution, solving for the transient thermal performance of the O-HPES system. A description of the program functionality and the validation of the same program are provided in Section 4.1 and Section 4.2, respectively.

4.1. Program Description

The mathematical model of Section 3 was implemented into a time-marching problem resolved via Euler’s first-order ordinary differential equation given in Equation (19), where a given property at a given time step $\left(i\right)$ was determined from the computations of the previous time step $\left(i-1\right)$. In Equation (19), $x_{\left(i\right)}$ would be a parameter on which $y_{\left(i\right)}$ is dependent, and $\Delta t$ is the time step or time difference between $t_{\left(i+1\right)}$ and $t_{\left(i\right)}$.

$$y_{\left(i+1\right)}=y_{\left(i\right)}+f\left(x_{\left(i\right)},y_{\left(i\right)}\right)\Delta t$$

(19)

The main flow of the execution of the program OpenHPES developed in Python© [28] is summarized in the flowchart of Figure 4. Following a set of user inputs (as established in Tables 3 and 4), the program enters into a dual-nested while loop process.

The outer-most while loop sets the ultimate criteria that determine whether the program will continue running or stops executing. In other words, the goal is to charge the AR to a desired final pressure $p_{f,ar}$ through a series of compression strokes (i.e., cyclic compression). Therefore, OpenHPES terminates only when the pressure in the AR $p_{ar}$ reaches the desirable maximum pressure $p_{f, ar}$ pre-defined by the user. If $p_{ar}<p_{f,ar}$, the program adopts a time-marching approach for one compression stroke of an LPAC and initiates the inner-most while loop where all the parameters during the CP and the RSP are evaluated as indicated in Figure 4.

A compression cycle is initiated assuming that atmospheric air intake has already taken place and that the air temperature inside the LPAC is in thermal equilibrium with the surrounding seawater and steel pipeline temperatures. Given the large volume of the LPACs, initially, $m_{lp}\gg 1 \mathrm{kg}$. Thus, all steps within the inner loop are executed systematically as per Figure 4. All thermodynamic and transport properties of the fluids (i.e., air and seawater) are extracted from the Python© in-built database CoolProp. The fluid properties are then utilized to carry out the transient thermal analysis. The $Nu$ number and the corresponding HTCs could thus be calculated for the different locations as stipulated in Table 1 of Section 3. Following the mass and energy balance calculations via Equations (1) and (2), the code proceeds to the next time increment. The inner-most while loop is repeated until the mass of air in the LPAC is less than 1 kg. The latter value was chosen after preliminary investigations to enhance the volumetric efficiency of the LPACs during compression while allowing some clearance to prevent water from being injected into the AR. If the condition $m_{lp}>1 \mathrm{kg}$ is unsatisfied, the program proceeds to the next compression stroke carried out by the alternate LPAC and re-performs the initial check for the pressure in the AR. If the condition $p_{ar}\ge p_{f,ar}$ is still not met, the program continues to iterate through the steps outlined in the previous text. The program execution thus mimics the cyclic charging process of the O-HPES system as a function of time.

4.2. Program Validation

Prior to extracting conclusive results, verification and validation tests were carried out. Validation procedures incorporated the comparison of outcomes from the Python© [28] model to the predictions made by Computational Fluid Dynamics (CFD). Due to the high computational effort required with CFD, a scaled model of 1:20 (relative to the dimensions listed in Figure 2) of one LPAC was modeled in Ansys^® Fluent version 2023 R2 [29]. For a like-with-like analysis, the conditions set in Fluent, as indicated in

Table 2. Geometric and operating parameters input in FLUENT and Python© for validation.

Parameter (Unit)	Value
Maximum hydraulic power—$P$ (kW)	5.50
Geometric cylindrical length—$L_{lp}$ (m)	7.55
Outer diameter—$D_{o,lp}$ (m)	0.080
Inner diameter—$D_{in,lp}$ (m)	0.075
Volumetric capacity—$V_{lp}$ (m3)	0.033
Initial air pressure—$p_{i,lp}$ (bar)	1
Maximum final operating pressure—$p_{f,lp}$ (bar)	$\approx$100
Initial air temperature—$T_i$ (K)	293.15

, were prescribed to OpenHPES in Python©. Figure 5, Figure 6 and Figure 7 compare the results from OpenHPES (i.e., Python) and CFD (i.e., Fluent), where the level of agreement is very good. Interestingly, from Figure 6, OpenHPES appears to underpredict the air temperature relative to the CFD outcome. However, considering the high uncertainty of existing convective heat transfer correlations when applied to LP mechanisms, the discrepancy of 3% between Python and CFD is considered to be relatively very small. The validation outcome thus confirms that the OpenHPES predictions are reliable.

5. Results

5.1. Default Test Run

The results were first output for what was selected as the default run with the parameter values for the LPACs, AR, and UL listed in

Table 3. Default values for input parameters attributable to the LPACs and AR.

Parameter (Unit)	LPACs	AR
Hydraulic power—$P$ (kW)	420	N/A
Geometric cylindrical length—$L$ (m)	149.44	95.04
Outer diameter—$D_o$ (m)	1.524	1.524
Inner diameter—$D_{in}$ (m)	1.420	1.430
Volumetric capacity—$V_c$ (m3)	237.69	154.53
Internal surface roughness—$\epsilon$ (m) [30]	4 × 10−5	4 × 10−5
Thermal conductivity—$k$ (W/ m K) [31]	64	64
Specific heat capacity of steel—$c_{st}$ (J/kg K) [31]	480	480
Vertical depth below MSL—$\mathrm{Z}$ (m)	10.50	200
Initial air pressure—$p_i$ (bar)	1	80
Maximum final operating pressure—$p_f$ (bar)	200	200
Design pressure—$p_d$ (bar)	220	220
Initial air temperature—$T_i$ (K)	293.15	288.15
Initial external seawater temperature—$T_{swo}$ (K)	293.15	288.15
Initial internal seawater temperature—$T_{swi}$ (K)	293.15	N/A
Initial steel temperature—$T_{st}$ (K)	293.15	288.15

and

Table 4. Default values for input parameters attributable to the UL.

Parameter (Unit)	UL
Geometric length—$L_{ul}$ (m)	190
Inner diameter of umbilical—$D_{in,ul}$ (m)	0.05
Internal surface roughness—${\epsilon }_{ul}$ (m) [30]	4 × 10−5
Secondary loss coefficient—sudden expansion—$f_e$ (-) [32]	0.998
Secondary loss coefficient—sudden contraction—$f_c$ (-) [32]	0.499
Secondary loss coefficient—valves—$f_v$ (-) [32]	2
Secondary loss coefficient—bends—$f_b$ (-) [32]	0.3
Number of valves—$N_v$ (-)	1
Number of bends—$N_b$ (-)	2

. Thus, whenever the term default setting/value is specified henceforth, it shall mean that the program inputs were prescribed at the magnitudes indicated in the aforementioned tables. Any changes to the input values incorporated for parametric investigations are indicated in Section 5.2.

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 show the time series plots simulating the operation of the subsea O-HPES at the default settings provided in Table 3 and Table 4. The outcomes incorporating both the CP and the RSP are presented in sets of three figures. The middle plots (i.e., labeled Figure X.b)) present the full time history for the whole duration of the cyclic charging. The plots on the LHS (i.e., Figures X.a)) and RHS (i.e., Figures X.c)) zoom in on the first and last hours of the charging process, respectively, to better visualize the trends in parameter variations. Furthermore, the outcomes attributed to the LPACs are plotted in blue, whereas the results representative of the conditions in the AR are plotted in brown.

Figure 8 is a square plot describing the state (i.e., closed = 0; opened = 1) of check-valves 3 and 4 (alternating), thus indicating the change from the CP to the RSP of the LPACs. It is also evident that, for the default settings, it takes 8.42 h and 74 LP strokes to fully charge the AR to 200 bar peak pressure.

Figure 9 presents the pressure variation in the alternating pair of LPACs. The pressure build-up over the whole charging cycle may be clearly observed from Figure 9b. Taking a closer look at Figure 9a,c reveals how the pressure rises incrementally with every other stroke due to the corresponding pressure rise in the AR, which is depicted in Figure 12. It is worth noting how the pressure in the LPACs still continues to increase when the check-valve is opened since the rate of compression remains higher than the rate of release of air into the AR. However, when the check-valve is opened, the rate of increase in pressure in the LPACs slows down. Furthermore, since a higher pressure is required to be reached after every other stroke, the check-valve opening is delayed by a few seconds with every cycle. For example, the first CP takes 0.067 h (4 min) to complete. Moving forward to the 74th and last stroke, the check-valve opens 0.090 h (5.50 min) after the start of the CP.

The variation in air temperature in the LPACs is presented in Figure 10, portraying a sharp deviation from the initial temperature of 293.15 K. Subsequently, the polytropic index during the CP, computed using Equation (7), deviates from the value of 1, which is achieved for ideal isothermal conditions. The index reaches a maximum value of 1.17, as presented in Figure 11. An important observation is that, initially, the CP starts as an isothermal process. Then, $n$ increases gradually due to the decreasing contact area with the compressed air as the LPACs fill up with water. Interestingly, the final value of $n$ for all strokes in the CP remains constant at 1.17, despite every consecutive stroke reaching a higher peak pressure. During the investigations, it was noted that, with a change in $p_{f,lp}$ of a given stroke, the volumetric ratio $r_v$ (refer to Equation (7) in Section 3.4) changes in a manner such that the peak value of $n$ for every stroke remains unaltered.

The pressure in the AR is observed to remain initially steady when no air is injected into the vessel and the LPACs are in the CP and draining phase (the latter is not modeled). However, when comparing the final hour (i.e., Figure 12c) to the first hour (i.e., Figure 12a), it is observed that the $p_{ar}$ drops during the CP of the active LPAC. In essence, when the AR is not receiving a flow of air, heat exchange still occurs between the stored compressed air in the AR and the subsea environment. Consequently, the fluid inside cools down as noted in Figure 13. Consequently, a pressure drop is encountered in the AR (refer to Figure 12).

It is also shown that the mass conservation principle is being complied with through the plots in Figure 14, Figure 15, Figure 16 and Figure 17. Figure 14 and Figure 15 show how, at any instance, both sets of plots are equivalent in magnitude, consistent with Equation (2). In line with the aforementioned equation, Figure 14 and Figure 15 also represent the air flowrate along the UL. Figure 16 illustrates the mass in the LPACs where during the CP, the mass of air is constant. For the time slots corresponding to an open check-valve, $m_{lp}$ drops instantly for the fluid to flow towards the AR at a flowrate as indicated in Figure 14. Additionally, Figure 17 presents the augmentation of the mass of air in the AR that rises in a staircase-like manner. The regions with a constant $m_{ar}$ correspond to the CP, whereas the rising $m_{ar}$ regions denote the RSP of the active LPAC. Performing a mass balance calculation by adding up the values in Figure 16 and Figure 17 will always results in a constant value of total air mass for a given stroke, confirming further that the mass is being conserved within the system.

Figure 18 shows the heat transfer rate from the LPACs, where initially, thermal energy is input (+ve) into the system. The total rate of heat transfer increases as the expeditious rise in the compressed air temperature (refer to Figure 10) becomes the driving parameter for thermal exchange. The total heat transfer rate adds up to an order of magnitude of approximately 250 kW. Note that, from further investigations, it was found that for the LPACs, the heat transfer rate from the hemispherical interface I2 remains less than 15% of the total heat transfer rate illustrated in Figure 18. The statement is valid for all instances during all compression strokes. The remaining 85% of the heat exchange rate initially occurs through the cylindrical interface I1, and the percentage is then reduced during the CP as the internal seawater in the LPACs provides an additional cooling effect.

The thermal transfer rate from the AR is portrayed in Figure 19. Qualitatively, the rate of heat transfer from the AR takes the same characteristic shape of the air temperature variation in the AR (compare to Figure 13), confirming that the temperature change is the main parameter driving heat exchange.

5.2. Parametric Study

Further time simulations were carried by changing a number of variables from the default values to assess the effects on the KPIs.

Table 5. Test matrix for a parametric study indicating the parameter values that varied from the default settings.

Test Case	Input Power $\mathit{P}$ (kW)	Ideal ESC of AR $\mathit{E}\mathit{S}{\mathit{C}}_{\mathit{i}\mathit{d}\mathit{e}\mathit{a}\mathit{l}}$ (kWh)	Ideal ESD 1 of System $\mathit{E}\mathit{S}{\mathit{D}}_{\mathit{i}\mathit{d}\mathit{e}\mathit{a}\mathit{l},1}$ (kWh/m3)	Ideal ESD 2 of System $\mathit{E}\mathit{S}{\mathit{D}}_{\mathit{i}\mathit{d}\mathit{e}\mathit{a}\mathit{l},2}$ (kWh/m3)	Volumetric Capacity of AR ${\mathit{V}}_{\mathit{c}, \mathit{a}\mathit{r}}$ (m3)	Outer Diameter of AR ${\mathit{D}}_{\mathit{o}, \mathit{a}\mathit{r}}$ (m)	Length of AR ${\mathit{L}}_{ \mathit{a}\mathit{r}}$ (m)	Length-to-Diameter Ratio of AR ${\mathit{L}}_{\mathit{a}\mathit{r}}/{\mathit{D}}_{\mathit{o}, \mathit{a}\mathit{r}}$ (-)
A (Default)	420	2500	16.20	3.97	154	1.52	95	62.5
B	420	2500	16.20	3.97	154	0.76	387	509.2
C	420	2500	16.20	3.97	154	2.03	53	26.1
D	105	2500	16.20	3.97	154	1.52	95	62.5
E	420	5000	16.20	6.37	309	1.52	191	125.7
F	420	10,000	16.20	9.14	618	1.52	383	252.0

presents the test matrix with the main variables, whose values were varied one at a time, from the default settings (Test A). The main variables were the length-to-diameter (L_ar/D_o,ar) ratio of the AR (Tests B and C), the input hydraulic power (Test D), and the ideal ESC or sizing of the AR (Tests E and F). Operation was maintained up to 200 bar in the AR at all instances. Any other parameters that are not indicated in Table 5 were maintained at the default values stipulated in Table 3 and Table 4. The outcomes for every test in Table 5 are provided in

Table 6. KPI values and additional outputs for the tests listed in Table 5.

Test Case	Time to Fully Charge the AR $\mathit{t}$ (hrs)	Number of LP Strokes (-)	Maximum Polytropic Index ${\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}$ (-)	Maximum Air Temperature in LPACs ${\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}$ (K)	Work Ratio WR (-)	ESCR or ESCD (-)
A (Default)	8.4	74	1.17	626.6	0.72	0.85
B	8.7	77	1.17	626.3	0.69	0.90
C	8.1	71	1.17	626.3	0.75	0.80
D	30.8	79	1.08	436.1	0.80	0.94
E	17.4	153	1.17	627.0	0.70	0.89
F	35.5	313	1.17	627.1	0.68	0.92

.

From Table 5, it is initially evident that the $ES{D}_{ideal,1}$ is by far higher than the 2 kWh/m³ [4,9,10] quoted in Section 1 for C-HPES systems. However, for a fair comparison, $ES{D}_{ideal,2}$ is more appropriate as it also caters to the volume of LPACs that are essentially also part of the system, despite not being part of the pressure containment unit. Yet, even when considering the volume of the LPACs, $ES{D}_{ideal,2}$ increases beyond 2 kWh/m³ [4,9,10] and is optimized with an increase in the ESC of the O-HPES system (or with an increase in the volumetric capacity of the AR). Considering both $ES{D}_{ideal,1}$ and $ES{D}_{ideal,2}$, it can be stated that the ESD of an O-HPES with a maximum working pressure of 200 bar is augmented by a factor of between 2.00 and 8.10 relative to a maximum ESD of 2 kWh/m³ [4,9,10] for C-HPES solutions also operating at a peak pressure of 200 bar.

Evaluating the outcomes in Table 6 leads to a number of observations. Firstly, considering Tests A to C, it can be said that, on average, it takes 8.41 h to charge an AR of a 2500 kWh ideal ESC with a pair of LPACs and an inputted constant hydraulic power of 420 kW. Furthermore, the tabulated values imply that the CP in the LPACs is independent of the geometry of the AR as the values of $n_{max}$ and $T_{max}$ are identical for the three test cases. Moreover, a trade-off can be realized between the WR and ESCR. Increasing the value of L_ar/D_o,ar is predicted to reduce the WR while simultaneously boosting the ESCR. Back-tracking the cause of the trends in the WR and ESCR can be explained as follows: A higher L_ar/D_o,ar value provides a larger surface area for heat exchange to occur between the AR and the subsea environment. Consequently, the air in a more slender AR will be able to cool down faster towards thermal equilibrium with the surrounding seawater. Simultaneously, the air temperature drop will also cause the air pressure in the AR to drop, as exhibited through the plots of Figure 12 and Figure 13 and as discussed in Section 5.1. Therefore, additional LP strokes would be required to charge the AR to the desirable 200 bar peak. More compression cycles thus incur higher losses, justifying the lower WR obtained for higher L_ar/D_o,ar values. At the same time, the AR would be occupying a larger mass of air at the final state, explaining the fact as to why the ESCR is improved with larger magnitudes of L_ar/D_o,ar. Evaluating how the different values of L_ar/D_o,ar change the thermal transfer across the AR boundaries requires a detailed investigation, which is beyond the scope of the present work but should be considered for future analyses. The variations predicted in WR and ESCR with a change in L_ar/D_o,ar are, however, minimal. For instance, when factoring L_ar/D_o,ar by up to 20 times from Test B to Test C, the WR and ESCR are observed to change by only 7 and 10%, respectively. Therefore, with the same reasoning of altering the heat transfer rates by varying the geometry of the vessel, future work should consider the geometric optimization of the LPACs where the thermodynamic compression work is performed.

Comparing Test D to the default Test A reveals how changing the input hydraulic power has a considerable effect on all KPIs. The maximum polytropic index reached by the end of the charging cycle is reduced by almost 8% from 1.17 to 1.08 when only one-fourth of the default hydraulic power value is supplied (i.e., by changing from Test A to Test D). The reduction in $n_{max}$ leads to a $T_{max}$ that is 30% lower than the 636.60 K reached during Test A, which is thus closer to the initial temperature of 293.15 K (refer to Table 3). Subsequently, the O-HPES system operated as per Test D can be said to be 11% more efficient than that for Test A, as the WR is augmented to almost 0.80. Interestingly, when varying the input hydraulic power, no clear trade-off between the WR and ESCR is observed. On the contrary, the ESCR increases with the same percentage as the WR when lowering the input power. The improvement in WR and ESCR with a reduction in $P$ stems from the fact that more time is allocated per compression stroke in the LPACs. Indeed, the total charging time increases by almost the same factor as the power reduction factor (which equates to four when reducing from 420 to 105 kW). The prolonged charging time allows for better heat exchange between the compressed air and the surrounding seawater, thus contributing to a charging process that is closer to isothermal conditions. Based on the test conditions in Table 5 and the outcomes in Table 6, it can be concluded that to achieve a WR and an ESCR higher than 0.80 and 0.95, respectively, the system requires a charging duration of at least 30 h (i.e., 1.25 days). Alternatively, the time required for every LPAC to complete a stroke in the CP should be around 15 and 22 min to allow for better heat exchange and to lead to higher values of the WR.

Tests E and F reconfirm that any geometric changes performed on the AR relative to the default Test A are trivial to the CP and the RSP in the LPACs in terms of $n_{max}$ and $T_{max}$. Interestingly, the charging time $t$ varies linearly with the $ES{C}_{ideal}$ of the AR for a fixed hydraulic power input. Again, a noticeable trade-off is observed between the WR and the ESCR. Increasing the volumetric capacity of the AR leads to a larger ESC. Yet, the process to charge to a higher ESC can be somewhat less efficient. The outcome follows the same reasoning explained earlier in terms of Tests A to C, whereby a larger number of compression strokes leads to higher inefficiencies but allows for more energy to be stored. The changes observed in the KPIs when doubling the $ES{C}_{ideal}$ of the AR add up to between 3 and 5% for both the WR and ESCR.

6. Conclusions

This article has presented a novel concept of a subsea HPES system operating in open-cycle configuration for integration with FOWTs. Through a time-marching approach, the numerical solution incorporating transient thermal effects has generated trends for the variation in the main thermodynamic parameters with time. The following have been established:

i.

The LDES system will take over 8 h to charge a 2500 kWh subsea AR to 200 bar peak pressure. This finding is subject to the test conditions including the geometric and operating conditions of the LPACs;

ii.

Notwithstanding the use of slender LPACs located subsea, the cyclic compression deviates substantially from the ideal, isothermal conditions. The maximum polytropic index was predicted to reach a value of 1.17, leading to compressed air temperatures in the LPACs that exceed 600 K.

An additional parametric investigation has also revealed the following:

i.

The ESD of an O-HPES system increases beyond the 2 kWh/m³ quoted in the literature [4,9,10] for C-HPES systems and is improved further with an increase in the volumetric capacity of the AR;

ii.

The augmentation factor for the ESD of an O-HPES system with respect to a C-HPES system falls in the range of 2.00 and 8.10. The actual augmentation value of a given system depends on whether the calculation of the ESD for the O-HPES system accounts for the volume of the AR only or the volume of the AR and LPACs;

iii.

The ESD of an O-HPES system was estimated to reach 16.20 kWh/m³ when based on the volume of the AR only. When the volume of the LPACs was also considered, the ESD reached was found to drop to approximately 4–9 kWh/m³, depending on the volumetric capacity of the AR;

iv.

When changing the geometry of the AR, a trade-off between the WR and ESCR occurs. That is, if a change in geometry causes the WR to increase, the ESCR decreases, and vice-versa;

v.

The variations in WR and ESCR with a change in L_ar/D_o,ar up to a factor of 20 were found to not exceed 10%;

vi.

Lowering the input hydraulic power for a given LPAC geometry is a key aspect to simultaneously improve both the WR and ESCR of the O-HPES system;

vii.

Subject to the test conditions considered in this study, a WR higher than 0.80 and an ESCR above 0.95 are technically achievable when the O-HPES is allowed to charge slowly over a period of 30 h (i.e., 1.25 days).

This paper has provided further insight on the thermal performance constraints of subsea O-HPES systems catering to utility-scale applications and designed to sustain the storage of energy for long durations. Future work should incorporate experimental campaigns on scaled prototypes in water basins to further validate the numerical outcomes presented in this article. The physical experiments shall also assist in gathering sufficient data to develop $Nu$ correlations specific to subsea LPACs. A design optimization exercise focusing on the geometry of the LPACs shall also be considered to seek alternative ways to attain a WR higher than 0.80. The concept introduced in this article shall be investigated and exploited further as it can be a gamechanger in the offshore and storage industry. In addition to the additional ballasting to the FOWTs, the use of subsea pipelines for the AR can prove to be beneficial for both energy storage and energy transmission between offshore substations.