Firouz Ardeshirian,Mehdi MansuriNamvaran Consulting Engineers

Tehran

An analysis proved controllability and determined the appropriate control valve to use to increase the oil retention time in an offshore oil separator.

The analysis used the Nyquist stability theory for determining the controllability of the separator system against three types of control valves.

This analysis was performed on oil/gas/water separators on the Aboozar offshore project in the Persian Gulf.

A dynamic compensator in the form of a PID (proportional integral derivative) controller in a unity feedback system is used to enhance the closed-loop performance.

Time and frequency response of the three control valves in a single-input, single-output feedback control system using the PID feature were analyzed in detail for the level control of an oil production separator, taking into consideration the limitation in storage capacity.

It was found that by both correct tuning of the PID controller and utilizing a linear control valve, the time response of the system will satisfy the process needs, although the equal percentage control valve demonstrates a rather desirable level control.

It was also shown that a quick-opening control valve will create an unstable control loop.

### Alternatives

The client requested ways to increase the volume of retained oil in two offshore oil production separators, each 2.8 m OD by 10 m length. The separators are in parallel and had been designed for specified residence time to provide adequate time for gravity separation of oil, gas, and water in a smooth flow pattern.One solution was to increase the separator length or the diameter. Because of the limited space available, an increase in weir plate height instead of an increase in separator dimensions was an alternate solution.

With this alternative, higher liquid storage is maintained upstream of the weir plate, hence longer residence time to separate oil and water. This meets the project requirements.

On the other hand, with an increased weir plate height the normal oil level downstream of the weir plate had to be kept at a lower height than the weir plate height for gas/oil separation purposes.

This will result in a constraint for the oil level control system because the oil storage capacity downstream of the weir plate becomes limited, and the sensitivity of the oil level due to change in input and output flow will increase.

This problem initiated the study of the level control system and subsequent analysis of control valve characteristics and process controllers to meet the requirements.

### Separator system

Fig. 1 [104,441 bytes] illustrates the separator piping and instrumentation and the closed-loop control system.The inlet liquid to the separator arrives from the oil field as a two-phase flow regime, mainly in slug form. Therefore, the gas, oil, and water volumes vary according to the natural flow.

The outlet oil from the separator is sent ashore with pumps downstream of the separator through a 70 km submarine pipeline.

The instrumentation and control system has to maintain control of flow, pressure, and level in smooth operations with minimum turbulence.

From a control point of view, the process variables are pressure, gas flow, liquid flow, and liquid level. The variables interact with each other.

Instrumentation and interactive control loops using split-range technique were established to meet the control requirement.

### Liquid level

From a separation point of view, because the liquid level control is the main concern, it was unnecessary to study an elaborate multi-input/multi-output system for assessing the detailed behavior of the interactions and their dynamic response.Therefore, a feedback control system for the liquid level only is analyzed, taking into consideration the split-range features of the transmitters and the controllers.

The split-range signals are incorporated into the block diagram for further consideration of signal deviation from the normal operating point.

Sudden change in input flow due to slug flow entering the separator was included as a disturbing signal to the closed-loop system.

Because of the system complexity, the dynamics of each part of the system are considered individually and after combining them together, the overall process dynamics of the system are derived.

### Calculations

The separator dynamics are determined using a material balance in the vessel as shown by Equation 1 (Equations box).The variables in the equation are as follows:

- V = Volume of oil downstream of the weir plate
- r = Oil specific gravity, assumed constant in this study
- Q
_{in}and Q_{out}= Separator oil input and output mass rates.

Using partial differentials, Equation 1 can be rewritten as Equation 2.

In Equation 2, dV/dh represents oil volume change with respect to the oil level. This is a nonlinear function of h and can be calculated for each vessel numerically. It is referred to as "vessel characteristic."

In normal operations, the input flow to each separator is a constant 50,000 b/d. Process investigations and sampling tests show that almost 70% of the flow is crude oil and the remainder is water.

Therefore in normal operations, the oil input flow is 35,000 b/d. Deviations from this value will be regarded as disturbance. The disturbance effects will be rejected during the controller action.

Separator output flow is affected by a control valve located downstream of the pump and upstream of the pipeline. Fig. 2 [56,628 bytes] shows the configuration.

In this figure, Q refers to the output mass flow rate from the separator. The P_{3} refers to pump suction pressure, and P_{2} and P_{1} refer to control valve inlet and outlet pressures, respectively.

P_{3} can be related to P_{i}, the separator operating pressure, by Equation 3, in which g is the gravitational acceleration constant.

The relation between the changes in flow and system pressures can be modeled in the form of Equations 4-6.

Pump characteristics are generally expressed by pump curves. These curves relate the pump head (difference of pump discharge and suction pressures) to the flow through the pump. This can be expressed by a nonlinear function f_{1}(Q) as defined by Equation 4.

Fig. 3 [49,242 bytes] depicts the pump curve considered in this study. The flow through the control valve is generally specified in the form of Equation 5.

In Equation 5, _{x} is the control valve stroke as a percent of the full open position of the valve. It is the command signal by the controller (control variable).

C_{v} is the valve sizing coefficient, and K_{i} determines the control valve characteristic that in this study is limited to equal percentage, linear, and quick-opening cases.

Fig. 4 [69,287 bytes] illustrates the curves of flow percentage vs. percentage of valve opening.

The remaining part of the system that has to be modeled is the submarine pipeline to onshore storage tanks. This pipeline was designed to transfer the produced oil of six separators.

In the sequel, it is assumed that the oil outlet of the other five separators remains unchanged during the time that the studied separator rejects the disturbance or changes its set point. Therefore, the flow through the pipeline is equal to 175,000 b/d plus the oil outlet of the studied separator, which may vary during the control action.

To compute the pressure drop over the pipeline, some hydraulic investigations from a hydraulic curve demonstrate the pressure drop vs. Q. This curve will be referred to as the pipeline characteristic and can be expressed by a nonlinear function f_{2}(Q) as shown in Equation 6.

The pipeline pressure drop varies when flow Q or pipeline upstream pressure changes. Therefore, during transient response analysis, the pipeline upstream pressure is considered as a time-dependent variable.

For this, a ramp is considered as a function for P_{1} against change in flow.

To compute the flow Q for a given control valve at a specified stroke _{x} and at a level h, Equations 3-6 are solved simultaneously. The resulting Q will be applied to Equation 2 that determines the separator level h.

The full nonlinear characteristic of each part is considered in the present study, and all of the time response simulations are based on these models.

But to design a PID controller for the system, the characteristics of all the components are linearized around their operating points. Then the general block diagram of the closed loop system and disturbance effect is as shown in Fig. 5.

In Fig. 5 [66,029 bytes], R(s), C(s), and D(s) represent the Laplace transform of set points, controlled output, and disturbance, respectively. G(s) is the open-loop linearized transfer function of the system that is defined as a second order transfer function with time delay as shown by Equation 7.

The variables in Equation 7 are as follows:

- 1. Control valve gain that has a nonlinear characteristic. The value of this gain depends on the control valve type and its operating point. From the control valve C

_{v}/stroke curves (Fig. 4), the corresponding characteristic is used.

2. Separator horizontal cross-section area in the downstream of weir plate that varies with respect to the liquid height.

3. Pump hydraulic gain at an operating point corresponding to the pump hydraulic curve and pump suction pressure.

4. Pipeline pressure drop against flow.

K(s) is a feedback compensator of the conventional PID form with the transfer function considered as shown by Equation 8.

The variables in Equation 8 are as follows:

- K
_{p}= Proportional gain of the compensator - T
_{i}= Integral time - T
_{d}= Derivative time - n = Index for quality of the derivative action. The n varies from 1 to 100 depending on the quality of the controller. A higher n value represents a higher controller quality.

### Tuning the parameters

To tune the parameters of the PID feedback controller, an approach is used that was originally proposed by John G. Ziegler and Nathaniel B. Nichols. This technique has maintained its validity because of its simplicity and applicability to a wide range of processes.In the Ziegler-Nichols frequency response method, the dynamics are characterized by two parameters that are the frequency where the open-loop dynamics have a phase shift of 180° and the gain at that frequency.

The design is based on knowing the point on the Nyquist curve of the process transfer function where the Nyquist curve intersects the negative real axis. This point is characterized by the parameters M and v_{c}, which are called ultimate gain and the ultimate frequency.

The Ziegler-Nichols design method gives simple equations for the parameters of the PID controller.

The information of the ultimate point is computed for the three control valves being considered. Table 1 [30,275 bytes] summarizes the results.

As shown in Table 1, the v_{c}is the same for the three valves and the only difference is the M. This point was predictable because the three control valves only differ on a nonlinear gain according to their characteristic. Other parts of the system, including all of the process dynamics, are the same. Using the values v

_{c}and M and incorporating them in the Ziegler-Nichols formula, one derives the PID controller parameters K

_{p},T

_{i}, and T

_{d}.

To assess the closed-loop system stability while taking into consideration the nonlinearity of the overall system, Lyapunov's linearization approach was adopted.

This method is concerned with the local stability of a nonlinear system. It is a formalization of intuition that says that a nonlinear system should behave similar to its linearized approximation for small motion ranges.

Because all physical systems are inherently nonlinear, Lyapunov's linearizaton method serves as the fundamental justification of using linear control techniques in practice, and therefore shows that stable design by linear control guarantees local stability of the original physical system.

Based on this theoretical discussion, the Nyquist criterion is used to examine the closed-loop system stability. Fig. 6 [54,893 bytes] shows the computed Nyquist frequency response diagram of the closed-loop system with predefined PID parameters.

Because the Nyquist contour does not enclose the critical point (-1,0), this implies that the feedback system is stable.

The instability of the level control system can be assessed by changes in the tuning parameters of the PID controller. It can be shown that poor tuning could result in unwanted throttling of the control valve and potential shutdown of the system.

### Loop analysis

Simulink software was used to develop the closed-loop system time response. For this, the block diagram in the overall configuration as shown in Fig. 7 [107,390 bytes] was designed and the details were introduced in the software.The closed-loop system time response of the liquid level and of the control valve stroke with respect to a change in input flow, and also with respect to a change in set point, are computed and shown in Figs. 8a-c [123,320 bytes].

These figures show the time response against a sudden change in input flow for the three types of control valves (linear, equal percentage, and quick opening) at time t = 10 sec.

Figs. 9a-b [90,995 bytes] show the time response against change in the set point for linear and equal percentage control valves at t = 10 sec. For all three control valves the normal operating point was considered at 70% of valve opening.

It is seen that both linear and equal percentage control valves demonstrate a desirable time behavior and steady-state condition. The linear control valve is superior over the equal percentage control for controlling the liquid level valve.

Although both linear and equal percentage control valves demonstrate a desirable level control, the linear control valve responds quicker and settles faster that the equal percentage control valve.

The quick opening control valve demonstrates an unstable condition and the liquid level cannot be controlled with this type of control valve.

The reason for this instability is because at the operating point and its vicinity the quick opening control valve introduces low gain in the system dynamics.

To obtain a desirable control response, a PID controller with higher proportional gain, in comparison to the other type of control valves, has to be employed (Table 1).

When a decrease in input flow (20% in this study) is introduced into the separator, the control system commands the control valve to close. At the greater closing positions, the quick-opening valve demonstrates higher gain.

This higher gain multiplied with that of the PID gain produces very high gain. (For example, the product of K(s) and G(s) transfer functions will have very high gain.) This will cause the relevant Nyquist diagram to encircle the critical point (-1,0), proving that the unity feedback system will become unstable.

### Bibliography

- 1. Astrom, K.J., and Hagglund, T., "Automatic tuning of PID controllers," Instrument Society of America, 1988.

2. Simulink-A program for simulating dynamic systems, User's Guide, Mathworks Inc., 1994.

3. Deshpande, P.B., and Ash, R.H., "Elements of computer process control with advanced control applications," Instrument Society of America, 1981.

4. Slotine, J.J.E., and Li, W., Applied nonlinear control, Prentice-Hall, 1991.

5. Mansuri, M., and Nikravesh, S.K.Y., "Optimal setting of commercial controllers," Iasted Conference, Innsbruk, Austria, 1991.

6. Kaya, A., and Scheib, T.J., "Tuning of PID controls of different structures," Control Engineering, July 1988.

7. Van Doren, V.J., "Basics of Proportional-Integral-Derivative control: Tuning fundamentals," Control Engineering, March 1998.

### The Authors

Firouz Ardeshirianis managing director of Namvaran Consulting Engineers Co., Tehran. He has been involved in project engineering management and design activities in the oil and gas industries since 1974.Ardeshirian holds a BS in mechanical engineering from Sharif University of Technology, Iran, and an MS in Automatic Control Systems from the University of Manchester Institute of Science & Technology.

Mehdi Mansuriis senior instrument and control engineer with Namvaran Consulting Engineers Co. Mansuri holds a BS in electrical engineering from Sharif University of Technology, Iran, and an MS in electrical engineering from Amir Kabir University of Technology, Iran.

*Copyright 1998 Oil & Gas Journal. All Rights Reserved.*