## 2 The Baseline ECU Design and Numerical Models
### 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]. \[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]  
where \(\eta_{1}\) and \(\eta_{2}\) are the hydraulic efficiencies and \(Re_{1}\) and \(Re_{2}\) 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 [PERSON] 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]. Figure 1: Schematic showing the offshore HPES system, highlighting the ECU and PCS. Figure 2: Centrifugal pump (**left**) pressure versus flow rate characteristic curves; (**right**) power versus flow rate characteristic curves. 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]. \[V_{F_{t}}=V_{F_{t-1}}+(Q\times dt) \tag{2}\]  
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. \[p_{G}=\frac{(p_{pr}+p_{A})}{((V_{T}-V_{F_{t}})\times V_{T})}-p_{A} \tag{3}\]  
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 SimscapeTM 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)  
Figure 3: The numerical model simulation procedure of the Alpha model.