Monitor

The different regimes can be divided into five: (1-2) normal regimes where all riser nodes are close to their desired positions, or the deviations are small and the tension variations have been small for a given time period, (3) the risers have a medium deviation from the desired configuration due to slow variations from LF TLP motions and tide, (4) the risers are subject to fast and large disturbances due to TLP motions or currents, (5) the risers are in a (near) collision situation called an error state.

R1 R2 R2_d

∆x_d

∆e_rel

∆x_R12

Figure 6.5: Actual and desired riser positions.

The switching conditions are mainly defined by the deviation from the desired horizontal distance between the risers

∆e_rel = ∆x_d− ∆x^R12, (6.18)

∆x_R12 = x_R2− xR1, (6.19)

where ∆x_d is the vector of the desired distance between the risers in each node,

∆x_R12 is the horizontal distance between two neighboring risers, and x_Rj is the horizontal position vector of riser j. For a riser segment, the scalar values are found in Fig. 6.5. ∆e_rel is the (scalar) deviation from the desired R2 position, R2_d, in the riser segment shown here. The maximum deviation along the riser is defined as k∆erelk^∞. The max deviation is given a value in diameters, D, depending on the regime. The monitor comparing values in Table 6.1 is based on a desired relative riser distance ∆x_d= 15D, which is the distance at the top and bottom end points used in the simulations here at 1200m water depths. The model error vector e_p used in the calculations of the monitoring signals is mainly found from the maximum deviation from the desired riser distance. In some cases additional conditions on the horizontal R2 velocity, ˙xR2, or the maximum tension variations for a given period of time |∆T (t, t0)|max are included. The switching conditions for the error calculations are found in Table 6.1.

The relative velocity vector ∆ ˙e_reltells how fast the risers are moving towards or away from each other, where how fast they are moving towards each other is of main importance. This number will be small in most cases since the risers move

Regimes Monitor comparing values µp ρ σ

Normal 1 k∆erelk^∞= 0D µ1 1 1

Normal 2 k∆erelk^∞= 1.5D

µ₂ 2 1

& |∆T (t, t0)|max≤ 20kN

Slowly varying k∆erelk^∞= 2.5D µ3 3 2 Fast changing k∆erelk^∞= 2.5D

µ₄ 4 3

& | ˙xR2|max≥ 0.6m/s

Error state k∆erelk ≥ 13D µ₅ 5 2

Table 6.1: The switching conditions.

slowly, such that the variance in position will be captured by considering the relative distance. Furthermore, if both risers are moving fast, for instance due to WF TLP motions, ∆ ˙e_rel would not necessarily capture these motions, since the relative motion might be small. On the other hand, if ∆˙e_rel is small, it may not lead to an instant collision. By considering only ˙xR2, the fast TLP motions are captured. If R2 is exposed to varying drag forces due hydrodynamic interaction, this could be captured both by considering ∆˙e_rel and ˙x_R2. For activation of the dynamical controller, we will consider ˙x_R2.

The scalar model error e_p for each regime p ∈ P is given below. The model error closest to the definition of each regime is the smallest, and its index number is used to decide the controller.

Normal 1, p = 1

This regime is defined as when the risers are parallel, with the desired horizon-tal distance between the nodes at the each water depth. Using the monitoring comparing values from Table 6.1 gives the following model error and γ-function:

e₁ = ∆e_rel− 0D, (6.20)

γ₁ = γ(k∆erelk) = k∆erelk^∞. (6.21) Hence, the maximum deviation from the desired position determines the error and the monitoring signal.

Normal 2, p = 2

In addition to the deviation for the desired relative horizontal distance shown above, the variation in tension is also included. The function T (t, t₀) is the tension for a given time window, ses Fig. 6.6. t is the current time, and t₀ is the oldest time for which information is kept, for instance t₀ = t − 100s. If the

T_max

T_min

∆|T (t, t0)|max

t₀ t

ToptensionT(t)

Time Time window

Figure 6.6: Time window for the top tension.

tension difference between maximum and minimum tension in the time window is smaller than the boundary, here 20kN, the error contribution from tension is set to zero. Otherwise, the errors are weighted and summarized.

|∆T (t, t0)|max = |T (t, t0)|max− |T (t, t0)|min, (6.22) e_2,T = 0 for |∆T(t, t0)|max≤ 20kN, else (6.23) e_2,T = k_T|∆T (t, t0)|max, (6.24)

e_2,∆x = k∆erel− 1.5Dk^∞, (6.25)

e₂ = [ e_2,T e_2,∆x ]^T, (6.26)

γ2 = γ2(ke²k) = q

(e2,T)²+ (e2,∆x)², (6.27) where k_T = 1e⁻⁵, such that for |∆T (t, t0)|max> 20kN, the error contribution for tension will be e_2,T > 0.2, whereas the contribution from the relative horizontal distance with deviation 0.5D gives e_2,∆x= 0.15. For a smaller tension deviation, the error from the relative horizontal distance is the only contribution.

Slowly varying, p = 3

The slowly varying regime is for LF motions and tide giving medium deviation from the desired relative distance. Like for the regime normal 1, this regime does

only consider the relative horizontal distance. Hence,

e₃ = ∆e_rel− 2.5D, (6.28)

γ₃ = γ₃(ke3k) = ke3k^∞. (6.29) Fast changing, p = 4

For the fast changing regime, the risers experience faster motions due to the TLP motions or current. This can be monitored through the horizontal velocity vector of R2, ˙x_R2, which is included in this error model in addition to the relative riser distance. Since we want to capture the fast motions, we use a high-pass filter, corresponding to the WF motions with periods T = 5 − 20s. We are interested in the largest velocity and how fast it changes. The largest horizontal velocity is found by

˙x_R2,max= k ˙xR2k^∞. (6.30)

If ˙x_R2,max is larger than the boundary value given in Table 6.1, we have e_{4, ˙x} = 0.

Otherwise the error is given by e_{4, ˙x} = 0.6m/s − ˙xR2,max. Since the risers are changing the direction according to the specified motion from the TLP, the re-sulting error will be chattering between zero and the calculated value. A low-pass filter is therefore introduced to give a mean error contribution from the veloc-ity. The contribution from relative distance is weighted to be of less importance.

Hence,

e4,∆x = 1

2k∆erel− 2.5Dk^∞, (6.31)

e₄ = [ e_4,∆x e_{4, ˙x} ], (6.32)

γ₄ = ke4k1 = |e4,∆x| + |e4, ˙x|. (6.33) Error State, p = 5

Instead of finding the max-deviation from a desired configuration, we here inves-tigate if the closest node is in a (near) collision situation. This could be written

e₅ = ∆e_rel− 13D, (6.34)

γ₅ = γ₅(e₅) = |e5|min. (6.35) This regime will also be activated if the risers are far apart, for instance if the second riser has a large deflection.

6.4.3 Switching Logic

The switching logic decides which controller to use through the controller switch-ing signal σ, based on the value of the monitorswitch-ing signal µ_p in (6.15). The

scale-independent hysteresis switching logic from Section 6.3.3 is used. The state-diagram (Fig. 6.3) shows how the process switching signal ρ is found based on the smallest monitoring signal µ_p. The mapping σ = χ(ρ) is found in Table 6.1.

λ_p is the forgetting factor for µ_p. A larger factor λ gives faster forgetting, while a smaller λ gives longer memory. To decrease the detection time when active control is needed, the memory is slightly decreased by choosing λ_p slightly bigger for p ∈ {3, 4, 5}. Rewriting (6.15) on vector form we have

˙

µ= −λµ + γ(kepk), (6.36)

where ˙µ is the vector of monitoring errors, λ is the diagonal matrix of non-negative constants (here λ = diag

1 1 1.5 2 1.5 T

), and γ(kepk) is the resulting vector of the class K-functions. The initial monitoring value is set to µ(0) =

0.9 1 1 1 1 T

> 0.