A Numerical Model Comparison of the Energy Conversion Process for an Offshore Hydro-Pneumatic Energy Storage System

Abstract

Energy storage is essential if net zero emissions are to be achieved. In fact, energy storage is a leading solution for reducing curtailment in an energy system that relies heavily on intermittent renewables. This paper presents a comparison between two numerical models which simulate the energy conversion unit performance of a hydro-pneumatic energy storage system. Numerical modelling is performed in Python^TM (Alpha Model) and Mathworks^® Simulink^® and Simscape^TM (Beta Model). The modelling aims to compare the time-series predictions for the simplified model (Alpha Model) with the more physically representative model (Beta Model). The Alpha Model provides a quasi-steady-state solution, while the Beta Model accounts for machinery inertias and friction within hydraulic flow circuits. Results show that the energy conversion performance simulations between the two models compare well, with a notable difference during system start-up due to the inclusion of transients in the Beta Model. Given its simplicity, the Alpha Model has high computational efficiency, while the Beta Model requires more computational time due to its complexity. This study showed that, despite its simplicity, the Alpha Model is able to generate results that are very similar to those from the Beta Model (with the average RMSE being less than 5%).

1. Introduction

According to DNV’s Energy Transition Outlook [1], offshore wind alone is expected to make up approximately 11.4% of the world’s electricity generation if net zero greenhouse gas emissions are to be achieved by 2050. This value implies an average annual growth in offshore wind of approximately 13% up to the mid-21st century, without considering other renewable energy sources (RES), such as offshore solar photovoltaics (PVs) [1]. An increase in RES installations directly implies increases in energy curtailment, given that the supply of intermittent wind and solar power often does not match the actual demand by energy consumers. Energy storage is a primary solution for reducing intermittency issues associated with renewables and, thus, avoiding curtailment. The European Commission stated that the only way that renewable energy (RE) generation can reach full capacity is if energy storage systems (ESSs) are applied in tandem with renewable energy sources (RESs), with an increasing relevance set on long-duration energy storage (LDES) requirements due to the potential for avoiding up to 88% of generation curtailment [2,3].

Wang et al. [4] provided a comprehensive literature review of different marine renewable energy storage technologies, stating that no individual technology has emerged as a solution to all storage requirements. Nguyen et al. [5] provided a review of a variety of software tools used for ESS valuation, emphasising that research gaps exist in simulation models accounting for non-linear operating characteristics and in the provision of better capability to the tool users. Li et al. [6] carried out a cost and performance assessment on coupling an offshore compressed air energy storage (CAES) system with an offshore wind farm, with findings suggesting that viability is highly dependent on system conditions. Qin et al. [7] created and implemented a cost model to test a CAES system with a hydraulic power transmission, concluding that such a concept may reduce project costs for up and coming wind turbine technology during the 2020–2030 decade, yet expressing that further software and hardware technology research is heavily required. Fan et al. [8] simulated the performance of a battery energy storage system (BESS) co-located with a wind farm with the aim of providing enhanced frequency response service and storing the otherwise-curtailed energy generated. Their results indicated that cost saving due to avoiding an independent connection for the BESS was significant compared to obtaining approximately 1% less in revenue from the system’s enhanced frequency response service. Maisonnave et al. [9] developed an electrical power conversion system for an isothermal CAES system which operated via kilowatt-scale centrifugal pumps. Söderäng et al. [10] created a digital twin of a combustion engine-based power plant combined with a battery energy storage system in Simulink^® with the aim of providing a computationally time-efficient numerical model, while also considering the system’s governing physics. Möller et al. [11] modelled a hybrid energy storage system made up of a combination of BESS and hydrogen with solar PVs as the primary RES. Hutchinson et al. [12] numerically modelled a BESS for verification of results with experimental testing, highlighting that numerical models can provide quick and realistic projections of how the same system would perform in real life. Fiaschi et al. [13] simulated RES energy production combined with three ESSs, where one of the findings was that an undersized battery system can lead to curtailment due to batteries not being able to store all RES energy produced when the generation is high for long periods of time. Camargos et al. [14] experimentally tested a combined pumped-hydro CAES system made up of two storage tanks, one with compressed air and the other with water in order to generate electricity via a Pelton turbine. The research found that the system obtained competitive power conversion efficiencies, while having a more advantageous setup in terms of power control and costings.

The Energy Transition Expertise Centre (EnTEC) [15] reported that power market arbitrage became profitable in 2021 and 2022, with profitability only expected to increase with further cost reductions due to technological advancements in the energy storage sector. EnTEC highlighted that an economic key performance indicator is the cost variation based on the type of storage technology and any geological limitations which may impact the ESSs viability [15]. Arellano et al. [16] developed a methodology which assesses the feasibility of storage technologies for offshore applications due to the benefits of offshore ESSs, primarily being that offshore storage can provide a secure and large-scale energy supply. The research also stated that offshore energy storage solutions are advantageous since they minimise land usage and re-use already-deployed offshore structures instead of decommissioning them. Hazim et al. [17] proposed an innovative re-purposing solution which utilises offshore pipelines as energy storage based on a hydro-pneumatic energy storage (HPES) system. Their research found a compromise between providing a feasible energy storage solution while also saving time and decommissioning costs by applying existing infrastructure.

Experimental testing of a kilowatt (kW) scaled version of a HPES system was performed by Buhagiar et al. [18], resulting in a promising thermal efficiency of >93%, while also utilising well-proven components, providing high safety levels crucial for offshore deployment and operation and a long lifetime. Such high thermal efficiencies are achievable given the excellent heat sink and heat source characteristics of seawater in which the HPES system is submerged and also because of the system’s low compression and expansion ratios, which are restricted to <2.5 [18]. Cutajar et al. [19] presented a software tool which determines an optimised design for pressure vessels used for offshore HPES systems. Odukomaiya et al. [20] developed and tested a hybrid compressed gas–hydro energy storage prototype, finding that high thermal efficiencies were obtained and that their HPES system round-trip efficiency was dependent on correctly selecting auxiliary components making up the system.

While previous work in offshore HPES has focused on kW-scale numerical modelling or prototype-based analysis [18,21], this paper presents and compares two numerical models which simulate a megawatt (MW) scale Energy Conversion Unit’s (ECU) performance of an offshore HPES system. The offshore HPES system is described in detail in previous literature [18,22]. The ECU consists of hydraulic machinery in the form of a variable speed, multi-stage centrifugal pump and a Pelton turbine utilised in the charging and discharging processes of the ESS. The changes made to upscale the HPES from kW-scale to MW-scale involve the selection of hydraulic machinery which satisfy MW power conditions. Additionally, the machinery must handle much larger pressures and flowrates. The volume capacity of the pressure containment system (PCS) must be increased considerably.

The numerical models are dubbed the Alpha and Beta models, modelled in Python^TM (version 3.9.13) and Mathworks^® Simulink^®/Simscape^TM (R2020b), respectively. The Alpha model is a quasi-steady-state numerical model which simulates the operation of the aforementioned hydraulic machinery and the PCS of the offshore HPES system. The PCS is responsible for housing the compressed air being compressed via a liquid piston, created through the injection of seawater. The novelty of the work is the development of a reliable numerical code which is able to model the performance of the ECU and overall HPES system at MW-scale. The Beta model is a more detailed and computationally demanding numerical model which aims to provide a more realistic operational scenario by accounting for transients, hydraulic component inertias, hydraulic pipe losses, and includes a number of electrical black boxes. The aim of the study is to compare and contrast the results of the two novel numerical models to evaluate the Alpha model’s accuracy and to understand the effect of transients on the overall system performance and efficiency. The creation of such numerical models provides an approximation on how a MW-scale system would operate without the need for physical prototyping, which can lead to great expenditure.

The paper is organised as follows. Section 2 provides the HPES system modelled and describes the methodology applied for developing both numerical models. Section 3 presents simulation results from the simulations run and compares the errors between the two models. Finally, Section 4 summarises the work performed and presents the research findings and contributions.

2. The Baseline ECU Design and Numerical Models

The following section describes the set-up of the ECU system hydraulic machinery and also provides a brief explanation of the numerical models used in the study being presented here.

2.1. The Baseline ECU and PCS Design

Initially, the setup of the ECU design to be used throughout testing and analysis was to be specified. Figure 1 presents a schematic of the ECU and PCS system. The MW-scale ECU consists of a variable speed, multi-stage centrifugal pump and a Pelton turbine. The multi-stage centrifugal pump must operate at variable speeds depending on the PCS state of charge since, unlike in pumped-hydro energy storage, the centrifugal pump will have to operate under large variable-head conditions required by the HPES system [23,24,25].

The characteristic curves of the centrifugal pump assumed throughout this study are presented in Figure 2. These curves were derived from an existing kW-scale centrifugal pump by applying dynamic similarity rules based on the Head, Flow rate, and Power numbers. The Reynolds number has been ignored; however, Reynolds number effects were considered to obtain the efficiency changes due to upscaling by means of Equation (1) [26].

$$\frac{1-{\eta }_2}{1-{\eta }_1}={\left(\frac{R{e}_1}{R{e}_2}\right)}^{0.1}$$

(1)

where ${\eta }_1$ and ${\eta }_2$ are the hydraulic efficiencies and $R{e}_1$ and $R{e}_2$ are the Reynolds numbers of the kW-scale and MW-scale pumps, respectively.

Similar to the pump, the Pelton turbine operates at a variable speed and with a large variable head, whereby both speed and nozzle orifice are varied to supply the output power required, while also maintaining an optimal bucket to jet speed ratio of approximately 0.48 [27,28]. This value is taken from the literature, where the theoretical optimal ratio is 0.5; however, it becomes 0.48 due to viscous friction and windage losses due to bucket aerodynamic drag [27]. The ECU components were designed and selected based on the PCS requirements of the full-scale HPES system having an operating pressure range of between 80 and 200 bar [19]. The pressure range selected is based on previous work by Cutajar et al. [19], who found that the most cost-effective design to minimise material requirements in terms of kilogram of steel per unit of stored energy (kg/kWh) occurred at an approximate pressure ratio of 2.5 and for a maximum compressed air pressure of 200 bar, which is the typical working pressure for offshore pipelines. Additionally, the compression and expansion processes of the PCS were assumed to be isothermal since the PCS is considered to be located in a subsea environment, thus using the surrounding seawater as an excellent heat sink and heat source [18,29].

Table 1. Main parameters of the variable speed, multi-stage centrifugal pump.

Centrifugal Pump Parameter	Value
Rated Power ($P_{Hp}$)	4.30 MW
Rated Speed ($\omega$)	3250 RPM
Number of Stages	9
Maximum Hydraulic Efficiency (${\eta }_{pump}$)	82%

summarises the main parameters of the variable speed centrifugal pump.

Table 2. Main parameters of the Pelton turbine.

Pelton Turbine Parameter	Value
Rated Power ($P_{Ht}$)	5.00 MW
Rated Speed (${\omega }_{Pelton}$)	426 RPM
Maximum Efficiency (${\eta }_{Pelton}$)	92%
Runner Diameter (D)	3.78 m
Bucket to Jet Speed Ratio ($\varphi$)	0.48
Generator Efficiency (${\eta }_{gen}$)	0.95
Number of Nozzles (N)	2
Skin Friction Coefficient (k)	0.97
Jet Angle ($\beta$)	165

summarises the main parameters of the Pelton turbine wheel and, finally, the PCS parameters are also shown in

Table 3. Main parameters of the PCS.

PCS Parameter	Value
Pre-charge (Minimum) Pressure ($p_{pr}$)	80 bar
Pressure Limit ($p_{max}$)	200 bar
Total Volume ($V_T$)	4080 m${}^3$

[19,30].

2.2. The Alpha Model

As previously mentioned in Section 1, the Alpha model is a quasi-steady-state numerical model (developed in Python^TM) which simulates the operation of the ECU charging and discharging the PCS of the offshore HPES system. The flowchart presented in Figure 3 shows the procedure which the Alpha model implements for the numerical solution. The simulator requires pump, Pelton turbine, and PCS model inputs, mainly based on the parameters shown in Table 1, Table 2 and Table 3 in Section 2.1. Additionally, the pump requires polynomial curve inputs, taken from the equations of the curves shown in Figure 2, as a function of rotational speed. The simulation duration and timestep may also be manually adjusted, where a 1-s timestep was considered throughout this study since opting for a smaller timestep only led to a maximum Normalised Root Mean Square Error (RMSE${}_N$) of 1.39% across all criteria analysed in Section 3 when implementing finer timesteps of 0.5 and 0.1 s. Finally, a constant power input or file with a power time-series must be inputted for the system to run. If power is positive, meaning that the RES power generation is greater than a set moving average for scheduling power or greater than grid demand, this implies that the centrifugal pump will be in operation and, thus, the PCS is being charged. Meanwhile, if power is negative, which means that demand is greater than supply, PCS discharge is implied. The numerical simulation working in the time domain is automatically stopped when the time limit set is reached, or when the PCS reaches its upper or lower pressure limits (200 and 80 bar, respectively).

The PCS modelling involved a time marching approach, where the volume of air within the pressure vessel and its pressure are established at each timestep using Equations (2) and (3) [31].

$$V_{F_t}=V_{F_{t-1}}+(Q\times dt)$$

(2)

where $V_{F_t}$ and $V_{F_{t-1}}$ are the PCS fluid volumes at the current and previous timesteps, Q is the flow rate, which is positive or negative based on if fluid is entering or exiting the PCS, respectively, and $dt$ is the timestep.

$$p_G=\frac{(p_{pr}+p_A)}{((V_T-V_{F_t})\times V_T)}-p_A$$

(3)

where $p_G$ is the PCS gauge pressure at the current timestep, $V_T$ is the total PCS volume and $p_{PR}$ and $p_A$ represent the gauge pre-charge (minimum) pressure and atmospheric pressure, respectively. This equation is a simplified version of the equation utilised by the Simscape^TM gas-charged accumulator model [31], based on the assumption that the HPES system is isothermal since it uses the surrounding seawater as a heat sink [18,29]. The methodology for operating the pump within the Alpha model is summarised in Figure 4. The inertia-less pump operation is regulated via a PID controller (P = 7, I = 20, D = 0.04) which translates the differential between desired and shaft power, $∆P$, into a speed output due to the centrifugal pump’s requirement of operating at variable speed. With speed being a function of both PCS pressure and desired power, interpolation of the pump characteristic curves (Figure 2 in Section 2.1) is required to obtain the pressure-flow and power-flow curves at the output speed provided by the PID controller. The Ziegler–Nichols method [32] was used to obtain baseline PID values. Manual tuning was then applied to fine-tune the controller. Such tuning was acceptable since the study focuses on analysing the quasi-steady-state Alpha model’s accuracy for constant power charging and discharging. Once the curves are obtained, a flow rate satisfying both pressure and power requirements is found. Since the pump curves present shaft (electrical motor power output) power ($P_S$), the pump hydraulic power, $P_{Hp}$, is determined using Equation (4). Equation (5) is used to find hydraulic efficiency, ${\eta }_{pump}$.

$$P_{Hp}=p_G\times Q$$

(4)

$${\eta }_{pump}=\frac{P_{Hp}}{P_S}$$

(5)

For the discharging process of the ECU, the Pelton turbine was modelled as a function of pressure as well as power. The jet velocity, $v_{jet}$, was calculated using Equation (6).

$$v_{jet}=c_v\sqrt{\frac{2\times p_G\times {10}^5}{{\rho }_{sw}}}$$

(6)

where $c_v$ is the constant coefficient of velocity, ${\rho }_{sw}$ is the seawater density and $p_G$ is the PCS pressure in bar. Apart from operation based on the PCS pressure, the desired power had to be attained. This was performed by adjusting the needle valve position, h, once again interpreted based on the Simscape^TM needle valve model [33]. Equation (7) was applied at the start of an iterative process, presented in Figure 5, within the same timestep.

$$A_{nv}\left(h\right)=\pi \times \left(d_{nv}-hsin\left(\frac{\alpha }{2}\right)cos\left(\frac{\alpha }{2}\right)\right)\times hsin\left(\frac{\alpha }{2}\right)$$

(7)

where $A_{nv}$ is the needle valve area, $d_{nv}$ is the needle valve diameter, and $\alpha$ is the needle angle. The total flow rate (Q), based on the number of nozzles (N), was then calculated by applying Equation (8) [33,34], such that

$$Q=N\times C_D\times A_{nv}\left(h\right)\times \sqrt{\frac{2}{{\rho }_{sw}}}\times \frac{p_G}{{(p_G^2+p_{CR}^2)}^{1/4}}$$

(8)

where $C_D$ is the constant discharge coefficient and $p_{CR}$ is a constant value which represents the minimum pressure for turbulent flow [33,34].

The sequential increase in needle valve position h (where $∆h$ was 0.01 mm) was continued until the Pelton turbine wheel yielded the desired power generation by comparing the desired power with the Pelton turbine’s hydraulic power ($P_{Ht}$), computed by applying Equation (9).

$$P_{Ht}={\rho }_{sw}\times Q\times (1+kcos\left(\beta \right))\times (v_{jet}-v_{bucket})\times v_{bucket}$$

(9)

where k is the bucket skin friction coefficient, $\beta$ is the bucket jet angle and $v_{bucket}$ is the bucket velocity.

2.3. The Beta Model

The Beta model uses Mathworks^® Simulink^®/Simscape^TM to model the same system described in Section 2.2, while including electrical components, such as a three-phase source and a VSD, a piping system, and other built-in Simscape models which simulate the PCS as well as the centrifugal pump. The Pelton turbine was modelled using an amalgamation of mathematical functions via Simulink^® while also applying some built-in components from the Simscape^TM library [31,33]. Due to computational challenges, the ECU hydraulic machinery and PCS modelling was split into two models, (i) a charging model including the variable speed, multi-stage centrifugal pump and PCS and (ii) a discharging model including the mathematical model of the Pelton turbine and PCS. The following Sections (Section 2.3.1 and Section 2.3.2) will describe the two individual models making up the Beta model.

2.3.1. The Charging Model

While the control system for this model is identical to the system presented and explained previously in Figure 4 in Section 2.2, a primary difference was the inclusion of electrical parameters and pump’s motor inertia within the Variable Speed Drive (VSD) block [35,36]. As a result of the introduction of inertia, the pump’s speed had a ramp up which was different from the quasi-steady-state solution of the Alpha model since the controller gains affect speed transients and, thus, require tuning through the in-built MATLAB^® PID tuner.

Figure 6 presents the system modelled in Simscape^TM for the operation of the variable speed, multi-stage centrifugal pump charging the PCS at a constant, desired power. The output from the VSD is directly connected to the pump’s motor, where the pump takes seawater in and pumps it to the PCS. The pump drivetrain (motor-shaft-load) is governed by Equation (10) presented hereunder [35]

$$T_e=J\frac{d}{dt}\omega +F\omega +T_m$$

(10)

where $T_e$ is the electromagnetic torque, J is the motor inertia, F is the friction factor, $\omega$ is the pump’s motor rotational speed, and $T_m$ is the mechanical torque. The centrifugal pump is modelled based on the charts presented in Figure 2 in Section 2.1. As a result, once the VSD outputs the desired speed (motor speed restriction between 0 and 3250 RPM) to obtain the desired power, Equations (11) and (12) are then used by Simscape^TM to obtain the pressure ($p_G$) and power ($P_S$) values produced by the pump at the desired speed, respectively.

$$p_G=p_{G_{ref}}\times {\left(\frac{\omega }{{\omega }_{ref}}\right)}^2$$

(11)

$$P_S=P_{S_{ref}}\times {\left(\frac{\omega }{{\omega }_{ref}}\right)}^3$$

(12)

where ${\omega }_{ref}$ is the angular velocity obtained based on the inputted pump curves, $p_{G_{ref}}$$P_{S_{ref}}$ are the pump curve pressure and power values at ${\omega }_{ref}$ and $\omega$ is the speed outputted from the PID controller and input into the VSD. A pipe flow system was also added, with one pipe connecting the seawater to the pump’s input and a second pipe connecting the pump’s output to the PCS. The PCS was assumed to be at a sea depth of 100 m. The piping considered friction-generated pressure losses ($p_{loss}$) by implementing Equation (13) [37].

$$p_{loss}=f\times \frac{(L+L_r)}{D_p}\times \frac{{\rho }_{sw}}{2A_p^2}\times Q^2$$

(13)

where f is the friction factor, L and $L_r$ are the pipe length and equivalent length of resistances, respectively, [38], $D_p$ is the pipe diameter and A${}_p$ is the pipe cross-sectional area. The friction factor was evaluated through an iterative process by using the Moody Chart to evaluate head losses and selecting a pipe diameter. The latter was selected such that the head losses did not exceed 1%.

A check valve was added at the PCS’s inlet to prevent backflow, thus preventing the system pressure from dropping below the stipulated pre-charge pressure of 80 bar. The check valve was modelled based on the needle valve, shown in Equation (8) in Section 2, where the only difference is that the check valve opens fully once the pre-charge pressure is reached, while the needle valve has a variable position based on the needle valve position.

2.3.2. The Discharging Model

Equations (6)–(9) in Section 2.2 are used in both the Alpha and Beta models. The Beta model introduces a nested PID control loop, based on previous work by Buhagiar et al. [39], to regulate power requirements and maintain optimal speed based on the bucket-to-speed ratio previously mentioned in Section 2.2 [27]. Moreover, the Pelton turbine’s inertia and damping are also considered to calculate the bucket torque to be matched to the generator’s torque for power generation, a step which is omitted in the Alpha model. Equation (14) presents the governing equation for the Pelton turbine wheel implemented in the Beta model.

$$M_B-M_G=J_{Pel}{\dot{w}}_{Pelton}+D_{Pel}{w}_{Pelton}$$

(14)

where $M_B$ is the bucket torque, $M_G$ is the generator torque, $J_{Pel}$ is the Pelton wheel inertia and $D_{Pel}$ is the viscous damping of the turbine. Figure 7 presents an overview of the Simulink^® model of the Pelton turbine, specifically highlighting the aforementioned nested PID control loop and also presenting the inertia and damping transfer function block [40]. Finally, the Beta model also considers frictional ($P_{Friction}$) and windage ($P_{Windage}$) losses. Equations (15) and (16) show how the said losses were calculated [27,41,42].

$$P_{Friction}=0.000105\times M_F\times {\omega }_{Pelton}$$

(15)

$$P_{Windage}=15\times {\left(\frac{{\omega }_{Pelton}}{(2\times \pi )}\right)}^3\times D^5\times {\left(R_1\right)}^{1/4}{\left(R_2\right)}^{3/4}{\left(R_3\right)}^{5/4}{\left(R_4\right)}^{7/4}$$

(16)

where $M_F$ is the frictional moment, ${\omega }_{Pelton}$ is the Pelton wheel speed, D is the Pelton wheel diameter and $R_1$ to $R_4$ are Pelton turbine wheel geometric approximations [27].

2.4. Simulation Tests Performed

As previously explained in Section 2.3, the Beta model charging and discharging models were split due to computational restrictions, particularly within the charging model. Since a full charging cycle from 80 to 200 bar was not possible due to the high computational resources required, the charging cycle was divided into six 10 bar increments across the pressure range between 80 and 190 bar.

Table 4. The charging runs performed in the Alpha and Beta models at constant power (MW).

Charging Runs	Input Power (MW)	Pressure Range (bar)
Run 1	1.5	80–90
Run 2	2.0
Run 3	2.5	100–110
Run 4	3.0
Run 5	3.0	120–130
Run 6	5.0
Run 7	3.5	140–150
Run 8	5.0
Run 9	4.0	160–170
Run 10	5.0
Run 11	3.6	180–190
Run 12	3.8

and

Table 5. The discharging runs performed in the Alpha and Beta models at constant power (MW).

Discharging Runs	Input Power (MW)	Pressure Range (bar)
Run 1	1.0	200–80
Run 2	2.0
Run 3	3.0
Run 4	4.0
Run 5	5.0

summarise all the simulation runs performed using the Alpha and Beta models. The runs were selected based on input powers which were within the operating capabilities of the pump, in accordance with Figure 2 in Section 2.1. For example, charging the system at 1.5 MW at a pressure of 160 bar would not be possible according to the pump curves, even when applying variable speeds, since the pump reaching such high pressures would require greater speeds as well as a greater power input. Similarly, if a power of 5.0 MW were to be inputted to charge the system when the PCS is at a pressure of 80 bar, the flowrate limit would be reached, thus also reaching a pump power output limit and not satisfying the requirement of charging the system at the desired 5.0 MW. The Root Mean Square Error (RMSE) and Normalised RMSE (RMSE${}_N$) were used as the statistical tools to evaluate the accuracy in results obtained between the Alpha and Beta models. Equations (17) and (18) present the equations applied for the different RMSE evaluations.

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{(X_{Alpha}-X_{Beta})}^2}{n}}$$

(17)

$$RMS{E}_N=\frac{RMSE}{(X_{Max}-X_{Min})}$$

(18)

where n is the number of observations, $X_{Alpha}$ and $X_{Beta}$ are the Alpha and Beta model recorded values and $X_{Max}$ and $X_{Min}$ are the maximum and minimum recorded values of the dataset being analysed.

3. Results and Discussion

The results section summarises the results of the runs selected for the comparison of the Alpha model to the more comprehensive Beta model, highlighting the key similarities and differences, and assessing the Normalised Root Mean Square Error (RMSE${}_N$) generated by the two numerical models, both for charging and discharging cycles.

3.1. HPES System Charging

Figure 8 presents a portion of the results obtained specifically from Runs 2 and 7, directly comparing the Alpha and Beta models for the same simulation runs. On observation, the results are highly comparable and follow the same trends, with minimal time differences between the two models. A primary visual difference of the power results between the Alpha and Beta models is the effect of the PID controller and motor inertia within the Beta model, where the controller gain parameters play a crucial role in the pump’s power and speed ascent time and overshoot. A finding obtained during simulation runs was that if a full charging cycle were to be performed in the Beta model (from 80 to 200 bar), PID active control would be required. Firstly, due to the large variation in PCS pressure and secondly, owing to the large variation in powers, the system at 120 bar could be charged at powers of between 3.0 and 5.0 MW, thus having a direct influence on the pump’s required speed, which is an output of the PID controller, as previously shown in Figure 6 in Section 2.3.1. In Figure 8c,d, the response of the Beta model in the two cases indicates different PID controller gain parameters since the former experiences undershoot and a longer settling time, while the latter is more aggressive, thus experiencing an overshoot, yet obtains a relatively faster settling time. It must be noted that while the PID controller gain parameters are satisfactory for the purposes of this study, further optimisation of the control system’s response is not to be excluded.

Table 6. The controller gain parameters for all charging runs performed in the Beta model.

Charging Runs	Proportional (P)	PID Parameter Gains Integral (I)	Derivative (D)
Run 1	0.004	0.0050	$5\times {10}^{-7}$
Run 2	0.008	0.0045	$5\times {10}^{-7}$
Run 3	0.005	0.0050	0.50
Run 4	0.005	0.0050	0.00
Run 5	0.006	0.0050	0.15
Run 6	0.003	0.0100	0.02
Run 7	0.200	0.0004	0.15
Run 8	0.003	0.0100	0.02
Run 9	0.009	0.0100	$1\times {10}^{-4}$
Run 10	0.003	0.0100	0.02
Run 11	0.0095	0.1000	$1\times {10}^{-4}$
Run 12	0.0095	0.1000	$1\times {10}^{-5}$

presents the controller gain parameters implemented for every run.

Table 7. The controller regulation quality indicators for all charging runs.

Charging Runs	PID Regulation Quality Indicators
	Maximum Overshoot (%)		Settling Time (s)		Steady State Error (MW)
	Alpha Model	Beta Model	Alpha Model	Beta Model	Alpha Model	Beta Model
Run 1	0	0	8	366	0.001	0.011
Run 2	0	0	21	85	0.001	0.004
Run 3	0	8	19	257	0.002	0.003
Run 4	0	0	21	150	0.002	0.002
Run 5	0	0	18	90	0.002	0.004
Run 6	0	0	18	103	0.002	0.023
Run 7	6.57	5.7	20	50	0.003	0.003
Run 8	0	0	15	67	0.003	0.022
Run 9	6.25	20	15	53	0.004	0.002
Run 10	1.2	0	15	104	0.03	0.022
Run 11	17	16.7	30	139	0.003	0.056
Run 12	15	15.7	30	56	0.004	0.0006

presents a comparison between the controller regulation quality indicators for all the charging runs across the two numerical models. While steady state error remains low across all runs, the overshoot for both models towards the upper pressure limit is large. The Alpha model shows a lot less sensitivity to the controller regulation quality across all runs due to the exclusion of inertia and its quasi-steady-state modelling approach. It must be noted that while the PID controller gain parameters are satisfactory for the purposes of this study, further optimisation of the control system’s response is not to be excluded. In reality, better-optimised PID control response can lead to even better RMSE results (presented later in

Table 8. Summary of all results for the Normalised Root Mean Square Error (RMSE${}_N$) comparing the predictions for the Alpha and Beta model PCS charging process.

Charging Runs	Normalised RMSE (%)
	Input Power	Pump Speed	Flow Rate	PCS Pressure	Pump Efficiency
Run 1	6.94	3.51	10.50	2.51	8.31
Run 2	5.76	4.26	6.38	1.73	7.29
Run 3	4.18	3.93	4.99	2.56	2.13
Run 4	6.79	5.66	7.74	1.99	8.83
Run 5	5.74	5.67	6.01	1.29	6.36
Run 6	9.11	7.73	10.30	1.68	12.20
Run 7	4.18	4.59	6.53	1.12	7.51
Run 8	9.32	8.29	10.30	2.57	11.30
Run 9	5.61	4.84	8.64	2.33	8.55
Run 10	4.81	5.75	6.37	2.37	5.43
Run 11	4.52	4.43	9.07	3.57	13.90
Run 12	6.36	4.36	11.70	2.97	11.20

) since the results comparison also considers the pump speed ramp up difference between the two numerical models due to the inclusion of machinery inertia in the Beta model.

Figure 9 presents more results comparing the same runs within the Alpha and Beta models. The two numerical models follow the same trend in each plot. Table 8 presents the RMSE${}_N$ results to provide a more reliable representation on the models’ comparative accuracy. The results obtained clearly illustrate the Alpha model’s capabilities of reproducing the Beta model simulation predictions for the same prescribed operational conditions. A particularly important trend is that, despite the maximum RMSE${}_N$ across all runs reaching only 13.9%, Runs 6, 8, and 10 experienced the highest errors in the majority of parameters. While variables such as pump speed and flow rate are expected to differ due to the different factors considered in the numerical calculations, such a result also coincides with the fact that the three mentioned runs consisted of the highest constant power, implying that RMSE${}_N$ has a direct proportionality with input power.

Although the power was a system input, the RMSE${}_N$ was still calculated since minor variations occurred which were regulated by the speed controller to maintain power as close to the desired (input) power as possible. The calculation was thus also used to prove that the Beta model controller gain variations across all runs all provided a stable system, since the power difference between the desired and shaft power was the controller’s input value.

3.2. HPES System Discharging

While the Alpha and Beta models for the Pelton turbine and PCS are very similar as previously discussed in Section 2.2 and Section 2.3.2, the Beta model includes transients and integrates a nested PID control loop. Additionally, the Pelton wheel’s inertia and damping are also considered together with power loss calculations in the form of windage and friction. Figure 10 presents an example comparison of Run 3 between the typical results yielded from each numerical model. As may be observed, the plots all follow the same trends, where a minimal difference in simulation time may be noted. Similar to the comparison between the numerical models in Section 3.1, the time difference results from the transients experienced in the Beta model, where an overshoot or undershoot within the control systems impacts the time required to fully discharge the PCS. Figure 10f presents the overall efficiency of both models, where a key observation is that while the efficiency is constant across the entire simulation for the Alpha model, a slight variation in efficiency occurs in the Beta model due to the inclusion of the Friction and Windage power losses, since these are directly proportional to the Pelton Wheel speed and the cube of Pelton Wheel speed, as shown in Equations (15) and (16), respectively.

Table 9. Summary of all the Normalised Root Mean Square Error (RMSE${}_N$) results comparing all the Alpha and Beta model for discharging.

Discharging Runs	Normalised RMSE (%)
	Input Power	Pelton Speed	Flow Rate	PCS Pressure	Gen. Torque	Nozzle Position
Run 1	3.10	4.52	2.83	4.27	4.39	1.51
Run 2	3.45	2.78	3.80	2.67	3.50	2.26
Run 3	3.76	3.53	4.54	2.13	3.69	2.84
Run 4	3.96	3.98	5.15	1.87	4.08	3.38
Run 5	4.57	4.46	5.66	1.72	4.51	3.91

presents the results for the RMSE${}_N$ across the five considered runs for the main parameters of the numerical models. While all error results are reasonably low (<6%) and show the codes’ comparability and reliability, several parameters show a trend. The Input Power, Flow Rate, and Nozzle Position all increase with increased constant power input, while the PCS Pressure decreases with increased constant power input. The reason for a PCS Pressure decrease may be due to reduced overshoot and settling time compared to the other runs, since operating at 5 MW is the rated power of the Pelton turbine considered. The increase in error across the former three variables, however, results from their correlation through the nested PID controller, whereby a higher power requirement seems to increase error within the Beta model, while the Alpha model continues to provide a steady-state solution across all runs. Since the Beta discharging model does not include electrical parameters, the computational time between both models is negligible, only taking a maximum of 45 s to complete a full discharging cycle.

4. Conclusions

The scope of this research article was to present the design, testing, and comparison of a simplified numerical model, dubbed the Alpha model, to a more comprehensive numerical model, namely the Beta model. Both models simulate the charging and discharging process of a megawatt-scale offshore HPES system’s ECU and PCS. While the two numerical models simulate the same system, the degree of complexity of the integrated models differs, thus resulting in significant differences in the computational resources required. The Alpha model is a computationally efficient, quasi-steady-state process which provides modularity and flexibility since numerous centrifugal pump, Pelton turbine, and PCS combinations can be simulated by changing the input parameters. A shortcoming of the Alpha model is that its simplified approach disregards transients due to machinery inertia, control system response, and system pressure losses. Meanwhile, the Beta model, as shown through Figure 4 and Figure 7 in Section 2.2 and Section 2.3, is a more complex model which mimics the physical system and aims to implement as many components as possible at the expense of computational efficiency. The Beta model is more suitable for analysing phenomena such as hydraulic machinery ramping, time response over short periods of time due to varying controller parameters, and transient flow analysis through the inclusion of hydraulic piping systems. While experimental data of such a system are not available, a validation step performed within each model was that of setting the pump speed to one of the speeds from the pump curves, setting a fixed head and observing that the power, efficiency, and flow rate values outputted matched the points on the pump curves. Nonetheless, further validation would be beneficial and is, thus, a limitation of the present study. Despite the model differences, the results produced were promising and have demonstrated the reliability of the simplified Alpha model compared to the Beta model by proving that the RMSE results are all consistently exceptionally low across all runs, both for the charging and the discharging simulations. Apart from the aforementioned reason, the runs performed resulted in the following key findings:

• The use of a single, variable-speed centrifugal pump under variable head conditions with a pressure ratio of approximately 2.5 means that the operating power (input power) is dependent on the pressure within the PCS. Pump compatibility between input power and pressure must be observed to ensure the pump is operable and further ensure that operation occurs at points as close to the pump’s best efficiency point as possible.
• The aforementioned power and pressure relationship leads to the need for an active PID control system when operating the centrifugal pump. This occurs since the large variations in power and pressure requirements directly impact the centrifugal pump’s operating speed as a result of the control system implemented, as depicted in Figure 4 and Figure 6 in Section 2.2 and Section 2.3.1, respectively.
• The mismatch in simulation duration time for the runs performed between the Alpha and Beta models is primarily caused from an amalgamation of the Beta model commencing from a complete stop, as well as the PID controller’s overshoot and undershoot. An Undershoot caused the Beta model’s results to indicate that charging or discharging of the system took longer. Meanwhile, an overshoot caused the opposite, where the Beta model simulation charged or discharged the system during the respective run quicker. The time described here refers to the time within the simulation, not the computational time required.
• The longest computational time taken by the Beta model was approximately 180 min as opposed to the Alpha model’s 40 s. When considering that the largest Normalised RMSE across all Runs was 13.9%, the Alpha model is a clear front runner in terms of speed and usability. Nevertheless, the Beta model offers an overall system understanding but is more applicable for analysing transients in the design of any HPES or CAES system.

5. Outlook

Future research must focus on the possibility of applying the same system but implementing a number of smaller pumps operating in series or in parallel to meet the variable power and head requirements demanded by the RES and PCS, respectively. Furthermore, additional work to transform the Beta model into a complete digital twin would be beneficial to improve the results’ accuracy, with a special focus on system start-up.