Control Problem Definition

Participants are tasked with designing a control strategy that maximizes a defined performance index over the scoring interval for three predefined sea state scenarios. The following details clarify the control problem formulation.

Key Time Intervals for Performance Index Evaluation

To clearly define the control problem, it is important to distinguish the key time intervals, illustrated in Figure 7. These intervals are consistent across all three sea state scenarios:

../_images/startup_int_vs_scoring_int.png

Figure 7 Key time intervals for the performance index evaluation.

Important

  • Full simulation time span: \(t \in [t_{init}, t_{end}]\)
    The participant’s controller must be defined and operational for the entire simulation duration in all sea state scenarios.

  • Startup interval: \(t \in [t_{init}, T_0]\)
    Initial period during which the performance index is not evaluated.

  • Scoring interval: \(t \in [T_0, t_{end}]\)
    Period during which the performance index is computed for each sea state scenario.

  • Scoring interval start time \(T_0\):
    The scoring interval start time is identical across all three sea states.

  • Sea-state-dependent simulation horizon:
    Note that the simulation end time, \(t_{end}\), varies depending on the sea state scenario, whereas \(t_{init}=0\) s always.

For additional details on the startup interval, see Numerical Implementation.

WAPPAC Competition Performance Index

The performance index \(\mathcal{G}\) is designed to balance energy capture, physical constraint compliance, and capacity factor, and can be interpreted as a surrogate for the Levelized Cost of Energy (LCoE).

The objective is to maximize \(\mathcal{G}\) over the scoring interval \(t \in [T_0, t_{end}]\) by defining the control force \(F_{pto}(t)\) appropriately for all three predefined sea state scenarios:

\[\max_{F_{pto}(t)} \; \mathcal{G}\!\left(F_{pto}(t); t \in [T_0,t_{end}] \right) = \frac{\bar{P}_{pto}} {2 + \frac{\left[|x(t)|\right]_{98}}{x_{\max}} + \frac{\left[|F_{pto}(t)|\right]_{98}}{F_{u,\max}} - \frac{\bar{P}_{pto}}{\left[p_{pto}(t)\right]_{98}}}\]
\[\begin{split}\text{s.t.}& \quad \text{WavePiston dynamics (1),} \quad \\ & \quad p_{pto}(t) = F_{pto}(t) \dot{x}(t) \ge 0 \quad \forall t \in [t_{init},t_{end}]\end{split}\]

where:

  • \(\bar{P}_{pto} = \frac{1}{t_{end}-T_0}\int_{T_0}^{t_{end}} p_{pto}(t) \, dt\) — average PTO power during the scoring interval.

  • \(\left[|x(t)|\right]_{98}\) — 98th percentile of absolute sail displacement during the scoring interval.

  • \(\left[|F_{pto}(t)|\right]_{98}\) — 98th percentile of absolute PTO force during the scoring interval.

  • \([p_{pto}(t)]_{98}\) — 98th percentile of instantaneous PTO power during the scoring interval.

  • The constant 2 is a scaling factor.

Constraints:

  • \(x_\mathrm{max}\) — maximum allowable sail displacement (see Figure 6).

  • \(F_{pto,\max}\) — maximum allowable PTO control force.

  • \(p_{pto}(t) \ge 0 \; \forall t \in [t_{init},t_{end}]\) — PTO is passive; power cannot be negative at any time during the full simulation time span.

WavePiston Constraints Handling Considerations

The participant’s controller must be defined for the full simulation duration \(t \in [t_{init}, t_{end}]\). When designing the control strategy, the following aspects must be carefully considered:

Important

  • Position and PTO force constraints

    • Treated as soft limits: occasional exceedances may be tolerated but should be minimized to achieve higher performance index values.

    • Evaluated using the 98th percentile over the scoring interval.

  • Passivity constraint

    • A hard physical constraint: the PTO can only absorb power and cannot supply it.

    • Must be satisfied for the full simulation time span: \(p_{pto}(t) \ge 0 \; \forall t \in [t_{init},t_{end}]\).