## 2 The Baseline ECU Design and Numerical Models
### The Baseline ECU and PCS Design
which translates the differential between desired and shaft power, \(\Delta P\), 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 [PERSON] (1965) 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, \(\eta_{pump}\). \[P_{HP}=p_{G}\times Q \tag{4}\]  
\[\eta_{pump}=\frac{P_{HP}}{P_{S}} \tag{5}\]  
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). \[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]0  
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 SimscapeTM needle valve model [PERSON] (1965). Equation (7) was applied at the start of an iterative process, presented in Figure 5, within the same timestep. \[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]1  
where \(A_{mv}\) is the needle valve area, \(d_{mv}\) 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) [PERSON] (1965); [PERSON] (1965), such that  
\[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]2  
where \(C_{D}\) is the constant discharge coefficient and \(p_{CR}\) is a constant value which represents the minimum pressure for turbulent flow [PERSON] (1965); [PERSON] (1965). \[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]3 A schematic showing the variable speed centrifugal pump’s numerical modelling procedure. The sequential increase in needle valve position \(h\) (where \(\Delta h\) 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). \[\frac{1-\eta_{2}}{1-\eta_{1}}=\left(\frac{Re_{1}}{Re_{2}}\right)^{0.1} \tag{1}\]4  
where \(k\) is the bucket skin friction coefficient, \(\beta\) is the bucket jet angle and \(v_{bucket}\) is the bucket velocity.