## 1. Introduction

Since the late 1960s, the North Sea has been home to a booming Oil and Gas industry and has provided substantial quantities of petroleum products [

1]. Reservoirs that are located beneath the ocean seabed were initially pressurized with high concentrations of oil and gas and relatively small amounts of water. The long period of oil exploration has had its toll on the reservoirs, wherein natural water leakage into the reservoirs and water re-injection have changed the balance of oil, gas and water concentrations [

2]. Presently, the fluids pumped from the reservoirs has in many cases a

$98\%$ water fraction, and, due to conveyance costs, the water must be separated from the product on the offshore facilities [

3,

4,

5]. In the Danish sector, the water fraction has increased by

$350\%$ over the period of 2005 to 2015 resulting in an increasing strain on the separation facilities who must adhere to discharge regulations [

6]. The current regulation for the Oil in Water (OiW) concentration in the discharge into the North Sea is 30 mg/L [

7]. In most cases, the OiW discharges from the North Sea operators can be maintained far below the required limits; for example, the average OiW concentration for the year 2011 in the Danish sector of the North Sea was 4.8 mg/L [

8]. The limit is exceeded occasionally, for example, in 2008, there were 25 instances where the monthly average limit was exceeded in the Danish sector of the North Sea, although the number of such instances has fallen drastically and the limit was exceeded just once in 2012 [

8]. It is, however, vital to keep in mind that the OiW in the discharge is measured offline using a Gas Chromatography with Flame-Ionization Detection (GC-FID) instrument, where at least two samples per day in accordance with the OSPAR Convention’s OSPAR reference method (ISO 9377-2 GC-FID) are taken [

9,

10], and that above exceptions is a monthly average. Thus, statistically, the number of discharges that exceeded the limits is potentially greater and could be very high OiW concentrations at times. Another important factor is not merely the oil discharge concentration but the total bulk of oil discharged, which is increasing due to the increase in the total liquid volume that is produced. In 2015, the total mass of oil discharged into the North Sea was 193 tonnes, whereas the limit set by the Danish Government is 202 tonnes per annum [

11]. It is thus of high importance to continually improve the de-oiling facilities and their operation and control, to have a steady discharge and keep the OiW concentration below the required limit and approach a zero discharge. This article focuses on improving the performance of a de-oiling facility using a novel control solution.

The most common units of a de-oiling unit process are a three-phase gravity separator followed by a de-oiling hydrocyclone separator [

12,

13,

14,

15,

16,

17]. The control of such a conventional de-oiling system is performed by two Proportional-Integral-Derivative (PID) controllers that control the level in the gravity separator and the pressure drop ratio (PDR) across the hydrocyclone. A Piping and Instrumentation Diagram (P&ID) of the de-oiling system that is considered in this study can be seen in

Figure 1. In previous work, it was shown that such a system is sensitive towards fluctuating inlet flow rate [

15,

18,

19,

20], which is a re-occurring phenomenon in such installations and in most occasions it is caused by slugging flow regime in the upstream pipeline system [

21,

22,

23]. An investigation of the offshore data, as shown in

Figure 2, shows a typical performance of a conventional de-oiling controller, consisting of two individual PID controllers, a level and a PDR controller, as shown in

Figure 1, during a fluctuating inlet flow rate.

Here, the fluctuations are directly transmitted into an actuation of the underflow valve

$\left({V}_{u}\right)$ which aims a maintaining the level, and by doing so disrupts the PDR value, which in many cases exceeds its boundary, as indicated in

Figure 2 by the horizontal dashed lines. Due to the relatively larger physical size of the underflow valve and the physical coupling in the system, it dominates the overflow valve

$\left({V}_{o}\right)$ that is used to control the PDR [

24]. This leads to the overflow valve chattering at almost fully closed position for long periods of time, which results in excess oil being sent through the underflow along with the water, and if this is not recirculated through the de-oiling system, the oil will be discharged into the ocean. In the opposite case when the overflow valve is fully open, excess water will be sent through the overflow, which, again, necessitates recirculation in order to purify the overflow. It was shown in [

14,

25] that a hydrocyclone operates at a high efficiency as long as the pressure drop ratio (PDR) and the inlet flow rate to the hydrocyclone are kept within certain boundaries—1.5–3 with respect to the PDR [

26]—and the boundary of the inlet flow rate to the hydrocyclone is specific to each individual hydrocyclone set-up [

25]. Furthermore, it was shown that the PDR has an insignificant effect on the separation efficiency as long as it is kept above 1.7 PDR [

27], and that the inlet flow rate to the hydrocyclone must be kept above a system specific threshold in order to ensure high separation efficiency [

27], which in some cases has been reported to be as high as 99% [

25]. It is also noteworthy that the inlet flow rate to the hydrocyclone has no influence on the PDR, as was shown in [

27], and in most cases only the PDR is controlled, and the inlet flow rate, neither to the gravity separator nor to the hydrocyclone, is controlled. In addition, the explicit reference tracking of the PID, and especially the level control structure, amplifies the disturbance transmission through the system, i.e., the inlet flow rate to the gravity separator is transmitted to the hydrocyclone, which affects the de-oiling systems performance as discussed in [

24,

27,

28]. Thus, as in most cases, the inlet flow rate is not directly measured, and the flow rate in some cases is only a part of a secondary objective, the flow through the hydrocyclone varies with the gravity separator inlet flow rate [

18]. We believe that the performance of the de-oiling system can be improved upon by introducing a new control solution that addresses some of the aforementioned challenges.

The conventional de-oiling system consists of a multiple input and multiple output (MIMO) layout, but it is in most cases controlled by a set of single input single output (SISO) PID controllers that may or may not be governed by some sort of function based controller strategy [

15]. Most of these controllers are designed and tuned in an ad-hoc manner. This type of system would greatly benefit from an advanced model based MIMO control strategy, where the two control objectives are explicitly handled. As reference tracking in the de-oiling system results in poor overall system performance, we aim at reducing the emphasis on the reference tracking while ensuring that the system stays within its boundary. As was proposed in [

27], a robust H

${}_{\infty}$ control solution is employed, as it promises to handle the first objective and enables the relaxation of the reference tracking and meanwhile ensures that the system is kept within a boundary. The development of such a control strategy requires a process MIMO model of the system, and as such models are currently unavailable, a MIMO model was developed as part of this study. The article is organized as follows. The MIMO process model is described in

Section 1.

Section 2 describes the development of the robust control solution based on this MIMO model. A description of the PID benchmark controller can be seen in

Section 3. An analysis of the performance of both these controllers in simulations and as implemented on a scaled pilot plant is presented in

Section 4 and

Section 5, respectively.

Section 6 and

Section 7 contain a discussion of the results and conclusion of the study, respectively.

## 2. De-Oiling Process Model

The de-oiling process starts with a mixture of oil, water and gas entering the gravity separator at its inlet valve. Here, the gas and oil droplets larger than 150

$\mathsf{\mu}$m are separated and thereafter the effluent from the gravity separator flows into the hydrocyclone where the residual oil droplets up to 15

$\mathsf{\mu}$m are removed [

29,

30]. In the gravity separator, the oil, which is the lighter phase, collects on the surface of the water phase where the floatation time of the oil droplets is governed by stokes law. The residence time of the liquid in the gravity separator is controlled by the level of the interface between the oil and the water as shown in

Figure 1, which is controlled by the flow of liquid through the underflow valve

$\left({V}_{u}\right)$ located on the hydrocyclone see

Figure 1. As the level in the separator and the opening of the underflow valve are the main variables in the control loop that controls the gravity separator, these are included in our model. For more detailed information on the operations of the gravity separator, refer to [

13,

28,

31]. The hydrocyclone works on the principle of centripetal/centrifugal force, which is created by the cyclic motion of the liquid inside the hydrocyclone. The centripetal force pushes oil droplets towards the centre of the hydrocyclone and out through a narrow outlet on one side of the hydrocyclone called the overflow, and the water exits through a larger underflow valve [

32,

33,

34,

35]. The flow through the overflow is controlled by a valve referred to as the overflow valve

$\left({V}_{o}\right)$, as shown in

Figure 1. By adjusting this valve, the pressure drop over it is changed, and, consequently, the pressure drop ratio (PDR) of the hydrocyclone will change, which is defined below:

The PDR controller operates on the principle that the PDR is approximately linearly proportional to the flow split ratio

$\left({R}_{f}\right)$, defined as

$({R}_{f}={F}_{u}/{F}_{i})$, which is the ratio of the hydrocyclone’s inlet volumetric flow rate

$\left({F}_{i}\right)$ to the underflow volumetric flow-rate

$\left({F}_{u}\right)$ [

15,

19,

25,

36,

37]. If it is assumed that the flow inside of the hydrocyclone is sufficient to create an strong centripetal force, thus forcing all of the oil into an oil core, then in theory controlling the flow through the overflow will control the flow of oil through the hydrocyclone. The reason for the use of PDR control and not simply controlling the flow of liquid or oil through the overflow is the lack of flow measurements. In addition, steady state analyses have related the

${R}_{f}$ to the hydrocyclone’s efficiency

$\left(\u03f5\right)$, defined as (

$\u03f5=1-\frac{{C}_{u}}{{C}_{i}}$) [

15,

25,

34,

36], where

${C}_{i}$ is the concentration of oil in the inlet to the hydrocyclone and

${C}_{u}$ is the concentration of oil in the underflow of the hydrocyclone. Due to the common use of the PDR as the controlled variable, and the influence of the underflow and overflow valves on the hydrocyclone’s dynamics, they were chosen as the controlled and manipulative variables in our considered model.

#### 2.1. Model Development

A simplified model of the gravity separator is obtained by applying mass balance equations adopted from [

31].

where

A is the cross sectional area of the separators water section,

L is the length of the separators water section,

l is the water phase’s height (interface level),

${F}_{in}$ is the liquid feed rate,

${C}_{v}$ is the valve coefficient,

$f\left(u\right)$ is the valve’s characteristics of the openness area related to the openness percentage

u,

$\Delta {P}_{out}$ is the pressure drop over control valve, and

${\rho}_{w}$ is the water phase density. As

${V}_{o}$ has a small impact on the total output flow from the gravity separator, it is omitted. Under the assumption that the separator pressure, interface level and the pressure downstream of the valve are insignificant to affect the water level dynamic, the nonlinear model is linearised around an operating point of 0.15 m. Due to the complicated hydrodynamics of the hydrocyclone, a black-box model via system identification was proposed; refer to [

38,

39], where the model is designed from the PDR perspective, as this is the sole observable parameter in most of current the installations. The hydrocyclone model is described as a set of two identified models where the first model describes the input–output relationship from the overflow valve to the PDR and the second model describes the input–output relationship from the underflow valve to the PDR. The two sets of hydrocyclone models are second order linear transfer function models. The data collected for the parameter identification were obtained at our scaled pilot plant, where the PDR was kept around an operating point of 2 PDR. The complete model of the considered system is a MIMO model, with two control inputs:

${V}_{u},{V}_{o}$, and two outputs:

$l,\mathrm{PDR}$, presented in Equation (

3). The identified model parameters for the hydrocyclone model are shown in

Table 1, where the parameter of the separator model is also shown. The disturbance

d, which represents the inlet volumetric flow rate (

${F}_{in}$), is added to the system through a weighting matrix

E, which is adjusted through an ad-hoc method to account for the algebraic conversion from valve openness to volumetric flow rate. This conversion is justified by the fact that within the narrow operating range of the linear model the valve openness is linearly proportional to the flow rate variation.

#### 2.2. Model Validation

Figure 3 illustrates the validation of the de-oiling system’s model, divided into three plots, where the first plot represents the relationship between the underflow valve and the level, the second between the underflow valve and the PDR, and the third between the overflow valve and the PDR. The simulated

l decreases linearly after the step input on

${V}_{u}$ as expected, although the model has a faster data dynamic than the experimental data One of the reasons for this is the natural drift in the pressure inside of the gravity separator, which slowly decreases with a full opening of the underflow valve. This part is neglected in the model development, as during nominal operation of the system the underflow valve will not be actuated to a fully open position for excessive periods and from this experiment we see that the deviation from the experimental data is small around the operating point but exceeds as we move away from the operating point. Both the hydrocyclone models that were identified have a good fitness to the real data, although the simulation has a slower dynamic than the real system, in particular the model describing the relationship between the underflow valve and the PDR. This is partially due to the linear model’s performance in wide operating ranges, as the underflow valve is set to actuate relatively more than the overflow valve with respect to its impact on the PDR. This is reflected in the large deviation in the PDR, from approximately 2 to approximately 1.4.

## 3. Synthesis of the Robust Control Solution

We aim at designing a controller

$K\left(s\right)$, which governs the plant as shown in

Figure 4a. Following the standard representation of the robust H

${}_{\infty}$ control solution shown in

Figure 4b, our system can be rearranged following this configuration, and the closed-loop inter-connected system

$P\left(s\right)$ is obtained based on the general definition shown in Equation (

4).

The external inputs to the system defined by

w, are the reference

$\Delta ref$ and the disturbance

$\Delta d$ and the

$\Delta $ represents the linear model’s deviation from the linearized equilibrium value. The output signals are defined as

z, and consist of

l and PDR. The vector of control signals defined as

u, consists of the input variables

${V}_{u}$ and

${V}_{o}$. The vector of the available measurements defined as

y, consists of the error signals

$\Delta {e}_{l}$ and (

$\Delta {e}_{\mathrm{PDR}}$). It can be denoted by:

where:

which leads to the matrix representation of our system’s interconnected system

P in Equation (

7).

#### 3.1. Numerical Solvability

We intend to develop the robust control solution using methods provided in [

40,

41], where the interconnected system

P has to satisfy several assumptions listed below:

#### 3.1.1. Assumption 1

First, the linear system must be controllable from the control inputs

u and observable from the measurement output

y, which is a requirement to find a stabilizing controller

K [

42]. The dynamic system consisting of the pair

$(A,{B}_{2})$ must be controllable [

43]. The rank of the controllability matrix

$Rank\left(\mathcal{C}\right)$ has been calculated using the system matrices, and it leads to a full column rank, indicating that the system is controllable. The dynamic system consisting of the pair

$(C,{A}_{2})$ must be observable [

43]. The rank of the observability matrix

$Rank\left(\mathcal{O}\right)$ has been calculated using the system matrices,

A and

C, and this leads to a full row rank, indicating that the system is observable.

#### 3.1.2. Assumption 2

${D}_{12}$ and ${D}_{21}$ must be full column and row rank, which ensures that the controller is proper, in our case ${D}_{21}$ is full row rank, while ${D}_{21}$ is not. Thus, a control weighting matrix $W=\left[\begin{array}{cc}0.01& 0\\ 0& 0.01\end{array}\right]$ is added to ensure that ${D}_{12}$ is a full column rank only for design purpose.

#### 3.1.3. Assumption 3

It must show that the matrix in Equation (

8) has full column rank for all

$\omega $ frequency range. The rank has been calculated for frequencies in the range 0–10 rad/s, with a step of 0.001 rad/s, and with respect to the considered system

P the rank of matrix in Equation (

8) is always full column rank, plus it is observed that this considered system doesn’t have any right half-plane transmission zero, thus this assumption is fulfilled.

#### 3.1.4. Assumption 4

It must show that the matrix in Equation (

9) has full row rank for all

$\omega $. The rank has been calculated for frequencies in the range 0–10 rad/s, with a step of 0.001 rad/s, and with respect to the considered system

P the rank of matrix in Equation (

9) is always full row rank; in addition, it is observed that this considered system does not have any right half-plane transmission zero, thus this assumption is fulfilled.

with all the above assumptions satisfied the controller

K can be designed.

#### 3.2. H${}_{\infty}$ Controller Synthesis

To ensure that an admissible controller $K\left(s\right)$ exists for a given $\gamma $, such that $\Vert {F}_{l}(P,K){\Vert}_{\infty}<\gamma $, i.e., to ensure the existence of the H${}_{\infty}$ suboptimal controller, a test is done which ensures that the following three conditions hold:

H${}_{\infty}\in dom\left(Ric\right)$ with ${X}_{\infty}=Ric\left({H}_{\infty}\right)\ne 0$;

J${}_{\infty}\in dom\left(Ric\right)$ with ${Y}_{\infty}=Ric\left({J}_{\infty}\right)\ne 0$; and

$\rho \left({X}_{\infty}{Y}_{\infty}\right)<{\gamma}^{2}$.

If the conditions above are satisfied then all rational internally stabilizing controller

K(

s) satisfying

$\Vert {F}_{l}(P,K){\Vert}_{\infty}<\gamma $ are given by:

where:

The H

${}_{\infty}$ controller solution is found by solving the two Riccati equations, i.e., finding the solutions to the Riccati equations

${X}_{\infty}$ and

${X}_{\infty}$, refer to Equation (

12), and the state feedback and output injection matrices

F and

L. For detailed definition of the Hamiltonian matrices

${J}_{\infty}$ and H

${}_{\infty}$, and

F and

L, refer to [

40].

The controller is numerically solved using the D-K iteration; for more information, refer to [

40,

44,

45]. The calculation of the controller

K was done using commercial software, using the

`hinfsyn` function in Matlab’s Robust Control Toolbox; refer to [

44,

46].

#### 3.3. The Designed H${}_{\infty}$ Controller

The numerically optimal upper bound

$\gamma $ value is calculated as 1.0059, and the singular values are plotted for both the open loop and closed loop systems from each of the two inputs, as shown in

Figure 5. From the singular values shown in

Figure 5, it can be seen that singular values are reduced for all frequencies with respect to both inputs. In addition, we see that the H

${}_{\infty}$ controller gives a relatively flat response as it minimizes the peak of the frequency response.

## 4. Benchmark PID Control Solution

The PID control structure is designed to emulate the offshore scenario and thus to function as a benchmark control solution for evaluation of our newly developed H

${}_{\infty}$ control solution. The most common control structure in the Danish sector of the North Sea is the one shown in

Figure 1, where there are two individual PID controllers for the two manipulated variables,

${V}_{u}$ and

${V}_{o}$, using the two feedback parameters,

l and PDR. Following this control structure, the PID solution tested in this paper was manually tuned with an aim to emulate similar performance as was observed from the real-life scenarios (we have the real-life data from the cooperated operators collected from a specific North-sea field shown in

Figure 2). In

Table 2, the offshore PID controller performance during slugging regime is compared to the benchmark PID in simulations and in our experiments; the respective data are shown in

Figure 2,

Figure 6 and

Figure 8.

From the pilot plant and offshore perspective the benchmark PID controller performs close to the offshore controller. The offshore system’s ${V}_{o}$ was saturated in a wide range of the data collected, which emphasizes the dominance of ${V}_{u}$ and the relatively small impact ${V}_{o}$ has on the amplitude of l and the PDR in comparison to the ${V}_{u}$. In general, the behaviour of the benchmark PID controller with regard to the pilot plant implementation replicates the behaviour of the offshore PID controller well, for example the PDR oscillates heavily during fluctuating flow while the level stays relatively stable. The benchmark PID control solution is thus suitable to be used for evaluating the H${}_{\infty}$, as it performs closely to the real offshore system.

## 8. Conclusions

Slugging flow, which causes the gravity separator inlet flow rate ${F}_{in}$ into the offshore de-oiling facilities to fluctuate severely, is an unmeasurable disturbance and has negative effects on the de-oiling process. The controllable parameters in the de-oiling system are the level of the gravity separator l and the pressure drop ratio (PDR) of the hydrocyclone, which are controlled by the actuation of the underflow valve (${V}_{u}$) and overflow valve (${V}_{o}$) respectively. An initial investigation showed that the current PID control paradigm and the inherent coupling of the unit processes of the de-oiling facilities, the gravity separator and the hydrocyclone, results in a system which has poor disturbance rejection. Thus, fluctuating ${F}_{in}$ is observed to propagate to the downstream hydrocyclone, which in previous studies was shown to have a direct impact on the hydrocyclone’s de-oiling efficiency $\u03f5$. It was therefore concluded that the de-oiling system would perform better if its disturbance rejection was improved. In this study, we have investigated the benefit of a robust H${}_{\infty}$ control solution in comparison to a benchmark PID control solution that is used for offshore de-oiling. A MIMO model of the de-oiling process was developed, based on which a robust suboptimal H${}_{\infty}$ control solution was designed. The H${}_{\infty}$ control solution was tested in simulations and then implemented and tested on the scaled pilot plant. In the simulations, where the H${}_{\infty}$ control solution was compared to a benchmark PID control solution, the H${}_{\infty}$ control solution facilitated disturbance attenuation and provided a more relaxed actuation of ${V}_{u}$ and ${V}_{o}$. The conventional PID control solution, however, saturated the ${V}_{o}$ while trying to track the level reference, a scenario similar to what was observed in data from offshore facilities during slugging flow.

The results from the experiments performed on the scaled pilot plant further proved that the H${}_{\infty}$ control solution was better at handling the disturbance ${F}_{in}$, and in all the circumstances that were tested, it kept the system stable. The benchmark PID control solution once again under certain circumstances saturated the controllable parameters. The reduced reference tracking of l by the H${}_{\infty}$ control solution, resulted in almost perfect damping of the transmission of the disturbance ${F}_{in}$ into ${F}_{i}$. In addition, in real life scenarios, the reduced valve actuation by the H${}_{\infty}$ control solution could reduce the wear and tear on the system.

In future work, the model of the de-oiling system will be extended to include the de-oiling hydrocyclone efficiency in the model and control design.