# Optimisation of a Gas-Lifted System with Nonlinear Model Predictive Control

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## Abstract

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## 1. Introduction

## 2. Gas-Lifted System: Models and Analysis

#### 2.1. Gas-Lifted System Models

#### 2.2. The Stable and Unstable Gas-Lifted System

#### 2.3. Bifurcation

#### 2.4. Linearised Gas-Lifted System

## 3. Terminal Equality-Constrained Nonlinear Model Predictive Control (NMPC) with Input Target and Control Zones

#### 3.1. Recursive Feasibility

#### 3.2. Convergence

## 4. Stabilisation of Gas-Lifted System Using Terminal Inequality Constrained NMPC with Input Target and Control Zones

#### 4.1. Undisturbed Gas-Lifted Well Stabilization Using NMPC with Input within Bound: One Input Case

#### 4.2. Undisturbed Gas-Lifted Well Stabilisation Using Terminal Equality Constrained NMPC with Input out of Bound: One Input Case

#### 4.3. Disturbed Gas-Lifted Well Stabilisation Using NMPC with Input within Bound: One Input Case

#### 4.4. Disturbed Gas-Lifted System Stabilisation Using Terminal Equality Constrained NMPC Having Input within Bound: Two Input Case

#### 4.5. Compare with PI Controller

#### 4.6. Comments on Validation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A gas-lifted system showing the states and key algebraic variables. Gas is delivered from the surface compressor station to the system through the gas-lifted valve, and the produced mixture is removed from the volume through the production choke. [Source: by author].

**Figure 2.**States of a gas-lifted system for $u=0.65$ in a stable mode. These states are the mass of the gas in annulus ${x}_{1}$, mass of the gas in tubing ${x}_{2}$ and mass of the oil in tubing ${x}_{3}$. All three states are positive, including other algebraic variables of the system. The states converged to fixed values as there was no oscillation.

**Figure 3.**States of a gas-lifted system for $u=0.95$ in an unstable mode. The states do not converge to fixed values but oscillate with a period of about 53 min. This oscillation is not healthy for the downstream equipment and it also reduces the mean oil production.

**Figure 4.**Bifurcation diagrams for GOR = 0.0001, 0.001, 0.01 and 0.1. Mean production (red solid) is reduced after the system goes into instability. A GOR of 0.1 shows no instability for any input value.

**Figure 5.**Step response of the linearised gas-lifted system in stable and unstable modes. The output is the downhole pressure, and the steady states correspond to input values from $u=0.65$ to $u=1.00$.

**Figure 6.**Linear and nonlinear states of the gas-lifted system for u = 0.70. The linear states do not converge to the nonlinear state.

**Figure 7.**Optimal and estimated states of the gas-lifted system. The optimal states are the true states of the system, while the estimated states are the EKF outputs. The states are within the state zones, and the states are stabilised by NMPC despite the input target falling within the unstable region.

**Figure 8.**Oil production rate at 95% valve opening (u = 0.95). The optimal production from the controller (black solid line), unstable production (red dashed line) and mean production (blue dash-dotted) were obtained at GOR = 0.01, while the stable production (green dotted line) was obtained at GOR = 0.4.

**Figure 9.**The states of the gas lift system when the desired input (${u}_{des})$ switches from the input bound to out of the input bound. The state’s bounds are still respected.

**Figure 10.**Desired input (blue dash-dotted), optimal input (black solid) and the input bounds (green dash) when the desired input switches from within bound to out of bound. The optimal input approaches the upper bound as far as possible but does not converge to it.

**Figure 11.**Control cost function for the gas lift system. The first part corresponds to ${u}_{des}$ = 0.65, which is in the stable region. The second part corresponds to ${u}_{des}$ = 1.25, which is out of the input bound. The cost function decreases to zero for the first part that the input is reachable but leaves an offset for the unreachable part

**Figure 12.**States of the gas-lift system with disturbance occurring between the 60th and 65th minutes. The NMPC restored the states to their zones after the disturbance made the states to temporarily violate the control zones.

**Figure 13.**Two inputs of the gas-lifted system. The second optimal input follows the desired input ${u}_{des}$ = 0.4 kg/s as close as possible.

**Figure 14.**Control cost function for the gas-lifted system. The first part corresponds to the time before the disturbance’s arrival, and the second part corresponds to the part after the disturbance’s arrival. The cost function decreased to a value permitted by time for the first part but decreased to zero for the second part

**Figure 15.**Optimal inputs when the desired inputs are out of input bounds. The two inputs saturated on the upper boundary of the control input even when disturbance arrived. This helped the attenuated disturbance effect very quickly.

**Figure 16.**States of the gas-lift system with disturbance occurring between the 120th and 125th minutes. The saturated optimal inputs ensure faster disturbance attenuation than when the input target is within the limit.

**Figure 17.**Stablised states of the gas-lift system with PI controller. The oscillation is damped out slowly compared to the use of NMPC.

**Figure 18.**Input and output of the gas-lifted system using PI controller. The PI controller opens the valve gradually from the equilibrium point to $u=0.95$, which is in the unstable region, making the oil production steady at 9.03 kg/s.

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**MDPI and ACS Style**

Adukwu, O.; Odloak, D.; Kassab, F., Jr.
Optimisation of a Gas-Lifted System with Nonlinear Model Predictive Control. *Energies* **2023**, *16*, 3082.
https://doi.org/10.3390/en16073082

**AMA Style**

Adukwu O, Odloak D, Kassab F Jr.
Optimisation of a Gas-Lifted System with Nonlinear Model Predictive Control. *Energies*. 2023; 16(7):3082.
https://doi.org/10.3390/en16073082

**Chicago/Turabian Style**

Adukwu, Ojonugwa, Darci Odloak, and Fuad Kassab, Jr.
2023. "Optimisation of a Gas-Lifted System with Nonlinear Model Predictive Control" *Energies* 16, no. 7: 3082.
https://doi.org/10.3390/en16073082