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Scientific Reports volume 15, Article number: 4356 (2025 ) Cite this article stainless steal butterfly valve
In shale gas extraction, bottomhole liquid loading reduces gas well efficiency. Traditional time-based plunger lift methods use reservoir energy to remove liquid, but model-based optimization has since emerged. However, these methods, deployed on remote servers, lead to inefficient data transfer and high server loads. This study proposes an Adaptive Particle Swarm Optimization Model Predictive Control (APSO-MPC) for plunger lift optimization, implemented via edge computing. APSO dynamically adjusts inertia weights and learning factors, while a microprocessor-based edge architecture localizes computations at the controller, eliminating transmission delays and reducing server load. Simulations show APSO-MPC improves gas production by 18% compared to traditional methods, while edge computing increases data transmission by 24%, reduces packet loss by 83%, and lowers server memory and computational delays.
Shale gas is an unconventional natural gas trapped in shale formations characterized by low permeability and low porosity, making it difficult for the gas to flow freely within the rock1,2. Hydraulic fracturing and horizontal drilling have become economically viable methods for extracting shale gas3,4,5. In the initial stages of extraction, fracturing disrupts the reservoir balance, allowing liquids and gases to flow out of the pores6,7. The high-pressure gas expels most of the liquid along with the gas to the wellhead, but a small amount of liquid accumulates at the bottom of the well. In the later stages of extraction, as gas production decreases and bottomhole pressure drops, liquids begin to enter the wellbore, causing liquid loading8,9. To address this issue, intermittent shut-in and plunger lift technologies are commonly used10,11. Intermittent shut-in involves closing the gas well for a period to increase wellbore pressure and expel the liquid, though the effectiveness is limited. The plunger lift system operates by closing the production valve, allowing the plunger to descend to the bottom of the well, where liquids accumulate above it. As the plunger rises, it pushes the liquid column out of the wellbore12. Despite advancements in deliquification technologies such as foam treatment13,14, electric submersible pumps15, and downhole atomization16, these methods are costly and complex to operate. In contrast, the plunger lift system has gained popularity over the past two decades due to its ease of installation and low cost17,18. However, the lack of information on downhole pressure and reservoir inflow rates makes it difficult to determine the optimal timing for opening and closing the production valve. Operators typically set the valve timings based on empirical models, which do not adequately account for dynamic changes in the reservoir, resulting in reduced production efficiency.
To optimize the control of plunger lift systems, extensive research has been conducted by scholars. In the 1960s, Foss and Gaul first proposed a semi-empirical model to estimate the minimum pressure required to ensure that the plunger and the liquid above it reach the surface19,20. However, it lacked a solid theoretical foundation and struggled to fully capture the complexity of plunger lift operations. In 1981, Lea introduced a dynamic model to more accurately predict the acceleration of the plunger21, improving the pressure prediction made by the earlier model and optimizing plunger lift control. In 1995, Baruzzi and Alhanati developed a model for the plunger’s ascent phase based on dynamics, which calculated the minimum pressure recovery required for the plunger to achieve maximum production rate22. This model effectively reduced well shut-in time and increased production, but it only considered the dynamics of the ascent phase. In 2013, Parsa et al. combined traditional control techniques with reservoir performance analysis to develop an algorithm that optimized the well open-close cycle23, aiming to reduce valve shut-in time and maximize production. In 2017, Gupta et al. further developed a dynamic plunger lift model based on a hybrid state model that dynamically describes the entire plunger lift process12. With the continuous improvement of models, compared to traditional error feedback control methods, the approach that predicts the future system behavior based on its open-loop model, current system state, input trajectories, or disturbances entering the system to select the optimal possible input actions has potential in optimizing plunger lift operations24.
In industrial control, similar control methods are not uncommon, and the emergence of Model Predictive Control (MPC) can be traced back to the 1990s. Its core concept is to use the system’s mathematical model to predict future behavior and optimize the control input at the current time to achieve the desired goal or minimize a performance metric over a specified future period. In 2018, Nandola et al. applied MPC for online optimization control of plunger lift, but the control algorithm was deployed on a remote server18. With further development, the combination of adaptive control and predictive control led to the emergence of Adaptive Optimization Predictive Control (AOPC). Compared to MPC, AOPC can dynamically adjust control strategies in response to system dynamics and uncertainties, thereby enhancing the system’s disturbance rejection capability and robustness25. However, the optimization performance of AOPC is generally slightly lower than traditional MPC. Subsequently, the application of neural networks, especially Long Short-Term Memory (LSTM) networks, has greatly increased. Due to their outstanding performance in prediction tasks involving long time-series, nonlinear relationships, and complex dynamic systems, when combined with optimization control algorithms, they can effectively retain long-term historical information, helping the system make more accurate predictions and decisions26,27. However, despite the superior predictive performance of neural networks, they require large amounts of data, long training times, and significant computational resources. As the extraction environment becomes increasingly complex and the demand for data security and high timeliness continues to rise, avoiding large-scale remote data transmission under limited bandwidth conditions has become particularly important.
With the development of the Internet of Things (IoT), 5G communication technology, and artificial intelligence, edge computing has emerged28,29,30. Initially, the processing of large amounts of data relied on centralized systems, but as the number of IoT devices surged, edge computing began processing data closer to the source and near the end-users. This approach significantly reduces data transmission, lowers latency, and enhances real-time data processing capabilities31. Consequently, edge computing offers a viable solution to the issues mentioned above. In industrial control, the collaboration between edge computing and remote servers can effectively improve computational efficiency and real-time control performance, while enhancing data security and reducing the load on remote servers28,32. Due to the low cost, high reliability, and strong stability of hardware control systems and microcontrollers, an increasing number of control algorithms are being offloaded to hardware control systems and microcontrollers for computation33, significantly alleviating the computational burden on remote servers. However, due to the limited performance of microcontrollers, larger neural networks cannot be deployed on them. To address this issue, large predictive models can still be deployed on remote servers, while control models are deployed on edge computing devices32. However, this approach still faces the challenge of transferring large amounts of data to the remote server, leading to issues with large-scale data transfer.
To address the challenges of large-scale data transmission and the intensive computations that result in high server loads in the aforementioned optimization control, this paper builds upon previous research and proposes an edge computing-based Adaptive Particle Swarm Optimization approach for solving Model Predictive Control (APSO-MPC), aimed at optimizing the control of plunger lift systems. By embedding the computing unit into the plunger lift controller and involving it only in the calculation of the control algorithm, the feasibility and effectiveness of this method are validated within the dynamic model of the plunger lift system.
The main contributions of this paper are as follows:
1) A control algorithm for APSO-MPC is proposed. The control objective of the plunger lift system is optimally modeled using the MPC algorithm, and the APSO algorithm is established to solve the optimized output of the MPC algorithm.
2) A microprocessor based edge computing architecture is proposed. By setting the computing unit at the edge of the controller, the control algorithm is placed in the edge computing unit for computation, which avoids remote transmission of data and remote computation.
The organization of this paper is as follows: In Section “Plunger lift dynamic model” the periodic modeling of plunger lift operations is conducted, and this model is used as the controlled object model in this study. Section “APSO-MPC and edge computing implementation” describes the control optimization algorithm used and its implementation at the edge. Section “Simulation experiments” presents the simulation experiments, where the optimization method proposed in Section “APSO-MPC and edge computing implementation” is compared with the traditional timed valve switching method, and the efficiency of the edge computing approach is further validated. Finally, Section “Conclusions” provides the conclusions and summary of the work.
A typical plunger lift system is shown in Fig. 1. In Fig. 1, the controller is responsible for collecting sensor data and controlling the opening and closing of the production valves. The flow meter is used to measure the outlet gas flow.
Schematic diagram of the wellhead equipment.
Figure 2a shows the plunger lift cycle, divided into six stages triggered by manual (diamond) or autonomous (rectangular) events. The cycle starts with the valve closed, allowing the plunger to descend to the well bottom (Fig. 2b) as liquid accumulates above it and gas raises casing and tubing pressures. When the valve opens, gas flow drops tubing pressure (Fig. 2c), lifting the plunger and liquid slug. After the liquid reaches the wellhead, the plunger follows. The valve stays open for continued gas flow, then closes to repeat the cycle.
(a) Schematic diagram of various states and events in the plunger lift cycle (Gupta et al. 2017); (b)Position of the plunger and liquid slug during state changes; (c) Real data curves of pressure and flow rate during the plunger lift process.
Reservoir dynamics describe the relationship between gas and liquid flow rates from the reservoir and bottomhole pressure. During the dynamic modeling process, the IPR (Inflow Performance Relationship) model of the gas reservoir is utilized34, as follows:
In Eq. (1), \(\:{F}_{g,res}\:\) represents the gas outflow rate from the reservoir in\(\:\:kg/s\) ; \(\:{\rho\:}_{g,std}\) denotes the gas relative density in\(\:\:kg/{m}^{3}\) ; \(\:{C}_{res}\:\) is the reservoir outflow coefficient; \(\:{P}_{res}\:\) represents the reservoir pressure in\(\:\:\text{P}a\) ; and\(\:\:{P}_{wf\:}\) is the bottomhole pressure, measured in \(\:\text{P}a\) .
The relationship between the outflow of liquid and gas is as follows:
In Eq. (2), \(\:{F}_{l,res}\) represents the liquid flow rate from the reservoir in \(\:kg/s\) ; \(\:GLR\) denotes the gas-liquid ratio.
The gas then flows from the tubing through the production valve into the production line. A standard valve equation is used to simulate the flow rate through the production valve:
In Eq. (3), \(\:{C}_{\nu\:}\) represents the valve flow coefficient; \(\:{P}_{t}\) is the casing pressure in \(\:\text{P}a\) ; \(\:{P}_{l}\) is the pipeline pressure in \(\:\text{P}a\) ; and \(\:{\rho\:}_{g}\) is the gas density in \(\:kg/{m}^{3}\) .
A linear geothermal relationship is used to correlate temperature with well depth:
In Eq. (4), the formation temperature gradient \(\:{\Delta\:}T\) is approximately \(\:30k/km\) . \(\:{T}_{0}\) is the surface temperature, and \(\:x\) is the target formation depth in \(\:m\) . The static gas column pressure depends on whether it is closed.
Based on the six stages shown in Fig. 2a, a plunger lift model is established. Nandola et al. in 2017 proposed a dynamic model for plunger lift based on a hybrid state model using nine continuous state variables to capture the dynamic process18. As follows:
In Eq. (5), \(\:m\) represents mass in \(\:kg\) , with subscripts \(\:g\) and \(\:l\) denoting gas and liquid, respectively. The subscripts \(\:a\) , \(\:tt\) , and tb refer to the annulus, the tubing above the plunger, and the tubing below the plunger, respectively. \(\:{X}_{p}\) and \(\:{V}_{p}\) represent the plunger’s position and velocity in \(\:m\) and \(\:m/s\) , respectively, and are controlled by the force balance on the plunger. \(\:{A}_{r}\) represents the time from the valve opening until the plunger is detected by the sensor in \(\:\text{s}\) .
The model outputs five variables:
In Eq. (6), \(\:{P}_{c}\) , \(\:{P}_{t}\) and \(\:{P}_{l}\) represent the casing pressure, tubing pressure, and pipeline pressure, respectively. \(\:{F}_{out}\) denotes the production flow rate in \(\:kg/s\) .
Since plunger lift is a binary event, where control of the production valve switching manages the stage transitions, the model inputs are:
In Eq. (7), \(\:{u}_{1}\left(t\right)\) represents the condition for opening the control valve, and \(\:{u}_{2}\left(t\right)\) represents the condition for closing the control valve.
The plunger quickly reaches terminal velocity during descent and continues to fall at a constant speed until it reaches the liquid at the bottom of the well, where it then descends further within the liquid35. As follows:
In Eq. (8), \(\:i\in\:\{g,l\}\) , \(\:g\) and \(\:l\) represent the densities of the plunger in gas and liquid, respectively; \(\:{C}_{d}\) is the drag coefficient; \(\:{A}_{p}\) and \(\:{A}_{t}\) are the cross-sectional areas of the annulus and tubing in \(\:{m}^{2}\) , respectively; \(\:{m}_{p}\) is the mass of the plunger in \(\:kg\) ; and \(\:g\) is the gravitational acceleration in \(\:m/{s}^{2}\) .
The pressures directly above (\(\:{P}_{pt}\) ) and below (\(\:{P}_{pb}\) ) the plunger are calculated using the tubing pressure equations21. As follows:
In Eq. (9), \(\:{a}_{p}\) denotes the acceleration of the plunger rise, \(\:{P}_{fric}\) denotes the frictional force between the liquid slug of length \(\:{L}_{s}\) and the tubing wall, given by:
In Eq. (11), \(\:\text{R}\text{e}\) represents the Reynolds number, and \(\:\epsilon\:\) denotes the pipe roughness.
First, consider the mass balance in the tubing when the valve is closed (the first three stages). In the tubing, let \(\:{L}_{\left(j\right)}\) denote the height of the liquid column, which is calculated as follows:
In Eq. (12), \(\:j\in\:\{a,tb,tt\}\) represent the positions within the annulus, above the plunger, and below the plunger, respectively. \(\:{\rho\:}_{l}\) denotes the density of the liquid, and \(\:{m}_{{l}_{\left(j\right)}}\) is the mass of the liquid column.
Since the pressure in the annulus is unaffected by the position of the plunger and the annulus is closed at the top, the pressure at the surface of the gas column \(\:{P}_{c}\) is as follows:
In Eq. (13), \(\:H\) represents the height of the gas well.
Since the only flow rate into or out of the annulus is \(\:{F}_{a}\) , the mass balance equation for the annulus (in all modes) is given by:
During the plunger descent and pressure recovery phases, the mass balance on the tubing segment follows a similar reasoning as for the annulus. \(\:{L}_{tb}\) represents the bottom liquid level and is calculated as follows:
When the production valve is closed, the net mass in the tubing changes only due to the net inflow \(\:{F}_{g,tub}\) . The gas mass in the upper and lower sections of the plunger is distributed based on the position of the plunger, given by:
At the same time, the gas flow around the downward-moving plunger is described by:
When the plunger reaches the liquid, it continues to descend (within the liquid). Since there is no gas below the plunger:
When the plunger reaches the bottom of the well (i.e., during the build-up phase), there is no gas or liquid below the plunger. The liquid above the plunger increases due to the flow from the reservoir into the tubing.
During the plunger ascent phase, the plunger moves upward due to the pressure differential. Gas flows out from the tubing section above the plunger, while the section below the plunger is pressurized by the gas flowing into the tubing. Since the plunger acts as a barrier between the two sections, no gas flows from the bottom of the tubing to the top. Therefore, the mass balance for the two tubing sections is given by:
During the plunger ascent phase, liquid from the reservoir and the annulus flows into the tubing, which alters the volume of liquid at the bottom of the tubing \(\:{m}_{l,tb}\) . In an ideal plunger stroke, the liquid above the plunger remains unchanged. However, during the plunger ascent, there is liquid leakage, which can be modeled as a process with a loss rate of k_leak:
Therefore, the mass balance of the liquid in the tubing section is:
The following formulas calculate the pressures at the top and bottom of the tubing:
During the liquid slug arrival phase, the gas mass above the plunger is zero, and the liquid above the plunger is discharged through the production valve. Since the liquid slug ascends at the same speed as the plunger, the liquid flow rate \(\:{F}_{l,out}\) depends on the plunger’s velocity and is given by:
At this point, the gas above the plunger is evacuated, i.e., \(\:{\Delta\:}{m}_{g,tt}=0\) , and the change in liquid is as follows:
The mass balance equation for the tubing section below the plunger remains the same as before.
During the after-flow phase, the plunger remains stationary in the wellhead’s catcher or blowout preventer. The tubing section below the plunger now experiences gas and liquid flowing from the bottomhole and exiting through the production valve. Therefore, the outflow section of the model is similar to the gas flow above the plunger during the ascent phase. The overall balance equation is given by:
Use the flow gas equation in the tubing section below the plunger to calculate the pressure distribution:
The model was designed using C/C + + in Visual Studio, incorporating disturbances based on pipeline pressure variations to simulate the dynamic process of a real gas well.The parameters used in the model are shown in Tables 1 and 2. Table 1 outlines the gas well parameters, while Table 2 details the composition of the light gases.
Figure 3a shows pressure and flow rate data for a 155-minute plunger lift cycle with a 120-minute shut-in time. The plunger takes 906 s to reach the top at 3.31 m/s. After the valve closes at time zero, tubing pressure rises rapidly while casing pressure increases slowly. During the accumulation phase, casing pressure rises faster than tubing pressure, aligning with production trends in Fig. 2c. Figure 3b shows plunger speed variation. At the 120-minute mark, the valve opens, and the plunger accelerates to 2.5 m/s within 40 s. The speed gradually increases but decelerates near the wellhead due to the liquid slug, influenced by slug height and pressure differential.
Single cycle data of plunger lift: (a) Pressure and flow rate; (b) Plunger velocity.
In a 24-hour test with nine plunger lift cycles (Fig. 4), pipeline pressure increased by over 20%, causing higher backpressure and increased casing and tubing pressures. Figure 4a shows a decline in flow rate due to reduced reservoir inflow, leading to lower production. Figure 4b illustrates plunger speed variations, where rising pipeline pressure reduces the pressure differential, slowing the plunger’s ascent.
24-hour data of plunger lift: (a) Pressure and flow rate; (b) Plunger velocity.
The above data was obtained by simulating the plunger lift process of an actual gas well, and the data is consistent with real gas well production data.
For the plunger lift model described above, this section will present the optimization method for the plunger lift system. The Adaptive Particle Swarm Optimization-Model Predictive Control (APSO-MPC) algorithm will be developed based on the theory proposed by Nandola et al. in 201818. The APSO-MPC process is illustrated in Fig. 5.
In control optimization, where rolling horizon optimization is performed at the end of each cycle, both input and output data are continuous-time data. Therefore, continuous-time series data are converted into periodic time series data. The input and output data are denoted as \(\:{u}^{b}({T}_{j},i)\) and \(\:y({T}_{j},i)\) , respectively, where \(\:i\) is the cycle index and \(\:{T}_{j}\) represents the time within the cycle.
In Eq. (27), \(\:Y\left[i\right]\) represents the set of data for the \(\:i\) -th cycle, and the relationship between periodic data \(\:y({T}_{j},i)\) and continuous-time data \(\:y\left(t\right)\) , \(\:y\left(t\right)\) is as follows: \(\:t=\sum\:_{i=1}^{j-1}\:{T}_{i}+{T}_{j}\) .
Using the periodic data, two output values are computed to evaluate the net production and average plunger speed. First, the net flow is calculated. The net gas production can be defined as the total gas production over the cycle divided by the cycle time, as:
In Eq. (28), \(\:Fout\) represents the fluid flow rate, \(\:\sum\:_{k}\:Fout\left(k,i\right)\) denotes the total gas production during the \(\:i\) -th cycle, and \(\:{T}_{i}\) is the total duration of the cycle.
Next, the average arrival speed of the plunger is calculated. The average arrival speed of the plunger can be defined as the ratio of the well depth to the plunger’s ascent time, as follows:
In the control system, two control inputs are required. The first control input is to regulate the opening of the production valve. During the accumulation phase, when the casing pressure exceeds a preset threshold \(\:{P}_{max}^{i}\) , the production valve is opened. Therefore:
Similarly, the second control input regulates the closure of the production valve. When the flow rate drops below a set threshold \(\:{F}_{min}^{i}\) , the valve is closed. Therefore:
Therefore, the control inputs and outputs are transformed into \(\:u\left(i\right)={\left[\begin{array}{cc}{u}_{1}\left(i\right)&\:{u}_{2}\left(i\right)\end{array}\right]}^{\text{T}}\) and \(\:y\left(i\right)={\left[\begin{array}{cc}{y}_{1}\left(i\right)&\:{y}_{2}\left(i\right)\end{array}\right]}^{\text{T}}\) .
Further, the ARX model is used to model the system with the transformed inputs and outputs for model predictive control (MPC), which is crucial for effective control. The form can be expressed as:
In Eq. (32), \(\:y\left(t\right)\) represents the output at time \(\:t\) ; \(\:u\left(t\right)\) denotes the control input at time \(\:t\) ; \(\:{a}_{i}\) and \(\:{b}_{i}\) are the autoregressive and input coefficients, respectively, where \(\:i\in\:(\text{1,2},\dots\:,n)\) ; \(\:e\left(t\right)\) represents the noise term, reflecting the unexplained part of the model; \(\:k\) is the delay of the input; \(\:n\) and \(\:m\) are the orders of the autoregressive and input terms, respectively.
In practical control systems, each output model may be influenced by multiple input controls, forming a Multiple Input Single Output (MISO) model. By linearly stacking these models, a Multiple Input Multiple Output (MIMO) model is created. The state-space representation of such a model is given by:
In Eq. (33), \(\:x\left(t\right)={\left[\begin{array}{cccccccc}y(t-1)&\:y(t-2)&\:\cdots\:&\:y(t-n)&\:u(t-1)&\:u(t-2)&\:u(t-2)&\:u(t-m)\end{array}\right]}^{\text{T}}\) .
In the rolling optimization process, two objectives need to be addressed36. The first objective is to maximize the net production:
In Eq. (34), the first term represents the sum of squares of the net production over \(\:N\) operational cycles of the plunger lift system, while the second term represents the sum of squares of the control input increments. \(\:{q}_{1}\) and \(\:{r}_{1}\) are the weight parameters that adjust the priority between the two optimization objectives.
The second objective is to ensure that the average plunger speed reaches the desired value:
In Eq. (35), the first term represents the sum of the squared errors of the average plunger speed over \(\:N\) cycles of the plunger lift system, while the second term represents the sum of the squared increments of the control variables. \(\:{q}_{2}\) and \(\:{r}_{2}\) are weighting parameters that adjust the priority between the two optimization objectives.
By combining Eq. (34) and Eq. (35) into a comprehensive optimization objective, the complete optimization form is as follows:
In Eq. (36), \(\:Q=\left[\begin{array}{cc}-{q}_{1}&\:0\\\:0&\:{q}_{2}\end{array}\right]\) , \(\:R=\left[\begin{array}{cc}{r}_{1}&\:0\\\:0&\:{r}_{2}\end{array}\right]\) , \(\:{y}_{ref}=\left[\begin{array}{c}0\\\:{\nu\:}_{ref}\end{array}\right]\) .
Due to the typically large disturbances encountered in shale gas wells, upper limit \(\:{y}_{2,max}\) and lower limit \(\:{y}_{2,min}\) are set for the average plunger speed, with the average value serving as the expected speed \(\:{\nu\:}_{ref}\) , as follows:
This study employs PSO to solve the aforementioned optimization problem. To prevent the algorithm from prematurely converging to a local optimum, adaptive adjustments are made to the inertia weight and learning factors within the PSO algorithm. The detailed flowchart of the algorithm is shown in Fig. 6.
Flowchart of the improved particle swarm optimization algorithm used in this study.
The principle of PSO involves initializing a group of random particles (random solutions) and iteratively finding the optimal solution. In each iteration, particles update their positions by tracking two “extreme values”: the first is the best solution found by the particle itself, known as the personal best position, denoted as \(\:{P}_{i}\) ; the second is the best solution found by the entire swarm, known as the global best position, denoted as \(\:{P}_{g}\) . Assume the particle’s spatial dimension is \(\:\text{D}\) , and the position of the \(\:i\) -th particle is represented as an \(\:\text{D}\) -dimensional vector:
The velocity of a particle is represented as an \(\:D\) -dimensional vector:
The representations for the particle’s best historical position and the group’s best position are as follows:
In each iteration, the particle’s velocity and position are updated. The velocity update formula is as follows:
In Eq. (42), \(\:k\) represents the iteration count, \(\:i=\text{1,2},\cdots\:,N\) ,and \(\:N\) is the population size, \(\:{r}_{1}\) and \(\:{r}_{2}\) are random numbers within the interval \(\:\left[\begin{array}{c}\text{0,1}\end{array}\right]\) , \(\:{c}_{1}\) and \(\:{c}_{2}\) are the learning factors, and \(\:\omega\:\) is the inertia weight.
The position update is given by:
In PSO, the algorithm lacks flexibility and adaptability due to fixed inertia weights and learning factors during iterations. Additionally, to enhance the global search capability of the algorithm in its early stages and avoid local optima, adaptive adjustments are made to the inertia weight.
The adjustment of the inertia weight dynamically adjusts based on the convergence state of the particle swarm, calculated as follows:
In Eq. (44), \(\:\alpha\:\) is the adjustment parameter, \(\:{\sigma\:}_{fit}\left(k\right)\) is the standard deviation of the fitness at the \(\:k\) -th iteration, and \(\:{\sigma\:}_{fit}\left(0\right)\) is the standard deviation at the initial moment. The standard deviation is calculated as follows:
In Eq. (45), \(\:N\) represents the number of particles in the swarm, \(\:{f}_{i}\left(k\right)\) denotes the fitness of the \(\:i\) -th particle at the \(\:k\) -th iteration, and \(\:\overline{f}\left(k\right)\) is the average fitness at the \(\:k\) -th iteration. The calculation is as follows:
In fixed learning factors, maintaining a balance between local and global search is possible to some extent. However, in complex problems, fixed learning factors might cause the algorithm to prematurely converge to a local optimum rather than the global optimum. Dynamic adjustment of the learning factors can enhance the ability to escape local optima. Therefore, in adaptive optimization, learning factors are dynamically adjusted by using the current fitness and global fitness to guide the adjustment process.
When the fitness difference is large, increase the learning factor \(\:{c}_{1}\) to enhance individual learning capability. When the fitness difference is small, increase the learning factor \(\:{c}_{2}\) to enhance global learning capability. The adjustment calculation is as follows:
In Eq. (47), \(\:{c}_{1,min}\) , \(\:{c}_{1,max}\) , \(\:{c}_{2,min}\) and \(\:{c}_{2,max}\) are the minimum and maximum values of the individual learning factor, while \(\:{f}_{max}\) and \(\:{f}_{min}\:\) are the maximum and minimum values of the global learning factor.
The above MPC optimization problem is a standard unconstrained quadratic programming problem. Such problems are often solved using numerical methods. By using the gradient vector and Hessian matrix, the quadratic programming optimization problem can be reformulated as follows:
In Eq. (48), \(\:H\) represents the Hessian matrix (a positive definite symmetric matrix), \(\:f\) denotes the gradient vector, \(\:U\) is the control input vector, and \(\:U={\left[\begin{array}{cccc}u\left[i\right]&\:u[i+1]&\:\cdots\:&\:u[i+m-1]\end{array}\right]}^{\text{T}}\) .
The APSO approach for optimizing this problem is as follows: First, initialize a population of random solutions, with initial positions and velocities generated randomly within a specified range.
In Eq. (49), \(\:{x}_{min}\) and \(\:{x}_{max}\) represent the minimum and maximum values for the positions, while \(\:{\nu\:}_{min}\) and \(\:{\nu\:}_{max}\) denote the minimum and maximum values for the velocities.
Once the particles are generated, the iterative optimization process begins, which involves fitness evaluation. The mathematical representation is:
For the above quadratic programming problem, the fitness function is typically expressed as: \(\:f\left(x\right)=\frac{1}{2}{x}^{\intercal}Hx+{f}^{\intercal}x\) .
After the solution is completed, the optimal solution is updated, where the individual optimal update is as follows, when the fitness of the individual particle is less than the optimal fitness, i.e. \(\:f\left({x}_{i}\right)<f\left({p}_{i}\right)\) , then \(\:{p}_{i}={x}_{i}\) , and the global optimal update is as follows, when the fitness of the individual optimal particle is less than the global optimal fitness, i.e. \(\:f\left({p}_{i}\right)<f\left(g\right)\) , then \(\:{p}_{g}={p}_{i}\:\) .
And then adaptively adjust the inertia weights as well as the learning factor, update the number of iterations after the adjustment is completed, and substitute into the next iteration. In the process of iteration, set the numerical change amount threshold of the objective function and the maximum number of iterations as the termination condition of iteration. Set the numerical transformation amount threshold of the control input as \(\:\delta\:\) , when the first two subsequent control inputs are less than this threshold, the iteration is terminated, which is mathematically expressed as:
In Eq. (51), \(\:\parallel\:{U}_{k+1}-{U}_{k}\parallel\:\) is the change in control input between the current \(\:k\) -th iteration and the next (\(\:k+1\) -th) iteration, and \(\:\delta\:\) is the set tolerance, which is positive.
Set the maximum number of iterations as \(\:{k}_{max}\) , when the number of iterations \(\:k\) reaches the set maximum number of iterations, the iteration is terminated, which is of the mathematical form:
In Eq. (52), \(\:k\) is the current iteration number and \(\:{k}_{max}\) is the maximum iteration number.
Figure 7 illustrates the architecture of the plunger lift control system based on edge computing. The intelligent controller collects pressure data, which is sent to the edge computing unit for processing using optimization algorithms. The resulting control instructions are sent back to the intelligent controller, bypassing the server platform. The edge computing unit uses an embedded microprocessor, with the chip model being STM32H750VBT6. This chip is based on a high-performance Arm® Cortex®-M7 32-bit RISC core, with a clock frequency of up to 480 MHz.
Edge computing based plunger lift control system architecture.
In the edge computing-based plunger lift optimization control, the design and optimization of algorithm deployment are crucial, as shown in Fig. 8. First, memory planning is carried out, where, in addition to the fixed data required for global calculations, the remaining data is temporarily stored through dynamic memory allocation. Next, the required variable data types are designed and normalized to avoid the impact of differences in variable magnitudes on optimization results. Some of the data types are shown in Table 3. For algorithm design, C/C + + is used to describe the entire optimization control algorithm. Finally, the parameters required by the algorithm are initialized, as listed in Table 4. The initialization of particle positions is dynamically set based on the control dynamics of the previous plunger cycle, while the velocity range of the particles can be set to 20% of the position range.
Algorithm deployment and operation fowchart.
During algorithm execution, the edge computing unit retrieves data from the controller at 5-second intervals. When a plunger lift cycle ends, the retrieved data is fed into the algorithm for calculation, and the results are output to execute the control, while the data input for the next time step begins. To evaluate the performance of the algorithm under different implementations, the execution efficiency of the algorithm for various deployment methods is compared, as shown in Table 5. The PC configuration for the Visual Studio software computation is as follows: 12th Gen Intel(R) Core(TM) i5-12400@2.50 GHz with 16GB of RAM.
In Visual Studio, when PSO optimization is not used, the time required to solve the MPC algorithm is relatively longer, whereas using PSO optimization results in shorter solve times. However, when APSO is employed to solve the MPC algorithm, the computational time of the algorithm is further optimized. When the algorithm is ported to a microprocessor and the runtime is measured using a 32-bit high-precision timer, the results show that the computation time was reduced by 2.5 times. Additionally, the test did not consider data transmission delays, while the edge computing unit located at the edge of the controller offers advantages in data transmission. It is also worth noting that microprocessors, compared to computers, are single-core processors and inherently have differences in computational performance.
In this section, the control optimizing algorithms and implementations proposed in Section III are simulated and experimented, in which the plunger lift system built in Section II is used as the controlled object. During the experiments, the local computing is used as the controller to control the above mentioned plunger lift model and connected to the edge computing unit using TCP communication. The real-time simulation platform is shown in Fig. 9.
For ARX models, the prediction order impacts both computational complexity and predictive accuracy. This section examines the relationship between prediction order and accuracy to find the optimal order. Using average plunger speed data, 5 cycles were used to initialize the ARX model, with predictions made for the next 20 cycles for MPC optimization (Fig. 10a ). Lower orders (second and third) capture small trends but show lag, failing to predict larger trends accurately. Increasing the order to fourth improves overall trend prediction but reduces small-scale accuracy. Higher orders exceed the estimation capacity of the 5 data points. Figure 10b shows similar results with an additional data point.
Comparison of ARX model prediction data with real data: (a) Comparison of ARX model prediction for different orders with 5 historical data; (b) Comparison of ARX model prediction for different orders with 6 historical data; (c) Comparison of prediction for the same order with different historical data.
Figure 10c compares ARX model predictions using different amounts of historical data. For a fourth-order ARX model, increasing historical data from 5 to 6 cycles improves the display of small-scale trends but results in larger deviations from actual data. Based on simulations, using 6 data points requires an additional uncontrolled cycle, increasing the initialization time. Therefore, this study uses 5 cycles of historical data to initialize the ARX model, with both input and output orders set to 4.
In this section, simulations compare the edge computing method based on the APSO-MPC algorithm with the traditional time-based production method. For the traditional method, the shut-in duration is set to 120 min and the well-opening duration to 35 min, resulting in a 155-minute cycle. In contrast, the APSO-MPC-based edge computing method uses past cycle data for predictions, with 5 cycles of historical data, as detailed in Sect. "ARX model validation" To ensure stability, both methods are compared over a 48-hour period, with the first five cycles following the traditional time-based method.
First, in the time-based production method, 18 cycles are completed in 48 h, as shown in Fig. 11a. The average plunger ascent speed is approximately 3.3 m/s, lower than the ideal 3.8 m/s, with a production rate of about 3400 m³/d. Figure 11b shows measurement data for the 10th cycle, where integrating the flow rate gives a total gas production of 361.68 m³ per cycle. Given the 155-minute cycle time, daily production is 3360.12 m³. From valve opening, it takes 910 s for the plunger to reach the surface at 3000 m, with an average speed of 3.29 m/s, below the ideal.
Output data based on timed production: (a) Cycle-by-cycle; (b) 10th cycle.
Figure 12 presents the control results of the APSO-MPC and MPC methods. Within the same 48-hour period, both approaches successfully optimized the control of plunger lifting. Compared with traditional time-based control methods, during the 48-hour timeframe, the first five cycles were conducted according to scheduled production. Starting from the sixth cycle, the APSO-MPC or MPC algorithm was applied for optimization. As shown in the figure, both algorithms generated 22 plunger lifting cycles subsequently. Figure 12a demonstrates that the optimized control maintained the plunger ascent speed within the required range, and compared to the initial five cycles, natural gas production increased. Additionally, Fig. 12b shows that the pressure threshold for valve opening was reduced by 0.43 MPa, and the flow rate during valve closing decreased by 20 cubic decimeters per second, significantly improving production efficiency compared to time-based methods. However, as illustrated in the figure, the APSO-MPC algorithm responded more rapidly than MPC, reaching the desired state more quickly. This phenomenon indicates that the incorporation of APSO enhances the rapid response capability of the MPC control system, allowing it to optimize control parameters more swiftly and adapt to system changes.
Based on APSO-MPC and MPC optimized control of plunger lifting’s outputs and inputs: (a) Output; (b) Input.
Figure 13 shows the 10th cycle of plunger lift under optimized control. The cycle time is 5856 s (97.6 min), nearly 58 min shorter than the timed control cycle. Integration of the flow rate yields a total production of 268.65 m3 of natural gas per cycle, corresponding to 3963.72 m3 per day, given the cycle time. The average plunger speed in this cycle is 3.94 m/s, close to the expected 3.95 m/s, and natural gas production increased by 18% compared to the timed control method.
Output data of the 10th cycle of optimizing control based on APSO-MPC and edge computing methods.
First is the transmission efficiency verification. Figure 14a shows that for data under 256 Kbyte, all methods have similar transmission rates. However, as data size increases, Modbus RTU slows significantly, while TCP to the server experiences delays beyond 1 Mbyte due to network latency. TCP to the edge unit is 24% faster, with a transmission time of 5.146 s for a 4 Mbyte cycle. Figure 14b shows TCP to the edge unit achieving 6.5 Mbps bandwidth, outperforming the server, while Modbus RTU struggles with large data. Packet loss tests in Fig. 14c reveal that Modbus RTU has low loss due to strong anti-interference, while TCP has minimal loss for fewer than 50,000 frames. Beyond this, TCP to the server shows a sixfold higher packet loss rate compared to the edge unit, which remains stable at 0.015%, demonstrating edge computing’s efficiency in large-scale operations.
Transmission efficiency test of three methods: (a) Transmission rate with different amount of data; (b) Actual bandwidth with different amount of data; (c) Packet loss rate with different data frames.
The second is computational efficiency verification. Figure 15a shows that APSO-MPC is more efficient and faster than PSO-MPC. Although APSO-MPC on edge computing is slightly less efficient than on servers, it still outperforms traditional methods. Figure 15b demonstrates that as the number of managed gas wells increases, server memory usage rises sharply, especially with embedded optimization algorithms. PSO-MPC uses over 95% of server memory when managing 128 wells, while APSO-MPC can manage 146 wells with the same memory usage. However, further increases in well management overload server memory. The proposed edge computing approach reduces server memory usage to 20% when managing 146 wells by offloading computation to edge units. Figure 15c shows that server-based methods experience growing delays as task numbers increase, while edge computing enables parallel processing, reducing latency and improving overall efficiency.
Computation efficiency in different ways: (a) Computation rate with different amount of computation data; (b) Memory utilization of the server as the number of accessed wells transforms; (c) Computation latency with different amount of computation tasks.
In multi-well operations, the optimization control system based on edge computing demonstrates superior scalability and reliability. First, edge computing effectively offloads computational tasks, alleviating the burden on remote servers and enabling efficient parallel computing. Through proper memory allocation and resource scheduling, the system is capable of managing more wells. Second, self-monitoring and fault isolation mechanisms enhance the system’s robustness and self-healing capabilities. At the same time, edge computing reduces data transmission volume, optimizes bandwidth usage, and eliminates the delay issues associated with remote data transmission, improving the system’s real-time performance and security.
The optimizing control of plunger lift using the APSO-MPC algorithm, as proposed in this paper, leverages Model Predictive Control (MPC) for forward-looking output predictions, improving decision-making. The APSO algorithm offers adaptive iterative solutions, enhancing computational efficiency. By deploying this method on an edge computing unit, it eliminates remote data transmission delays and reduces server computational load. Simulations show that this approach boosts daily natural gas production by 18% compared to timed production, while ensuring effective drainage and safe plunger rise rates. Additionally, the edge computing architecture improved data transmission by 24%, reduced packet loss by 83%, and significantly lowered server memory usage and calculation delays, optimizing overall control efficiency.
Finally, although the above method performs excellently in experiments, the edge computing unit still has some limitations, primarily due to limited hardware resources and insufficient computational power. These constraints make it difficult for the edge computing unit to support more complex models, especially the operation of large neural networks. However, with the continuous advancement of hardware technology, the resources and computational capabilities of the edge computing unit are expected to improve significantly, enabling edge computing to handle more complex computational tasks. Additionally, as network transmission speed and quality continue to improve, data transmission delays and bandwidth bottlenecks will be effectively alleviated. This provides strong support for the widespread application of edge collaboration, and the edge collaborative approach will be further promoted in more areas of industrial control, thereby further improving control efficiency and optimization outcomes.
The datasets used and analysed during the current study available from the corresponding author on reasonable request.
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This research was supported by the Key Research and Development Project of Sichuan Science and Technology Program (No. 2022YFG0073) and the Youth Fund of Sichuan Natural Science Foundation (No. 2024NSFSC0907).
Southwest Petroleum University, Chengdu, Sichuan, China
Qiu Zhi, Zhang Lei, Zhang He & Liang Haibo
PetroChina Changqing Oilfield Company Third Gas Production Plant, Uttarakhand, Inner Mongolia Autonomous Region, China
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Authors’ ContributionZhi Qiu: Investigation, Writing-original draft and Formal analysis. Lei Zhang: Methodology, Methodology and Formal analysis. He Zhang: Investigation and Formal analysis. Haibo Liang: Conceptualization, Supervision, Writing-review & editing, and Funding acquisition. Yinxian Li: Super-vision and Writing-review & editing.
The authors declare no competing interests.
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Qiu, Z., Zhang, L., Zhang, H. et al. A plunger lifting optimization control method based on APSO-MPC for edge computing applications. Sci Rep 15, 4356 (2025). https://doi.org/10.1038/s41598-025-87726-w
DOI: https://doi.org/10.1038/s41598-025-87726-w
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