4.3 Electrical Constraints
4.3.2 Power Constraint
5 6x 105
Significant wave height (m)
Power(W)
With field weakening @ 1050V Without field weakening @ 1050V With field weakening @ 3500V Without field weakening @ 3500V With field weakening @ 7000V Without field weakening @ 7000V
Figure 4.7: Average electrical power extracted from irregular waves modelled using a Bretschneider spectrum, with a peak wave period of Tp = 8 s, and a significant height ranging from Hs= 1 m to 6 m, using MPC with a prediction horizon of Np = 40, with and without field weakening at a DC-link voltage of 1050V, 3500V and 7000V
If the DC-link voltage is large enough, both MPC approaches (with and without field weakening) will share a common iq(t) range (at ˙z(t) 6 2.2 m/s); hence field weakening becomes obsolete. This is the case for a DC-link voltage of 7000 V, as shown in Fig.
4.7. The average power from the two methods, at a DC-link of 7000 V, are the same, hence id(t) = 0 for both cases.
Table 4.1: Measured peak velocities during an irregular wave, modelled using a Bretschneider spectrum, with a peak wave period of Tp = 8 s and a significant height Hs=5 m. Showing the increase in peak velocity when field weakening is incorporated into a voltage constrained MPC
Peak Velocities from System
DC-link Voltage With Field Weakening Without Field Weakening
1050 V 0.4700 m/s 0.3905 m/s
3500 V 1.3746 m/s 1.3156 m/s
7000 V 1.9026 m/s 1.9026 m/s
4.3.2 Power Constraint
The majority of research in the wave energy industry makes use of control systems which obey the general rule that the PTO force should ideally cause the velocity to be
4. THEEFFECT OFCONSTRAINTS ON THE
OPTIMISATION OFELECTRICALPOWER
FROM AWEC 4.3 Electrical Constraints
However, for this to occur the PTO force must oscillate, which causes large power swings on the grid. Ideally, electrical power should not be taken from the grid as it may lead to grid side DC-link voltage control instabilities, poor power quality and voltage flickering. Negative power flow will complicate the design of the grid side DC-link voltage controller, as it introduces a non-minimum phase zero, which significantly affects the achievable closed-loop bandwidth of the DC-link voltage control system (Huang et al. 2015). Therefore, there is a need for a constraint to be included within the optimisation, which restricts the instantaneous power flow to be uni-directional.
The formulation of the power constraint is shown in (4.22). This power constraint must be applied at every step over the prediction horizon.
P (k + i) = λ′f dπ
τiq(k + i) ˙z(k + i) − Ri2q(k + i) − Ri2d(k + i) ≥ 0
∀i = {1, . . . , Np}
(4.22)
The power constraint for positive velocity is plotted in Fig. 4.8; these circular con-straints shrink in diameter as the speed decreases. The corresponding concon-straints for negative velocity are these circles reflected about the D-axis.
-8000 -600 -400 -200 0 200 400 600
500 1000 1500
id(t) current
iq(t)current
Power constraint @ 0.1 m/s Power constraint @ 0.05 m/s Power constraint @ 0.025 m/s Current constraint
Figure 4.8: Positive power constraints for a range of slow positive velocities, superim-posed on the current constraint
The MPC with the power flow constraint (4.22) was simulated with and without field weakening. The two MPC types were tested using an excitation wave based on a Bretschneider spectrum with a range of significant heights and a peak period of Tp = 8 s. The average absorbed electrical power resulting from uni-directional MPC is shown in Fig. 4.9.
Electrical Power Optimisation of
Grid-connected Wave Energy Converters using Economic Predictive Control
117 Adrian C.M. O’Sullivan
1 2 3 4 5 6 0
0.1 0.2 0.3 0.4 0.5 0.6
Significant wave height (m)
AveragePower(MW)
With field weakening @ 1050 V Without field weakening @ 1050 V With field weakening @ 3500 V Without field weakening @ 3500 V With field weakening @ 7000 V Without field weakening @ 7000 V
Figure 4.9: Average electrical power extracted from irregular waves, modelled using a Bretschneider spectrum with a peak wave period of Tp = 8 s and a significant height ranging from Hs = 1 m to 6 m; utilising a uni-directional power flow MPC with a prediction horizon of Np = 40, with and without field weakening at a DC-link voltage of 1050 V, 3500 V and 7000 V
Fig. 4.10 shows the typical time responses when an MPC with a uni-directional power flow is utilised with a DC-link of 1050 V and excited by an irregular wave (Bretschnei-der spectrum Hs = 4 m, Tp = 8 s). It can be seen that the MPC with no field weak-ening struggles to find the necessary PTO forces to maintain feasibility under a low DC-link voltage, even though the velocity is low and is only a medium sized signifi-cant excitation wave height.
4. THEEFFECT OFCONSTRAINTS ON THE
OPTIMISATION OFELECTRICALPOWER
FROM AWEC 4.3 Electrical Constraints
140 142 144 146 148 150 152 154 156 158 160 -0.4
140 142 144 146 148 150 152 154 156 158 160 -1
140 142 144 146 148 150 152 154 156 158 160 -2
0 2
Time (s)
140 142 144 146 148 150 152 154 156 158 160 0
140 142 144 146 148 150 152 154 156 158 160 0
Figure 4.10: Results of Uni-directional power flow MPC with a prediction horizon of Np = 40, with and without field weakening using a 1050 V DC-link during a Bretschneider irregular wave with Hs = 4 m and Tp = 8 s, i)The instantaneous elec-trical power, ii)Velocity, iii)Scaled PTO Force, iv)Magnitude of id(t),iq(t) currents, v)Magnitude of vd(t),vq(t) voltages
Electrical Power Optimisation of
Grid-connected Wave Energy Converters using Economic Predictive Control
119 Adrian C.M. O’Sullivan
With the combination of the power constraint at low velocities and the voltage con-straint at high velocities, there is a very narrow feasible region for all velocities. This narrow feasible region reduces the PTO force range, hence, reducing power extraction and PTO capabilities. When the system struggles to remain below the current limit, the crowbar is enabled (O’Sullivan & Lightbody 2016, Morren & de Haan 2007), there-fore causing a short circuit across the LPMG terminals and an open circuit between the LPMG and the VSC. This protection method is repeatedly enabled in Fig. 4.10 when no field weakening is used. This process is further described in appendix B.
Introducing field weakening creates a larger feasibility region. From Fig. 4.11 it is shown that the crowbar was not enabled, hence no damage occurred, and the velocity was higher than the system without field weakening, therefore creating more power.
Fig. 4.11 shows the performance of the MPC with field weakening and uni-directional power flow with a DC-link voltage of 7000 V during the same irregular wave that was used in Fig. 4.10. Fig. 4.11 shows that the instantaneous electrical power is unidirectional, the velocity reaches its limits at ±2.2 m/s, the scaled PTO force is nearly reaching its limits at ±1.128 N/kg and the current magnitude is under the limit of 527 A. From these results it is shown that the system is pushed to the limits in absorbing the maximum possible amount of power from the system, whilst maintaining feasibility.
However, what is interesting is that the magnitude of the vd(t) and vq(t) voltages is well below the constraint of 6062 V (the radius of the circular voltage limit shown in Fig. 4.4). Fig. 4.6 shows that if the DC-link level is increased past a certain point, it does not necessarily mean that the overall power absorption will increase due to the velocity restriction; hence, it is essential that an appropriate DC-link voltage is chosen to balance the controlled power electronics with potential power extraction benefits.
4. THEEFFECT OFCONSTRAINTS ON THE
OPTIMISATION OFELECTRICALPOWER
FROM AWEC 4.3 Electrical Constraints
120 122 124 126 128 130 132 134 136 138 140 -0.5
120 122 124 126 128 130 132 134 136 138 140 -3
120 122 124 126 128 130 132 134 136 138 140 -1.5
120 122 124 126 128 130 132 134 136 138 140 0
120 122 124 126 128 130 132 134 136 138 140 0
Figure 4.11: Results of uni-directional power flow MPC with a prediction horizon of Np = 40, with field weakening using a 7000 V DC-link during a Bretschneider irregular wave with Hs = 4 m and Tp = 8 s, i)The instantaneous electrical power, ii)Velocity, iii)Scaled PTO Force, iv)Magnitude of id(t),iq(t) currents, v)Magnitude of vd(t),vq(t) voltages
Electrical Power Optimisation of
Grid-connected Wave Energy Converters using Economic Predictive Control
121 Adrian C.M. O’Sullivan