*3.8. Experiment 5. Adaptive Control VT.2–WCr, VT.3–PT.1*

The model reference adaptive controller was first tested on the recommended control configuration with VT.2 controlling WC*r* and VT.3 controlling PT.1. The results are shown in Figure 11. The pressure is controlled very well when using the MRAC. Note that the reference signal in Figure 11c is the output of the reference model given in Equations (22) and (23), hence the signal is filtered and the initial response has much less overshoot compared to Experiment 2. The WC*<sup>r</sup>* controller is quite slow, hence the extraction rate increases slowly to the level where the drops in WC*<sup>r</sup>* occur. When the drops do happen, they are approximately equal to the drops experienced in Experiment 2.

The adaptive gains are shown in Figure 11f,g. The gains for VT.2 are initialized at 0, but the gains for VT.3 are initialized at 50. This was found through trial and error to be a good initial value for VT.3. Numerical values for the performance can be found in Table 8.

**Figure 11.** Experiment 5. MRAC on VT.2 and VT.3 with VT.2 controlling WC*r* and VT.3 controlling PT.1, as recommended by the RGA analysis.

#### *3.9. Experiment 6. Adaptive Control VT.2–dPT.2, VT.3–PT.1*

The final experiment used MRAC for VT.2 and VT.3, with VT.2 controlling dPT.2 and VT.3 controlling PT.1. From Figure 12b, it is clear that the WC*r* and ER have similar behaviour to that shown in Experiments 3 and 4. The pressure PT.1 is controlled quite well, though with some increases in oscillations compared to that observed in Experiment 5, caused by the need for a lower adaptation gain in this experiment.

The pressure difference controller has trouble bringing dPT.2 to the reference after the first change in inlet conditions happens at *t* = 480 s. This could be caused by the large initial overshoot caused by the zero initialization of the adaptive parameters. As can be seen from Figure 12f,g, the adaptive parameters start changing direction at *t* = 480 s, but, since *γ<sup>k</sup>* and *γ<sup>l</sup>* had to be chosen to be quite small because of the slow valve, the adaptation takes a long time. Furthermore, the changes in PT.1 is causing the pressure difference dPT.2 to change. Since the VT.2 controller (and valve) is so slow, it is unable to bring dPT.2 to the reference as can be seen in Figure 12d.

Numerical values for the performance can be found in Table 8.

**Figure 12.** Experiment 6. MRAC on VT.2 and VT.3 with VT.2 controlling dPT.2 and VT.3 controlling PT.1, as recommended by the RGA analysis.

(**g**) Adapted parameters VT.3 controller

(**f**) Adapted parameters VT.2 controller

#### *3.10. Numerical Comparison*

Table 8 shows a numerical comparison of the values of interest from Experiments 1–6. The table includes the mean, root-mean-square (RMS), standard deviation (STD) and median, as well as the integrated absolute error (IAE) for the variables being controlled. The initial transient has an effect on all these numbers, hence the values are also calculated from *t* = 200 s rather than from *t* = 0 to exclude this effect. These values are showed in parentheses.

**Table 8.** Numerical comparison of Experiments 1–6. Values in parentheses are calculated after the initial transient is over, i.e., from *t* = 200 s.


#### **4. Discussion**

From the RGA analysis, it was found that VT.3 should control PT.1 in all cases. The numerical data from Experiments 1 and 2 (Table 8) shows that the control structure proposed by the RGA analysis improves the ER. The values for WC*<sup>r</sup>* is slightly worse in Experiment 2, but if the initial transient is ignored the difference is reduced. The slow valve VT.2 is controlling WC*<sup>r</sup>* in Experiment 2, which may explain why the values are worse, as it takes this valve more time to reduce the error compared to VT.3.

The numerical data from Experiments 3 and 4 (Table 8) show that controlling dPT.2 rather than WC*<sup>r</sup>* improves the separator performance. The mean, median and RMS of the WC*<sup>r</sup>* are higher and the STD is much lower in both Experiments 3 and 4 compared to Experiments 1 and 2. This comes at the cost of a lower extraction rate. This could probably be improved by finding a better setpoint for dPT.2. The differences between Experiment 3 (not RGA) and Experiment 4 (RGA) are very small when looking at WC*r*. Experiment 4 has slightly lower values in mean, median and RMS and slightly higher in STD, but, if the initial transient is ignored, the values are slightly better than in Experiment 3 (except for median). Overall, Experiment 3 has slightly better values than Experiment 4. This is the opposite of what one might expect based on the results of the RGA analysis. The RGA analysis, however, is only based on steady state behaviour and does not consider time-delays or transients. The results may indicate that the transfer functions used in the RGA analysis are significantly different from the real dynamics, i.e., the model identification in Section 2.4 may be insufficiently accurate.

The performance of the adaptive controllers are approximately equal to the performance of the PI controllers. Experiments 5 and 6 must be compared with Experiments 2 and 4, respectively.

The WC*r* is slightly higher in Experiment 5 compared to Experiment 2, but the ER is much lower in Experiment 5. This is caused by the very low adaptation gains chosen in Experiment 5, which causes VT.2 to close very slowly and, hence, less variations are present in WC*r*. The pressure control, however, is slightly improved when using the adaptive controller.

The values from Experiment 6 are very similar to those from Experiment 4. The WC*<sup>r</sup>* is slightly worse, but the ER is higher. The adaptive gains for the dPT.2 controller again had to be chosen very low, which causes sluggish control of dPT.2. This again affects the static pressure due to the interactions between the two CVs, and hence the performance is reduced for both dPT.2 and PT.1. Comparing Experiments 5 and 6, it is again clear that controlling dPT.2 rather than WC*r* improves the efficiency.

According to the information shown in Table 8, the performance of the PI controller and the adaptive controller is approximately equal. However, aspects such as implementation and ease of operation should also be considered. The PI controllers could be easily implemented in the LabView block diagram, but the tuning required step response experiments and some trial and error. The MRAC, however, did not require a step response model, but the implementation required a custom script and the tuning was largely based on trial and error and the experience of the operators. The adaptation gain for the controller operating the slow valve VT.2 had to be very low, which may have negatively affected the end result. Improving the performance significantly with tuning, however, would be difficult due to the constraints imposed by the slow valve. Finding suitable adaptation gains and initial values for the MRAC was not trivial.

The SIMC tuning rules was chosen for the PI controllers, due to its simplicity and proven efficiency for first-order plus time delay systems [16], but other tuning methods specifically designed for tuning decentralized PI controllers with two inputs and two inputs exist [22,23]. These methods may reduce the interactions between the control loops and lead to tighter control during transients, at the cost of a more complex tuning procedure.

A multivariable controller (adaptive or not) would probably outperform both controllers as it would better compensate for the interactions between the control loops. Implementing this is suggested as future work.

A model predictive controller (MPC) would also be a natural next step. The MPC can calculate the optimal setpoints and inputs while also handling the input and variable constraints. Implementing an MPC is also suggested as future work.

#### **5. Conclusions**

This paper presented a control structure analysis and controller design for a novel multi-pipe separator concept developed at the Department of Geoscience and Petroleum at the Norwegian University of Science and Technology. The control structure analysis gives an indication of which outputs to pair with which inputs, and the controller design for the conventional PI controllers is based on the well established SIMC tuning rules. Step response experiments were performed to gather data for the dynamic models of the different input/output relations in the separator. The dynamic models were assumed to be of first order with a time delay, but second order models and models accounting for the interactions between the states would probably yield better results, considering the measured system responses. Model reference adaptive controllers were also developed for the separator. The performances of the PI and adaptive controllers were quite similar, but the adaptive controller does not require a step response model in the tuning procedure. Due to a lack of tuning rules, however, the adaptive controller was quite difficult to tune. Furthermore, the adaptation gains in the MRAC had to be chosen very small due to the slow control valve VT.2. A faster valve would probably improve the results.

It was found that controlling the dP over the incline in the separator, and the water/oil interface level, by proxy, yielded a more stable water cut ratio on the water outlet, which was the primary control objective. This is due to dP control establishing a buffer volume of water in the incline, unlike when controlling WC*r* directly.

The separator is a multiple input, multiple output (MIMO) system and would probably benefit from a multivariable controller rather than two decoupled controllers. Model predictive control could potentially improve the results even more, as the separator is subject to several constraints and control objectives. Finding the optimal setpoints and outputs within the constraints is key for efficient operation. This is suggested for future work.

**Author Contributions:** Conceptualization, S.J.O., H.S.S., M.S. and C.H.; methodology, S.J.O. and H.S.S.; software, S.J.O. and H.S.S.; validation, S.J.O. and H.S.S.; formal analysis, S.J.O. and H.S.S.; investigation, S.J.O. and H.S.S.; resources, H.S.S. and M.S.; data curation, S.J.O. and H.S.S.; writing—original draft preparation, S.J.O. and H.S.S.; writing—review and editing, S.J.O., H.S.S., M.S. and C.H.; supervision, M.S. and C.H.

**Funding:** This research was carried out as a part of SUBPRO, a Research-based Innovation Center within Subsea Production and Processing. The authors gratefully acknowledge the financial support from SUBPRO, which is financed by the Research Council of Norway, major industry partners, and NTNU.

**Conflicts of Interest:** The authors declare no conflict of interest.
