# A Study about Performance and Robustness of Model Predictive Controllers in a WEC System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Generic Mathematical Model for Point Absorber Wave

#### 2.2. Forces Identification Using Simulation Based on BEM

#### 2.3. Treatment of the Mathematical Model for the Design of MPCs

#### Simplified Model for the Design

## 3. Model Predictive Control for the Point Absorber WEC

#### 3.1. $MP{C}_{1}$

#### 3.2. $MP{C}_{2}$

#### 3.3. $MP{C}_{3}$

#### 3.4. $MP{C}_{4}$

#### 3.5. MPC_{5}

#### 3.6. Conventional Controllers

## 4. Study about Performances and Robustness

#### 4.1. Performance Comparison

_{3}and MPC

_{5}) achieve more regular power than the MPCs most used for WEC systems, those whose optimization criteria maximize the extracted power directly (MPC

_{1}and MPC

_{4}). In addition, Figure 5 shows how MPCs with embedded integrators achieve less overshoot in the control force applied by the PTO system than all other MPCs (reducing actuator overstress).

_{2}to the system, it gives the most irregular power; while the power generated with the MPC

_{3}, designed from the same model and following the same optimization criteria, is much cleaner, with fewer occasional peaks. This is due to the fact that the MPC

_{1}is continuously applying soft constraints to the oscillation speed, because it does not carry out a good control of the force that the PTO system exerts on the WEC during the time (see Figure 5). On the other hand, Table 5 shows how the addition of the embedded integrator (MPC

_{3}) considerably improves the behavior of the system with respect to that obtained by applying the MPC

_{2}. Since, the MPC

_{3}does not exceed nominal limits of the position and oscillation speed on any occasion. Furthermore, the MPC

_{3}controller generates a clearer control signal than the MPC

_{2}(see Figure 5), and as a consequence, the underdamped response of the PTO system decreases greatly.

_{5}controller is the one that generates more power, followed very closely by the MPC

_{1}, MPC

_{4}and, with a bit more distance, the MPC

_{3}. On the other side, the indicators $ONLP$ and $ONLS$ quantify the area of overshoot from the nominal limits of the position and oscillation speed, respectively. In this sense, the MPC

_{3}(together with the RD control) provides the best behavior, since it does not apply slack to the nominal limits on any occasion, then it is followed by MPC

_{1}. This can be verified in Figure 6 and Figure 7, which show a comparison between the positions and oscillation speeds obtained by applying the designed controllers to the mathematical model (15). As can be seen, all controllers keep the WEC system within its physical operating limits. Finally, the last two columns of Table 5 show the maximum and minimum power peaks obtained when applying each controller. In this aspect, ignoring the resistive damping control, MPCs with embedded integrators are once again the best performers. Note that these power peaks will cause an oversizing of: electrical machines, power electronics, accumulators, etc.

_{1}and MPC

_{4}, the use of a complete model reduces the overshoot of the PTO system a bit. Finally, the variation of the extracted mechanical power as a function of the prediction time ${t}_{f}$ is studied. In particular, Figure 8 shows a comparison between MPC

_{3}and MPC

_{1}. As can be seen, in both cases, after a prediction time of 3 s, the power generation does not improve significantly, while the computation effort increases. In addition, it should be pointed that MPC

_{3}does not have a monotonously increasing behavior that relates ${t}_{f}$ to the power generated, as would be expected. Nevertheless, in this aspect, the MPC

_{3}is better than the MPC

_{1}, because it can generate more power with less information of the future excitation force (up to $2.5$ s).

_{3}keeps the system within its nominal operating limits. On the other hand, it is also verified that the MPCs are superior in performances than the conventional I-P controller, in terms of: average mechanical powers generated, diminution of instantaneous power peaks and reduction of mechanical fatigue (due to exceeding nominal limits).

#### 4.2. Robustness Comparison

_{2}is the least robust. Meanwhile, the MPC

_{1}offers the best features in a power-robustness ratio. However, it should be noted that when considering more reasonable added uncertainty values (interval $[-50,50]\%$), the MPC

_{5}extracts significantly more power than the others. It should also be noted that the controllers that directly maximize power in their cost function (MPC

_{1}and MPC

_{4}) have the most predictable behavior with respect to the added uncertainty in ${m}_{\infty}$. On the other hand, the MPC

_{5}offers the best features with respect to the uncertainty added to the dynamics of the radiation force (up to $400\%$). In contrast, the MPC

_{2}gives very bad results in this respect. The MPC

_{1}also gets good results, because it achieves a practically constant power production despite variations of ${\u25b5}_{B}$.

_{5}supports an added uncertainty of $25\%$, again being the one that provides the best robustness results even with the smallest prediction time ${t}_{f}$. Note that, for such added uncertainty, the excitation force becomes more than three-times the force that the PTO system can apply to the buoy; see Figure 10. After the MPC

_{5}, the MPC

_{4}and MPC

_{1}get the best results, in this order.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Mesh defined by openWEC for a cylindrical buoy (11 m long and $7.5$ m in diameter). Comparison of the identified transfer functions: (

**b**) Bode diagram for excitation force ${F}_{e}(s)$; (

**c**) impulse response for radiation force ${K}_{r}(s)$.

**Figure 3.**Comparison of models: complete ($z1$, $w1$) vs. simplified ($z2$, $w2$). The height of the wave is n.

**Figure 4.**Simulation of the instantaneous mechanical powers generated by the WEC system when applying the seven controllers to the mathematical model (15).

**Figure 5.**Simulation of: wave force, force setpoint demanded for the PTO and real force produced by the PTO, when applying the seven controllers to the mathematical model (15).

**Figure 6.**Simulation of the positions obtained in the WEC system when applying the seven controllers to the mathematical model (15).

**Figure 7.**Simulation of the oscillation speeds obtained in the WEC system when applying the seven controllers to the mathematical model (15).

**Figure 8.**Variation of the extracted mechanical power as a function of the prediction time for $MP{C}_{1}$ and $MP{C}_{3}$ (control parameters listed in Table 4).

**Figure 9.**Variations of the average mechanical power generated as a function of the added uncertainty to: added mass, damping coefficient and hydrostatic restoring coefficient.

**Figure 10.**Wave force, force setpoint demanded and real force produced by the PTO, by applying the MPC

_{5}to the model (15), which has an added uncertainty to the hydrostatic restoring coefficient of $25\%$.

Coefficient | Value | Coefficient | Value | Coefficient | Value | Coefficient | Value |
---|---|---|---|---|---|---|---|

— | — | ${b}_{9}$ | 8.8 × 10^{1} | — | — | — | — |

${a}_{8}$ | 2.4 × 10^{−6} | ${b}_{8}$ | 1.5 × 10^{5} | ${a}_{{K}_{8}}$ | 1.2 × 10^{3} | — | — |

${a}_{7}$ | 2.2 × 10^{−4} | ${b}_{7}$ | 6.7 × 10^{−6} | ${a}_{{K}_{7}}$ | 2.3 × 10^{5} | ${b}_{{K}_{7}}$ | 8.9 × 10^{1} |

${a}_{6}$ | 3.6 × 10^{−1} | ${b}_{6}$ | 4.5 × 10^{9} | ${a}_{{K}_{6}}$ | 1.9 × 10^{8} | ${b}_{{K}_{6}}$ | 1.5 × 10^{5} |

${a}_{5}$ | 1.6 × 10^{1} | ${b}_{5}$ | 1.3 × 10^{10} | ${a}_{{K}_{5}}$ | 2.6 × 10^{10} | ${b}_{{K}_{5}}$ | 6.7 × 10^{6} |

${a}_{4}$ | 1.1 × 10^{4} | ${b}_{4}$ | 6.4 × 10^{10} | ${a}_{{K}_{4}}$ | 6.3 × 10^{12} | ${b}_{{K}_{4}}$ | 4.5 × 10^{9} |

${a}_{3}$ | 3.3 × 10^{4} | ${b}_{3}$ | 8.6 × 10^{10} | ${a}_{{K}_{3}}$ | 5.6 × 10^{14} | ${b}_{{K}_{3}}$ | 1.3 × 10^{10} |

${a}_{2}$ | 1.3 × 10^{5} | ${b}_{2}$ | 1.8 × 10^{11} | ${a}_{{K}_{2}}$ | 1.7 × 10^{15} | ${b}_{{K}_{2}}$ | 5.3 × 10^{10} |

${a}_{1}$ | 1.4 × 10^{5} | ${b}_{1}$ | 1.1 × 10^{11} | ${a}_{{K}_{1}}$ | 4.7 × 10^{15} | ${b}_{{K}_{1}}$ | 5.5 × 10^{10} |

${a}_{0}$ | 1.6 × 10^{5} | ${b}_{0}$ | 1.3 × 10^{11} | ${a}_{{K}_{0}}$ | 1.4 × 10^{15} | ${b}_{{K}_{0}}$ | 6.5 × 10^{10} |

Symbol | Description | Value |
---|---|---|

${k}_{res}$ | Hydrostatic restoring coefficient | 809,325 N/m |

${B}_{aprox}$ | Stationary approximation to system damping | 21,497 N s/m |

m | Mass of water displaced by the buoy at rest | 241,601.9 kg |

${m}_{\infty}$ | Mass of water added | 167,700 kg |

Symbol | Description | Value |
---|---|---|

${F}_{PT{O}_{max}}$ | Maximum stationary force for the power take-off system | 450 $KN$ |

${F}_{PT{O}_{min}}$ | Minimum stationary force for the power take-off system | −450 $KN$ |

${z}_{{n}_{max}}$ | Maximum nominal limit for the buoy position | 1.25 m |

${z}_{{n}_{min}}$ | Minimum nominal limit for the buoy position | −1.25 m |

${w}_{{n}_{max}}$ | Maximum nominal limit for oscillation speed | 1 m/s |

${w}_{{n}_{min}}$ | Minimum nominal limit for oscillation speed | −1 m/s |

${z}_{{f}_{max}}$ | Maximum physical limit for the buoy position | 1.7 m |

${z}_{{f}_{min}}$ | Minimum physical limit for the buoy position | −1.7 m |

${w}_{{f}_{max}}$ | Maximum physical limit for oscillation speed | 1.3 m/s |

${w}_{{f}_{min}}$ | Minimum physical limit for oscillation speed | −1.3 m/s |

${\kappa}_{z}$ | Maximum slack applied to the position nominal limit | 0.45 m |

${\kappa}_{w}$ | Maximum slack applied to the speed nominal limit | 0.3 m/s |

**Table 4.**Control parameter set for the sea state defined by the JONSWAP spectrum (3 m of significant wave height and 11 s of peak period).

Controller | R | Q | ${\mathit{W}}_{{\mathit{\epsilon}}_{\mathit{z}}}$ | ${\mathit{W}}_{{\mathit{\epsilon}}_{\mathit{w}}}$ | ${\mathit{K}}_{\mathit{P}}$ | ${\mathit{K}}_{\mathit{I}}$ |
---|---|---|---|---|---|---|

MPC_{1} | 4.500 × 10^{−7} | — | 1.000 × 10^{10} | 1.000 × 10^{7} | — | — |

MPC_{2} | 1.000 × 10^{−7} | 9.150 × 10^{4} | 1.000 × 10^{7} | 2.000 × 10^{9} | — | — |

MPC_{3} | 5.000 × 10^{5} | 4.575 × 10^{4} | 5.000 × 10^{10} | 1.000 × 10^{5} | — | — |

MPC_{4} | 4.500 × 10^{−7} | — | 1.000 × 10^{8} | 1.000 × 10^{4} | — | — |

MPC_{5} | 5.000 × 10^{−5} | 4.000 × 10^{5} | 1.000 × 10^{7} | 1.000 × 10^{9} | — | — |

RD | — | — | — | — | 7.902 × 10^{5} | — |

I-P | — | — | — | — | 7.050 × 10^{5} | 2.228 × 10^{4} |

**Table 5.**Results obtained in the application of the seven controllers to the WEC system. The powers are expressed in kW. Note that $ONLP$ indicates overshoot of nominal limits for position and $ONLS$ indicates overshoot of nominal limits for speed.

Controller | ${\overline{\mathit{P}}}_{\mathit{gen}}$ | $\mathit{ONLP}$ | $\mathit{ONLS}$ | ${\mathit{P}}_{{\mathit{gen}}_{\mathit{Max}}}$ | ${\mathit{P}}_{{\mathit{gen}}_{\mathit{Min}}}$ |
---|---|---|---|---|---|

MPC_{1} | 127.60 | 0.0000 | 0.0023 | 437.48 | −356.33 |

MPC_{2} | 110.72 | 0.0000 | 0.0281 | 744.06 | −364.97 |

MPC_{3} | 125.53 | 0.0000 | 0.0000 | 444.24 | −183.62 |

MPC_{4} | 127.50 | 0.0000 | 0.0107 | 452.75 | −369.27 |

MPC_{5} | 129.01 | 0.0000 | 0.0413 | 481.53 | −182.85 |

RD | 67.02 | 0.0000 | 0.0000 | 258.88 | −1.05 |

I-P | 109.90 | 0.0073 | 0.0511 | 583.73 | −107.34 |

Component | Amplitude | Period | Phase | Component | Amplitude | Period | Phase |
---|---|---|---|---|---|---|---|

${s}_{1}$ | $0.420$ | $13.00$ | $-\pi $ | ${s}_{9}$ | $0.200$ | $9.00$ | $0.00$ |

${s}_{2}$ | $0.520$ | $12.50$ | $1.5\pi $ | ${s}_{10}$ | $0.180$ | $8.50$ | $\pi $ |

${s}_{3}$ | $0.420$ | $12.25$ | $0.40$ | ${s}_{11}$ | $0.200$ | $7.50$ | $0.10$ |

${s}_{4}$ | $0.520$ | $11.50$ | $0.20$ | ${s}_{12}$ | $0.150$ | $6.50$ | $-0.77$ |

${s}_{5}$ | $0.450$ | $11.25$ | $0.11\pi $ | ${s}_{13}$ | $0.100$ | $5.50$ | $0.5\pi $ |

${s}_{6}$ | $0.300$ | $10.50$ | $-1.50$ | ${s}_{14}$ | $0.075$ | $5.00$ | $0.00$ |

${s}_{7}$ | $0.500$ | $10.00$ | $-0.33$ | ${s}_{15}$ | $0.020$ | $3.70$ | $0.12$ |

${s}_{8}$ | $0.210$ | $9.50$ | $0.78$ | — | — | — | — |

**Table 7.**Control parameters set for the sea state recorded in Table 7.

Controller | ${\mathit{T}}_{\mathit{m}}$$\left[\mathit{s}\right]$ | ${\mathit{T}}_{\mathit{f}}\left[\mathit{s}\right]$ | R | Q | ${\mathit{W}}_{{\mathit{\epsilon}}_{\mathit{z}}}$ | ${\mathit{W}}_{{\mathit{\epsilon}}_{\mathit{w}}}$ |
---|---|---|---|---|---|---|

MPC_{1} | $0.05$ | $1.5$ | $1.1\times {10}^{-7}$ | — | $1.0\times {10}^{10}$ | $1.0\times {10}^{7}$ |

MPC_{2} | $0.04$ | $1.2$ | $1.0\times {10}^{-7}$ | $2.15\times {10}^{4}$ | $1.0\times {10}^{7}$ | $2.0\times {10}^{9}$ |

MPC_{3} | $0.04$ | $1.2$ | $5.0\times {10}^{-5}$ | $1.475\times {10}^{4}$ | $5.0\times {10}^{10}$ | $1.0\times {10}^{5}$ |

MPC_{4} | $0.05$ | $1.5$ | $4.5\times {10}^{-7}$ | — | $1.0\times {10}^{8}$ | $1.0\times {10}^{4}$ |

MPC_{5} | $0.02$ | $0.6$ | $5.0\times {10}^{-5}$ | $1.75\times {10}^{4}$ | $1.0\times {10}^{7}$ | $1.0\times {10}^{9}$ |

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**MDPI and ACS Style**

Guardeño, R.; Consegliere, A.; López, M.J. A Study about Performance and Robustness of Model Predictive Controllers in a WEC System. *Energies* **2018**, *11*, 2857.
https://doi.org/10.3390/en11102857

**AMA Style**

Guardeño R, Consegliere A, López MJ. A Study about Performance and Robustness of Model Predictive Controllers in a WEC System. *Energies*. 2018; 11(10):2857.
https://doi.org/10.3390/en11102857

**Chicago/Turabian Style**

Guardeño, Rafael, Agustín Consegliere, and Manuel J. López. 2018. "A Study about Performance and Robustness of Model Predictive Controllers in a WEC System" *Energies* 11, no. 10: 2857.
https://doi.org/10.3390/en11102857