5.4 MPC Algorithm Simplification
5.4.1 Linear Viscid Model Predictive Control
The NMPC shown in section 5.3.1 produces promising results, but at the cost of in-creased computational effort for this application. The NPMC uses approximately 2.5 times the computational time of the linear MPC for the unconstrained case and 5 times the computational time when constraints are included. A more efficient method for dealing with the viscous forces in the MPC model would be a linear MPC which in-cludes a linear estimate of the non-linear viscosity force, that is fine tuned for each sea state.
The linear viscid Model Predictive Controller is used to approximate the NMPC, where the linear viscous coefficient estimate ( ˜Cvis) is chosen to produce similar average elec-trical power at each sea-state when compared to the full NMPC, but without the com-putational complexity. Here a constant ˜Cvis is selected for each sea-state, to provide constant matrices ˜Acand ˜Ecas defined in (5.5) and (5.10). Hence, for a given sea-state this represents a linear MPC problem.
F˜v(t) = − ˜Cvis( ˙z(t) − ˙η(t)) (5.10) The results in Fig. 5.10 show how the electrical power extracted depends on the choice of ˜Cvis. Here the MPC system was tested for sea state 3 (Tp = 13.326 s Hs = 1.5 m) and sea state 8 (Tp = 13.326 s Hs = 6 m), with and without constraints. There is a unique ˜Cvis value that corresponds to the maximum extractable average electrical power; the optimum ˜Cvis value for sea state 3 is 1 × 105 kg/s and the optimum ˜Cvis
value for sea state 8 is 2.1 × 105kg/s.
When the linear constraints are included in the MPC, the relative velocity between the wave and WEC is restricted, especially at higher sea states where the WEC velocity constraints are frequently active. This relative velocity restriction causes the average electrical power to be insensitive to choices of ˜Cvis at higher sea states, as shown in Fig. 5.10(b).
The linear MPC, with and without constraints, was tested across the entire sea state
5. THEEFFECT OFMODELUNCERTAINTY, VISCOSITY ANDMPC SIMPLIFICATION ON
ELECTRICALPOWERPRODUCTION 5.4 MPC Algorithm Simplification
0 1 2 3
Figure 5.10: Linear MPC performance (assuming a linear viscosity model in the con-troller) with and without constraints. Average electrical power absorbed from (a) sea state 3 (Tp = 13.326 s Hs = 1.5 m), (b) sea state 8 (Tp = 13.326 s Hs = 6 m) using a Bretschneider spectrum
range, where the optimum ˜Cvisvalues corresponding to the maximum average power points are shown in Fig. 5.11. This also shows the ˜Cvis regions where the average power is above 98% of the maximum average power available ( ˜Cvis98%).
From Fig. 5.11, the optimal ˜Cvis value with constraints increases somewhat propor-tionally with the corresponding sea state, but at a much reduced rate than when con-straints are not included. It is also shown that when the concon-straints are included, the C˜vis98%regions broaden, which then causes a larger common ˜Cvis98% overlap across the sea states. Therefore, a sea state invariant ˜Cvisestimate value can be found, which allows the average electrical power for all sea states to operate between 96% to 100%
of the maximum power available, hence allowing a simple and efficient way of sub-optimally extracting acceptable electrical power. In this example, when the constraints are included, a constant ˜Cvis= 1 × 105kg/s would provide between 96 − 100% of the average electrical power extracted by optimally tuning ˜Cvisfor each sea-state.
Variations in the viscosity force could be caused by increasing growth of biofouling on the WEC itself (Wright et al. 2016), which could change the hydrodynamic properties of the WEC. Furthermore, it is important to analyse the effects that a mismatched control model has on the average power absorbed from the viscous system, since the viscous drag coefficient could vary around a certain value in practice (Nepf 1999).
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145 Adrian C.M. O’Sullivan
1 2 3 4 5 6 7 8 9 Sea State
0 0.5 1 1.5 2 2.5
3 105
Linear vicosity damping (C)vis~ vis
Linear sea state invariant C~
Optimum Linear C~vis with constraints Optimum Linear C~vis without constraints
Linear C range with constraints~vis
Linear C range without constraints~vis
s
Figure 5.11: The linear viscous term ˜Cvis versus the range of sea states. Showing the C˜visvalues which correspond to an MPC with an efficiency greater than 98%, ˜Cvis98%
(results found using an NMPC were taken as 100% efficiency). This figure shows that as linear mechanical constraints are included into the MPC algorithm, the ˜Cvis98%
regions decrease to a point where a single ˜Cvisvalue of ˜Cvis = 1 × 105 can be utilised to allow efficiency greater than 96% across all sea states.
The robustness of the MPC due to a viscous mismatch is tested by changing the non-linear viscosity drag coefficient Cdin the hydrodynamic system by ±10% and ±20%, where Cd = 1.8 when a mismatch of 0% occurs; the robustness analysis utilises 1 m high monochromatic excitation waves. The economic MPC with a constant linear vis-cous approximation ˜Cvis = 1 × 105kg/s, (O’Sullivan & Lightbody 2017c), is used in this section to analyse the robustness of the linear MPC when changes in the non-linear hydrodynamic model occur. Fig. 5.12 shows the resulting absorbed average electri-cal power found from a hydrodynamic system with a ±10% and ±20% mismatched viscous drag coefficient Cd. As shown in Fig. 5.12, as the Cdviscous drag coefficient increases (+10%, +20%) and the MPC’s linear viscous coefficient ˜Cvisstays constant, the average electrical power decreases. Furthermore, as the viscous drag coefficient Cd
is reduced (−10%, −20%), the average electrical power increases. Fig. 5.13 shows the power ratio (5.11) that each mismatched system has against its corresponding matched system; the corresponding matched system involves changing the non-linear drag coef-ficient Cdin the NMPC’s internal model to obtain maximum average electrical power levels from each hydrodynamic model variation,
5. THEEFFECT OFMODELUNCERTAINTY, VISCOSITY ANDMPC SIMPLIFICATION ON
ELECTRICALPOWERPRODUCTION 5.4 MPC Algorithm Simplification
Power Ratio = Pe,mismatch
Pe,match
, (5.11)
where Pe,mismatch is the average electrical power from a WEC system with a varying Cdcoefficient, controlled with an MPC with a constant ˜Cvis = 1 × 105 kg/s. Pe,match
is the average electrical power from a WEC system with a varying Cd, controlled with an NMPC with the same Cdvalue as the WEC system.
It is shown in Fig. 5.13 that the power ratios for both the ±10% and ±20% cases are all above 0.91; therefore the linear viscous approximated MPC is acceptable for use when the viscosity is uncertain.
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Frequency (rad/s) 0
0.05 0.1 0.15 0.2 0.25
Average Power (MW)
(a) +10% (C
d=1.98) (b) -10% (C
d=1.62) (c) +20% (C
d=2.16) (d) -20% (C
d=1.44) (e) 0% (C
d=1.8)
Figure 5.12: Average electrical power absorbed from 1 m high monochromatic excita-tion waves when an MPC with a constant viscous coefficient ˜Cvis= 1 × 105 is tested on a hydrodynamic system with a non-linear drag coefficient mismatch of (a) +10%
(Cd = 1.98), (b) −10% (Cd = 1.62), (c) +20% (Cd = 2.16), (d) −20% (Cd = 1.44) and (e) 0% (Cd= 1.8)
5.4.1.1 Performance of Linear MPC
The performance of the system under NMPC, linear MPC (with optimal ˜Cvisselected for each sea-state) and a linear MPC with constant linear viscous damping ( ˜Cvis = 1 × 105 kg/s) was compared over the 9 sea states as defined in Table 5.1. In Fig. 5.14, it is shown that in the unconstrained case, the average electrical power produced by the
Electrical Power Optimisation of
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147 Adrian C.M. O’Sullivan
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Frequency (rad/s)
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
Power Ratio
(a) +10% (C
d=1.98) (b) -10% (C
d=1.62) (c) +20% (C
d=2.16) (d) -20% (C
d=1.44) (e) 0% (C
d=1.8)
Figure 5.13: Power ratio from 1 m high monochromatic excitation waves when an MPC with a constant viscous coefficient ˜Cvis = 1 × 105 is tested on a hydrodynamic system with a non-linear drag coefficient of (a) +10% (Cd = 1.98), (b) −10% (Cd = 1.62), (c) +20% (Cd = 2.16), (d) −20% (Cd= 1.44) and (e) 0% (Cd = 1.8)
NMPC and the linear MPC (with optimal ˜Cvisselected for each sea-state) are similar up to sea-state 7. Furthermore, the average electrical powers produced when using the linear MPC with a constant linear viscous damping ( ˜Cvis = 1 × 105kg/s) diverged from the NMPC average electrical powers as the sea-state increased.
When examining the constrained case, where these MPC methods all included lin-ear mechanical constraints within their algorithms (4.1), as shown in Fig. 5.15, it is important to note that all three controllers provided similar average electrical power extraction, except at the energetic sea-state 9. This implies that mechanical constraints in this example, limit the relative velocity and allow for excellent performance of the linear MPC, with a fixed linear viscous damping model.