Energy-Efficient Optimization of Jaw-Type Blowout Preventer Activation Using Combined Experimental Design and Metaheuristic Algorithms

Abstract

This paper presents the optimization of the power required to activate a jaw-type blowout preventer (BOP) in the oil industry using an axial piston pump. Experimental and numerical methods were combined to analyze the effects of pressure, flow rate, volumetric efficiency, and clearance leakage on energy consumption. Taguchi methodology with an orthogonal array and the “smaller-is-better” criterion was used in the experiments, while regression analysis provided a predictive model. Optimization was performed using the Grey Wolf Optimizer (GWO) in Python 3.13. The results show that pressure and flow rate significantly affect power consumption, while higher volumetric efficiency leads to notable energy savings. The optimal configuration reduced the power demand to 5.0001 kW. Based on this, reliability models were created to assess deviations from optimal conditions. The study demonstrates the effectiveness of combining statistical and optimization techniques for improving safety systems in the oil industry. The key contribution of this study lies in the integration of experimental Taguchi-based modeling with Grey Wolf Optimizer (GWO) metaheuristic techniques to optimize the energy-efficient activation of jaw-type blowout preventers, representing a novel methodological approach in the field of hydraulic safety systems in the oil industry.

1. Introduction

Hydraulic systems represent a key component of modern industrial facilities due to their ability to transmit high power with precise control. By utilizing pressurized working fluid, these systems enable the generation of large forces within confined spaces, making them suitable for a wide range of applications, including high-risk operations in the oil industry [1,2]. The main components of a hydraulic system pumps, working fluid, pipelines, control elements, and actuators contribute to its reliability and broad applicability [3].

In addition to their primary function of energy transmission, hydraulic systems require precise control of the working fluid to minimize losses and prevent oil overheating, as elevated temperatures lead to reduced viscosity, increased leakage, and a decline in volumetric efficiency, which directly impacts system reliability and energy performance [4].

Safety in well operations largely depends on the reliability of hydraulic equipment, particularly ram-type BOPs, whose failure or improper selection can lead to severe accidents. Industry experience, exemplified by incidents such as the Deepwater Horizon disaster, underscores the critical need for accurate prediction of expected pressures and regular testing in compliance with API standards [5].

Within the oil industry, pressure control equipment plays a critical role in wellbore operations, with jaw-type BOPs serving as key components for preventing blowouts and preserving well integrity [6,7,8,9]. The activation of these preventers is most commonly achieved hydraulically, allowing for immediate and efficient sealing of the wellbore. Such systems must ensure a high level of reliability with minimal energy consumption, which is the primary focus of this study.

The importance of optimizing the power required for BOP system operation lies in the fact that drilling conditions are highly demanding, where the speed and efficiency of response are critical for ensuring the safety of personnel and infrastructure. In this context, the aim of the research is to evaluate the possibility of activating a jaw-type BOP using an axialpiston pump while minimizing the required power, based on relevant technical and mathematical models. To determine the optimal operating parameters for efficient preventer activation, this study employs a combination of statistical methods—Taguchi methodology, regression analysis, and the Minitab 20.4 software package. The Taguchi approach, as part of the broader Design of Experiments (DOE) methodology, enables the systematic examination of multiple factors through orthogonal arrays, significantly reducing the number of required experiments while ensuring robustness and resistance to external disturbances [10,11,12,13]. Through clearly defined steps from problem identification, selection of factors and their levels, to validation of the obtained results, this method provides precise insight into system behavior even under unfavorable variations, particularly due to the analysis of Signal-to-Noise (S/N) ratios [10,11]. Although methodologically more heuristic than strictly statistical in nature, its application in industrial practice has proven to be extremely useful, particularly in the early stages of developing technical solutions [12]. In addition, regression analysis was used to model the relationships between input factors and system output performance, enabling the quantification of the influence of individual parameters and the prediction of system behavior under various operating conditions [10,11,14]. In combination with the Response Surface Methodology (RSM), it enabled the construction of multidimensional models used to identify optimal operating conditions and facilitate engineering decision-making with a minimal number of experiments [12,14]. The use of the Minitab software package further facilitated the analysis process, as this tool enables efficient experimental design, execution of regression models, visualization of factor interdependencies through main effects and response surface plots, and confirmation of statistical model validity using Analysis of Variance (ANOVA) tests. In this way, more precise insights and greater reliability were ensured in the process of optimizing system performance with multiple input variables [15].

The basic operating principle of hydraulic systems in this context involves the conversion of hydraulic energy into mechanical energy through a piston within the preventer’s cylinder. Pressure is regulated by the pump unit, while the system includes components such as hydro-pneumatic accumulators, safety valves, and control valves, all designed to ensure rapid and reliable wellbore closure, typically within less than 30 s [16,17]. Independent power sources (such as air compressors and generators) are often used to maintain functionality in emergency situations.

Safety equipment is a complex pressure control system used during the drilling and completion of oil and gas wells [6]. The blowout prevention system performs functions such as fluid containment, fluid volume regulation, and wellbore sealing for safety purposes. This system includes annular and jaw-type BOPs, which use hydraulic power to seal the wellbore, while seals and joints ensure safe fluid management [16].

During drilling, formation pressure typically increases with depth [7,18]. Control of that pressure is achieved through the use of drilling mud [7,8,9,18,19], which creates a hydrostatic column above the formation pressure [7,8,18,19]. However, if the formation pressure exceeds the hydrostatic barrier, uncontrolled fluid release may occur [16,20]. In such situations, the BOP system serves as a critical safety mechanism [6,7,8,9].

The function and design of BOPs, as well as their classification (e.g., single or double models with flat or universal rams), are thoroughly addressed in the available literature [7,16,17,18,20]. Their closing is performed hydraulically, by activating a cylinder that drives the rams via a piston [7,17]. After closing, the rams remain locked in position thanks to additional pistons, and the system can be reset by redirecting the fluid to a reservoir. The preventer is connected to the system through an opening in the block, which is mounted on the wellhead flange [17].

Pumps, as the core of every hydraulic system, perform the transformation of mechanical energy into hydraulic energy [21,22,23]. They can be gear, screw, vane, or piston pumps, with the latter being particularly valued for their robustness and efficiency [21,23,24,25]. Piston pumps are further classified into axial and radial types, with axial pumps utilizing a parallel configuration of the piston and drive shaft, allowing them to operate under high loads [21,23,26].

The use of axialpiston pumps further contributes to optimization. These pumps offer numerous advantages, such as exceptional resistance to high pressures, high efficiency, the ability to transmit significant power, a wide range of operating speeds, and long service life [27].

These pumps can have either fixed or variable displacement and are known for their ability to efficiently transmit power, offering high reliability and operation across a wide range of pressures and flow rates. Variable displacement pumps enable precise control of the output flow by adjusting the swashplate angle, allowing efficient adaptation of flow based on varying pressure loads, which further enhances system performance. In contrast, fixed displacement pumps have a constant piston stroke that cannot be adjusted [21,23,26,28].Their construction includes a cylinder block, pistons, and a swashplate, enabling periodic suction and discharge of fluid, which ensures continuous operation [22,23,24,28,29]. Axialpiston pumps are commonly used in power supply systems as well as in hydrostatic transmissions, where they serve to generate flow [30].

From a historical perspective, the development of hydraulics began with Archimedes, Pascal, and Bernoulli, eventually culminating in inventions such as the Bramah press, which had a tremendous impact on industry and transportation [1,31]. Today, thanks to modern technologies, hydraulic systems enable significantly more efficient machine operation, along with control of emissions and energy losses [32,33,34,35].In the paper [36], the authors investigated the influence of flow rate and pipe rotation speed on the efficiency of hole cleaning during drilling operations, aiming to identify optimal parameters that reduce energy usage while maintaining operational effectiveness.

Metaheuristic algorithms, such as genetic and hybrid approaches, achieve superior results in optimizing complex processes compared to traditional methods, while their integration with statistical techniques, such as the Taguchi design, is employed to reduce the number of experiments and enhance overall efficiency [37].

In this study, a formula for calculating the power required to operate the motor that drives the pump will be addressed, based on flow rate, pressure, and component efficiency. Based on statistical analysis of the experiments, configurations enabling the most efficient activation of the BOP with minimal energy consumption will be identified.

The research results showed that flow rate and pressure are the most influential factors determining the power required for system operation, while an increase in volumetric efficiency significantly contributes to reducing energy consumption. By combining the Taguchi method and regression analysis, optimal operating conditions were identified, and through the application of the GWO algorithm, the parameter configuration was further refined to ensure minimum power demand with stable system performance. In this way, the high reliability of the analytical model was confirmed, demonstrating its practical applicability under real industrial conditions.

In addition, the study formulated a clear methodological basis for future approaches to the optimization of similar hydraulic systems. Through a multi-criteria approach combined with intelligent optimization algorithms, a framework was established that allows for further enhancement of jaw-type BOP control systems. This approach provides the foundation for the development of advanced control systems with self-adaptive capabilities and predictive diagnostics, further improving safety and efficiency in wellsite operations.

In ram-type BOP control systems, the most widely used hydraulic medium is a mineral-based oil with ISO VG 46 viscosity grade, conforming to international standards such as DIN 51524-2 (HLP)or DIN 51524-3 (HVLP) [38,39]. ISO VG 46 denotes a kinematic viscosity of approximately 46 mm²/s (cSt) at 40 °C, which translates to about 6.8 mm²/s at 100 °C. This grade is preferred because it provides a stable balance between lubricity, leakage control, and energy efficiency across the operating temperature range typical of BOP control units. Compared with lighter ISO VG 32 oils, ISO VG 46 offers greater film strength and reduced leakage under high-pressure conditions, while maintaining sufficient flow characteristics for reliable valve actuation [4,40].

Manufacturers of surface BOP control units, including NOV and Cameron, commonly recommend hydraulic oils within the 200–300 SSU range at 100 °F (≈38 °C), which corresponds closely to the viscosity of ISO VG 46. At operating temperatures of 40–60 °C, ISO VG 46 typically maintains a viscosity between 16 and 30 cSt, ensuring efficient pump operation without excessive internal leakage. The oil also includes additives for oxidation stability, rust inhibition, and anti-wear protection, which are critical for long-term reliability of accumulators, pumps, and valve assemblies in harsh drilling environments. By adopting ISO VG 46 as the reference hydraulic fluid, the experimental results reported here align with common industry practice and increase the transferability of laboratory findings to field BOP applications. In addition to the influence of fluid viscosity, hydraulic parameters can also be affected by potential cavitation occurring in hydraulic systems with suction issues at the pumps, leading to further operational complications, such as hydraulic shocks, pipeline damage, and similar issues. Although the novelty of integrating the Taguchi method and Grey Wolf Optimization (GWO) is highlighted, the introduction can be strengthened by situating this hybrid approach within the existing literature. To date, current research in BOP optimization often employs either robust design of experiments such as Taguchi’s method or metaheuristic algorithms like GWO, but rarely a deliberate combination of the two. While the Taguchi method has been successfully applied in diverse industrial settings—including chemical processing, material fabrication, and sensor optimization—for reducing experimental runs and enhancing robustness, its synergy with GWO remains unexplored in oil systems [41].

Meanwhile, GWO, known for its balance of exploration and exploitation and ease of implementation, has been used in domains such as power system optimization and engineering design, yet its hybridization with Taguchi methods for BOP control systems is not present in the papers [42,43]. A few exceptions exist in other fields—e.g., hybrid Taguchi–PSO approaches for flowshop scheduling problems, but these applications remain outside the context of oil systems.

2. Materials and Methods

In this study, simulation of industrial conditions refers specifically to the reproduction of hydraulic operating parameters and boundary conditions representative of ram-type BOP control systems. The test bench was designed to replicate the working pressure range (200–260 bar), the flow supply of 30–35 L/min, and the volumetric efficiency levels (0.92–0.95) typically encountered in field units. Oil temperature was maintained in the range of 40–60 °C, consistent with the thermal environment of surface BOP control skids. The experimental setup focused exclusively on the hydraulic subsystem, i.e., the energy transmission and valve actuation processes, while external field factors such as wellbore multiphase flow, formation pressure dynamics, and drill-string induced vibrations were not included. This definition ensures that the optimization results remain valid for the hydraulic control unit itself, while acknowledging the limitations and the gap that still exists between laboratory and full field operation.

To optimize the power required for activating the jaw-type BOP using an axial piston pump, a combined analysis was conducted using experimental data and advanced processing methods. The experiment was organized according to the principles of the Taguchi methodology, with four key input factors defined: pressure, flow rate, volumetric efficiency, and clearance leakage. Factor values were varied at two levels across a total of 16 combinations, with each configuration repeated twice to ensure statistical reliability and enable error estimation. Measurements were performed on an experimental setup simulating the operation of the jaw-type BOP under real industrial conditions.

The experimental data were processed using the Minitab software package, which enabled the execution of Taguchi analysis with the “smaller-is-better” criterion, the determination of S/N ratios, and the generation of response tables for each factor. In parallel, multiple linear regression analysis was carried out to formulate a predictive power model as a function of the input parameters. The resulting regression equation provided a quantitative assessment of the influence of individual factors, while statistical indicators (R², p-values, Variance Inflation Factors (VIF)) confirmed the validity and stability of the model.

In the final phase of the research, the GWO metaheuristic algorithm, inspired by the hierarchy and hunting behavior of grey wolves, was applied to identify the optimal parameter combination. The optimization was implemented in the Python programming environment with over 20 independent iterations, each involving 30 agents and 100 steps. The objective function was derived from the previously formulated regression equation, with variable bounds defined according to the experimental range. Statistical analysis of the GWO results included evaluation of the best, worst, and average solutions alongside standard deviation, ensuring a reliable assessment of the optimization’s stability.

Based on the obtained results, models were developed to evaluate the system’s operational reliability, accounting for deviations of actual operating parameters from the optimized values in accordance with defined weights and exponential reliability expressions.

2.1. Governing Equations for Hydraulic Power

The calculation of the required shaft power of the pump is based on the energy balance of the hydraulic control unit. In the simplest form, the theoretical hydraulic power can be expressed as Equation (1):

$$P={\displaystyle \frac{p·Q}{600}}$$

(1)

To account for internal losses, the practical shaft power must include both mechanical efficiency ${\eta }_m$ volumetric efficiency ${\eta }_v:$

$$P={\displaystyle \frac{p·Q}{600·{\eta }_m·{\eta }_v}}$$

(2)

The volumetric efficiency is defined as the ratio of useful delivered flow Q to the total displaced flow ${Q+Q}_{leak}:$

$$P={\displaystyle \frac{Q}{{Q+Q}_{leak}}}$$

(3)

By substituting into the shaft power expression, the leakage-explicit form of the governing Equation (4) is obtained:

$$P={\displaystyle \frac{p·\left({Q+Q}_{leak}\right)}{600·{\eta }_m}}$$

(4)

2.2. Schematic Representation of the Blowout Preventer Assembly

As part of the methodological framework, Figure 1 presents the technical sketch of the blowout preventer, along with the drive unit of the preventer assembly. This schematic representation supports the description of the experimental setup and provides a visual reference for the subsequent analysis. In Figure 2, a schematic of the control unit is presented.

3. Results

3.1. Experimental Design

Table 1. Experimental Design.

Pressure [bar]	Flow Rate [L/min]	Vol Efficiency [%]	Leakage [L/min]	Power [kW]
69	15	0.92	0.05	1.919643
69	15	0.92	0.05	1.919643
69	15	0.92	6.5	2.742347
69	15	0.92	6.5	2.742347
69	136	0.95	0.05	16.80532
69	136	0.95	0.05	16.80532
69	136	0.95	6.5	17.60204
69	136	0.95	6.5	17.60204
345	15	0.95	0.05	9.295113
345	15	0.95	0.05	9.295113
345	15	0.95	6.5	13.27873
345	15	0.95	6.5	13.27873
345	136	0.92	0.05	86.76658
345	136	0.92	0.05	86.76658
345	136	0.92	6.5	90.8801
345	136	0.92	6.5	90.8801

presents the experimental design conducted to determine the influence of key operating parameters on the power required to activate the jaw-type BOP using an axial piston pump. The study included four input factors:

• Pressure bar,
• Flow rate L/min,
• Volumetric efficiency %,
• Leakage L/min.

For each of the 16 combinations of these factor values, the corresponding required power was measured kW.

The experiment was conducted in accordance with the principles of DOE, with controlled variation in factor values at two levels. The selected values were chosen to represent typical operating regimes of the system under both high and low loads. Each combination was repeated twice, enabling the assessment of measurement repeatability and the identification of experimental error.

3.2. Taguchi Analysis

Table 2. Response table for signal-to-noise (S/N) ratios—“smaller-is-better” criterion.

Level	Pressure [bar]	Flow Rate [L/min]	Vol Efficiency [%]	Leakage [L/min]
1	−15.96	−14.06	−23.09	−22.08
2	−29.94	−31.84	−22.81	−23.83
Delta	13.98	17.78	0.28	1.75
Rank	2	1	4	3

presents the results of the Taguchi analysis for S/N ratios, using the “smaller-is-better” criterion with the aim of minimizing the required power. Each row corresponds to the average S/N ratio for a specific level of the observed factor.

The results show the following:

The greatest influence on power variability is exerted by flow rate (Δ = 17.78), ranked as the most dominant factor. It is followed by pressure (Δ = 13.98), then clearance leakage (Δ = 1.75). Volumetric efficiency has the smallest influence (Δ = 0.28), indicating its relatively stable role compared to the other factors.

Lower (less negative) S/N ratios indicate a more stable system with lower power consumption, whereas higher negative S/N ratios indicate greater variation in results and less desirable performance. This analysis is essential for identifying which factors need to be precisely controlled in order to optimize the system.

Table 3. Response table for mean output values (power) by factor levels.

Level	Pressure [bar]	Flow Rate [L/min]	Vol Efficiency [%]	Leakage [L/min]
1	9.767	6.809	45.577	28.697
2	50.055	53.014	14.245	31.126
Delta	40.288	46.205	31.332	2.429
Rank	2	1	3	4

presents the average power values in kW for each level of the analyzed factors within the Taguchi methodology. Unlike the S/N ratios, this table allows for a direct interpretation of the factors’ influence on absolute output values, which is particularly useful for engineering decision-making.

The most significant influence was observed for the flow rate factor, where the change from level 1 to level 2 leads to an increase in average power from 6.809 kW to 53.014 kW (difference Δ = 46.205). The following factors are:

• Pressure (Δ = 40.288),
• Volumetric efficiency (Δ = 31.332),
• Leakage (Δ = 2.429), which is the least pronounced effect.

Here, the positive impact of a higher efficiency level (0.95) also becomes evident, as level 1 for this factor yields an average power of 45.577 kW, while level 2 results in a significantly lower value of 14.245 kW. This confirms that increasing efficiency is associated with reduced energy consumption.

The ranking of factors shows the same order of importance as in the S/N analysis: flow rate > pressure > efficiency > leakage.

The results presented in Table 2 and Table 3 demonstrate a high degree of consistency, despite the application of different analysis criteria. In both cases, flow rate and pressure were identified as the most influential factors affecting the required power, while volumetric efficiency and leakage played a considerably less significant role.

This agreement confirms the robustness of the obtained results and justifies the selection of experiments conducted using the Taguchi method. The use of two independent metrics (S/N ratios and mean values) further validates the established priority ranking, which is essential for making reliable engineering decisions during the system optimization phase.

Figure 1 illustrates the main effects of the individual factors (pressure, flow rate, volumetric efficiency, and leakage) on the mean value of the required power. Each point on the graph represents the average power value for a given factor level, while the slope of each line reflects the strength of the factor’s influence.

Key interpretations:

• Flow rate shows the most pronounced effect—increasing from 15 L/min 136 L/min results in a rise in average power from around 7 kW to over 53 kW.
• Pressure also has a strong impact—increasing from 69 bar to 345 bar leads to a significant rise in power.
• Volumetric efficiency has a negative effect—higher efficiency (0.95) results in lower average power (which is desirable).
• Leakage has the least influence, with a slight increase in average power between 0.05 L/min and 6.5 L/min.

Such a graph is particularly useful as it allows for a quick visual assessment of factor priorities, which is a key strength of the Taguchi methodology. It indicates that minimizing power primarily requires controlling flow rate and pressure, while variations in leakage are of lesser significance (Figure 3).

Figure 4 illustrates how the average S/N ratios change depending on the level of each factor (pressure, flow rate, volumetric efficiency, and leakage), in accordance with the Taguchi optimization methodology aimed at minimizing the output variable—in this case, power.

Interpretation by factors:

• Flow rate and pressure show the most pronounced drop in S/N values when transitioning from lower to higher levels:
  • For flow rate: from approximately −14 dB (15 L/min) to approximately −32 dB (136 L/min),
  • For pressure: from approximately −16 dB (69 bar) to approximately −30 dB (345 bar),
  • These differences indicate high system sensitivity to these parameters, meaning that higher values significantly deteriorate stability and increase power variability.
• Volumetric efficiency exhibits a very slight positive effect—higher efficiency (0.95) results in a slightly better S/N ratio, implying reduced variability in power.
• Clearance leakage also has a small but noticeable negative impact—increasing leakage from 0.05 L/min to 6.5 L/min leads to a decline in the S/N ratio, indicating a slight increase in system instability.

It is important to emphasize that all conclusions are fully consistent with the tabular response analysis (Table 2), which further confirms the validity of the experimental results.

3.3. Regression Analysis

Following the identification of key factors using the Taguchi method, a multiple linear regression analysis was conducted to quantify the influence of each individual parameter on the output variable, the power required to activate the BOP. This method enables not only the assessment of the significance of each factor, but also the construction of a mathematical model that can be used to predict system behavior under various operating conditions.

The result of this analysis is a regression model that links power with the relevant input parameters. The model is presented by Equation (5):

Power = 946.1 + 0.14597 × Pressure + 0.38186 × Flow rate − 1044.4 × Vol efficiency + 0.3766 × Leakage

(5)

In this equation:

• Power represents the dependent variable in kW,
• Pressure in bar, flow rate in L/min, volumetric efficiency in %, and leakage in L/min are the independent variables.

The coefficients for each variable represent the average change in the output value per unit change in the corresponding factor, assuming all other parameters remain constant. For example, an increase in pressure by 1 bar is expected to result in an increase in power of approximately 0.146 kW. Conversely, an increase in volumetric efficiency leads to a significant reduction in the required power, confirming its role in the energy optimization of the system.

Table 4. Regression model coefficients and statistical indicators.

Term	Coef	SE Coef	T-Value	p-Value	VIF
Constant	946.1	15.2	62.10	0.000
Pressure	0.14597	0.00177	82.48	0.000	1.00
Flow rate	0.38186	0.00404	94.59	0.000	1.00
Vol efficiency	−1044.4	16.3	−64.14	0.000	1.00
Leakage	0.3766	0.0757	4.97	0.000	1.00

presents the coefficients obtained from the multiple linear regression analysis, along with their standard error of the coefficients (SE Coef), T-values, p-values, and VIF, which indicate the presence of multicollinearity.

Interpretation of key indicators:

• All predictors have p-values less than 0.001, confirming that they are statistically significant at a high confidence level (α = 0.05),
• The highest T-values were recorded for flow rate (T = 94.59) and pressure (T = 82.48), further confirming their dominant roles in the model,
• The SE Coef represents a measure of the variability of the estimated regression coefficients. A lower SE Coef. indicates greater precision in the coefficient estimation. In this model, all factors have very low standard errors (e.g., SE = 0.00404 for flow), contributing to the high T-values and confirming the model’s reliability. It is particularly important to emphasize that the ratio between the Coef. and SE Coef. is exactly what determines the T-value:

$$T={\displaystyle \frac{Coef}{SECoef}}$$

(6)

Based on this, it can be concluded that all factors in the model are not only statistically significant but also precisely estimated, without pronounced fluctuations that would undermine the reliability of the results.

• The negative coefficient for volumetric efficiency (−1044.4) indicates a strong inverse effect of this parameter—increasing efficiency results in a significant reduction in the required power,
• VIF values for all factors are 1.00, indicating the absence of multicollinearity and confirming the stability of the model.

This table provides a solid statistical foundation for the interpretation and further application of the regression model in engineering optimization.

Table 5. Statistical indicators of regression model quality (Model Summary).

S	R-sq [%]	R-sq (Adj) [%]	R-sq (Pred) [%]
0.976960	99.94	99.92	99.88

presents the key indicators of the regression model quality:

• S = 0.97696—standard error of the regression. This value represents the average difference between the experimental and predicted power values. A lower value indicates high model accuracy. In this case, the deviation is less than 1 kW, which is an excellent result considering the range of power values in the experiment.
• R-sq = 99.94%—coefficient of determination. This value shows that the model explains 99.94% of the total variation in the output variable (power). This indicates an extremely strong correlation between the input factors and the output variable.
• R-sq (adj) = 99.92%—adjusted coefficient of determination. It accounts for the number of parameters in the model and reduces the R² value accordingly to avoid artificially inflating accuracy. The high value confirms that no factor is redundant and that the model is not overloaded with unnecessary variables.
• R-sq (pred) = 99.88%—predicted R². This value indicates the reliability of the model when predicting new, unseen data. A very high percentage suggests that the model can be safely used for optimization and engineering decision-making.

All the parameters in this table confirm that the obtained model is exceptionally well-fitted, stable, and predictively robust.

Table 6. ANOVA—Analysis of variance by factors.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Regression	4	18,982.2	4745.55	4972.02	0.000
Pressure	1	6492.4	6492.43	6802.26	0.000
Flow rate	1	8539.4	8539.44	8946.97	0.000
Vol efficiency	1	3926.7	3926.74	4114.14	0.000
Leakage	1	23.6	23.60	24.73	0.000
Error	11	10.5	0.95
Lack-of-Fit	3	10.5	3.50	*	*
Pure Error	8	0.0	0.00
Total	15	18,992.7

presents the results of the ANOVA analysis, which examines whether the factors used in the model significantly influence the changes in the output variable—power.

Key indicators and interpretation:

• All factors (pressure, flow rate, volumetric efficiency, leakage) have p-values equal to 0.000, indicating that they are statistically highly significant (at a 99.9% confidence level). Their inclusion in the model is fully justified.
• Flow rate (F = 8946.97) and pressure (F = 6802.26) have the highest F-values, indicating that they contribute the most to explaining the variability in power.
• Volumetric efficiency also has a very high F-value (4114.14), but a negative coefficient (as previously explained), meaning it contributes to power reduction.
• Leakage has a smaller yet still significant F-value (24.73), making it the least significant factor compared to the others.
• The Lack-of-Fit test (The Lack-of-Fit test (see Table 6) suggests that no additional, more complex models were necessary—which is expected given the very high R², marked with asterisks) suggests that no additional, more complex models were necessary—which is expected given the very high R².

Figure 3 shows the standardized effects of the four analyzed factors:

• A—pressure,
• B—flow rate,
• C—volumetric efficiency,
• D—leakage.

The red vertical line at approximately 2.2 represents the critical t-test value for a significance level of α = 0.05. Only the bars that exceed this threshold are considered statistically significant.

Analysis:

• Flow rate (B) shows the highest standardized effect and the most significant contribution to the total variability of power,
• It is followed by pressure (A) and volumetric efficiency (C), which are also statistically significant, as their effects are to the right of the critical line,
• Clearance leakage (D) has a very small effect that falls below the significance level, meaning this factor has a weak contribution to the model in terms of power influence, although it was previously deemed significant in the ANOVA due to high measurement precision.

The Pareto chart confirms the previous findings obtained through regression analysis and the Taguchi method—flow rate and pressure are the most dominant factors, while leakage is marginal.

Figure 5 shows four residual plots used to verify the assumptions of regression analysis: normality, constant variance, and independence of residuals. Taken together, these plots provide an overview of the model quality and potential deviations.

• Normal Probability Plot (top left):

• Residuals are compared against the ideal normal distribution,
• The points approximately follow a straight line, indicating a normal distribution of residuals, which is a desirable condition for model validity.

• Versus Fits (top right):

• Displays the distribution of residuals relative to the model prediction,
• No discernible pattern is observed (e.g., curvature or funnel shape), indicating that the assumption of homoscedasticity (constant variance) is satisfied.

• Histogram (bottom left):

• Residuals are symmetrically distributed, further confirming normality,
• Although the distribution is not perfectly bell-shaped, clear bilateral symmetry is observed

• Versus Order (bottom right):

• Residuals are shown in the order of experimental measurements,
• There is no systematic pattern or trend, indicating that the errors are independent and that the order of measurements did not affect the model.

Based on the plots in Figure 6, it can be concluded that the regression model satisfies all key assumptions: normality, homoscedasticity, and independence of residuals. This further confirms the validity and reliability of the previously developed model.

3.4. GWO Optimization

In this study, the GWO algorithm was applied to determine the optimal combination of process parameters (pressure, flow rate, volumetric efficiency, and clearance leakage) with the aim of maximizing the power expression.

The GWO algorithm was executed a total of 20 times, with each run involving 30 agents over 100 iterations. This multi-run approach ensures a reliable assessment of optimization stability and contributes to the identification of the global optimum within the defined variable bounds, as presented in

Table 7. Boundary values of variables.

Parameter	Lower Limit	Upper Limit
Pressure [bar]	69	345
Flow rate [L/min]	15	136
Vol efficiency [%]	0.92	0.95
Leakage [L/min]	0.05	6.5

.

The objective function used in the optimization process is defined by the mathematical expression shown in Equation (7):

$$f\left(x\right)=946.1+{0.14597}_{X_1}+{0.38186}_{X_2}-{1044.4}_{X_3}+{0.3766}_{X_4}$$

(7)

where

• X₁—pressure,
• X₂—flow rate,
• X₃—volumetric efficiency,
• X₄—leakage.

During the execution of the algorithm, the evolution of the best solution is monitored, and after multiple iterations, the best, worst, average, and standard deviation results are statistically evaluated, enabling a reliable assessment of the solution’s stability.

Based on 20 independent runs of the GWO algorithm, stable and consistent results were obtained.

Table 8. Statistical overview of GWO optimization results.

Statistical Value	Result
Best fitness	5.0001
Worst fitness	5.0218
Average value	5.0051
Standard deviation	0.0057

presents the basic statistical indicators of the optimization, with the best obtained power value being 5.0001 kW. The optimization identified the input parameter values that result in this optimum, namely: pressure of 257.94 bar, flow rate of 33.20 L/min, volumetric efficiency of 0.95%, and leakage of 1.997 L/min. The obtained result confirms the high precision and efficiency of the optimization within the defined limits.

3.5. Reliability Model of Ram BOP Operation

In reliability analysis of systems, complex operational conditions are often encountered that cannot be adequately modeled using a simple binary classification such as “failure” (0) or “proper functioning” (1). Such a binary approach disregards the continuous nature of operational parameters, especially in complex technical systems where the operational status may vary significantly depending on load and other factors. Therefore, the concept of relative power is used as an indicator of system load and condition—defined as the ratio between the current system power and a predefined optimal value. This approach allows for the quantification of deviations in system operation through a continuous parameter, providing a more detailed insight into its condition. In practice, if the relative power remains within a specific optimal tolerance range—e.g., ±10% around the nominal or optimal power—the system is considered reliable, i.e., capable of functioning properly without significant deviations that could indicate potential failures or performance degradation. This criterion offers a more flexible and realistic framework for reliability assessment, as it reflects natural fluctuations in system operation while enabling timely detection of anomalies that exceed acceptable limits and may signal the onset or development of faults. The reliability of a ram BOP can be approximated as a function of deviations of operating parameters from their optimal values, where the deviation is quantified as a distance Din a multidimensional parameter space. Based on this, an exponential reliability model can be defined as a function of deviation, as shown in Equation (10). This approach allows for a quantitative evaluation of the impact of parameter variations on the overall system reliability. Specifically, the smaller the deviation value, the more reliable the system is considered, since its operating conditions remain closer to the optimal regime, thereby minimizing the risk of failure or performance degradation. Conversely, an increase in deviation D indicates a greater departure from optimal conditions, which may lead to reduced BOP effectiveness and increased likelihood of undesired events. Formulating reliability as a function of deviation represents an efficient method for quantifying the stability and functional capability of the system under real working conditions, thereby enabling the application of mathematical and statistical models for precise monitoring and performance improvement. performance:

$$R=e^{-\alpha D}$$

(8)

where

• R—the probability of reliable performance of the BOP under specific operating conditions (0–1),
• α—coefficient of system sensitivity,
• D—overall system deviation index from the optimal values.

Defining the deviation of key parameters from the optimal values identified by GWO optimization (P = 257.94 bar, Q = 33.20 L/min, η = 0.95, C = 1.997 L/min) enables precise quantification of the system’s overall deviation. For this purpose, a composite deviation index D is introduced, representing the weighted Euclidean distance of actual parameters from their optimal values, normalized for each parameter individually. Mathematically, the value of D is defined by Equation (9):

$$D={\omega }_1\times {\left({\displaystyle \frac{P-257.94}{257.94}}\right)}^2+{\omega }_2\times {\left({\displaystyle \frac{Q-33.20}{33.20}}\right)}^2+{\omega }_3\times {\left({\displaystyle \frac{\eta -0.95}{0.95}}\right)}^2+{\omega }_4\times {\left({\displaystyle \frac{C-1.997}{1.997}}\right)}^2$$

(9)

where ω_1,2,3,4—significance weights:

• ω₁ = 0.3—pressure weight,
• ω₂ = 0.4—flow rate weight,
• ω₃ = 0.25—vol efficiency weight,
• ω₄ = 0.05—leakage weight.

Based on the introduction of actual parameters from the conducted research, the deviation factor is:

$$D=0.0249$$

The reliability model of the jaw-type BOP is calculated using the exponential function defined by Equation (10), with specific weight values and parameters obtained from experimental results.

$$R=exp\left(-\alpha \times \left[0.3\times {\left({\displaystyle \frac{P-257.94}{257.94}}\right)}^2+0.4\times {\left({\displaystyle \frac{Q-33.20}{33.20}}\right)}^2+0.25\times \left({\displaystyle \frac{\eta -0.95}{0.95}}\right)+0.05\times {\left({\displaystyle \frac{C-1.997}{1.997}}\right)}^2\right]\right)$$

(10)

where

• α—sensitivity factor (α = 3).

Based on the obtained deviation value of the actual parameters from the optimal ones, the reliability of the jaw-type BOP operation is:

$$R=92.79\%$$

4. Discussion

The results obtained through the application of the Taguchi method, regression analysis, and the Grey Wolf Optimizer (GWO) indicate a clearly defined dependence of the power required to activate the jaw-type BOP on the technological parameters of the axial piston pump. The most influential factors—flow rate and pressure—were consistently identified as dominant across all applied analytical methods, ensuring multi-level validation of the findings.

The S/N ratio analysis based on the “smaller-is-better” criterion within the Taguchi approach showed that variation in flow rate has the most pronounced impact on power consumption (Δ = 17.78), confirming that this parameter must be prioritized in system design. Although clearance leakage was also identified as a factor, its relatively low contribution to variability (Δ = 1.75) suggests that the system is more robust to changes in this parameter, which is valuable engineering information in the context of tolerances and maintenance.

Further insight is provided by the regression model, which achieved an exceptionally high coefficient of determination (R² = 99.94%), quantitatively confirming the reliability of the system’s mathematical representation. Moreover, the negative coefficient for volumetric efficiency (−1044.4) is particularly significant: increasing efficiency substantially reduces the required power, with direct implications for energy optimization and pump system design. This result emphasizes that volumetric efficiency is not only statistically significant but also practically decisive for system performance. Even modest improvements in efficiency can yield considerable energy savings and simultaneously reduce internal leakage, thereby enhancing the long-term reliability of BOP control systems.

The Pareto chart and ANOVA tests additionally confirm that the contribution of flow rate and pressure is statistically significant at the 99.9% confidence level (p < 0.001), while the effect of clearance leakage is marginal yet consistently present. This differentiation in factor significance is especially useful for defining maintenance priorities—while flow and pressure must be precisely controlled, acceptable leakage tolerances do not require equally strict oversight.

Compared to existing studies that primarily focus on either empirical tuning or component-level analysis of BOP systems [6,7,19], this work presents a novel integrated methodology. While previous approaches often relied on structural reliability assessments or static models, the current study advances the field by coupling experimental design (Taguchi) with metaheuristic optimization (GWO), thereby enabling energy-efficient control parameterization. This holistic approach improves not only system performance but also design robustness under realistic constraints.

However, it is necessary to acknowledge the limitations of this research. The experimental setup was developed under controlled laboratory conditions that replicate industrial parameters but do not account for the full complexity of field operations. Real-world well environments introduce additional variables such as dynamic formation pressure changes, multi-phase flow behavior, vibration effects, and temperature fluctuations, all of which may influence the hydraulic system’s stability and energy demand.

To ensure the broader applicability of the developed models, future work should focus on experimental validation in operational drilling environments. Integration of real-time sensor data and historical failure information from actual BOP systems could support the development of adaptive or self-correcting control logic. Furthermore, incorporating nonlinear modeling techniques (e.g., neural networks or ANFISs) may enhance predictive capabilities in more complex or stochastic scenarios.

From an energy-efficiency perspective, the optimized configuration reduced the required power to just 5.0001 kW, in contrast to over 90 kW observed in high-demand scenarios. This represents a power reduction exceeding 90%, which is especially valuable in emergency situations where energy availability may be limited. Such reductions contribute to system autonomy and operational continuity during critical failure events, directly supporting safety and environmental goals. Moreover, when translated into long-term operation, these savings may result in significant cost reductions and decreased emissions, thus aligning with the broader principles of sustainable energy management in the oil sector.

The optimized hydraulic parameters obtained in this study (pressure 257.94 bar and flow rate 33.20 L/min) were compared with industry standards and practical ranges used in surface control systems of ram-type BOPs. According to API Standard 53 [44], surface control systems must provide sufficient hydraulic capacity to close each ram preventer within 30 s, which is achievable at flow rates in the range of 6–9 gpm (≈23–34 L/min). Therefore, the optimized flow of 33.20 L/min is consistent with typical field practice. On the other hand, most surface control manifolds are designed for a working pressure of 3000 psi (≈207 bar), while many ram BOP actuators require only 1200–1500 psi (≈83–103 bar) to close. Thus, although the experimental optimum of 257.94 bar is technically feasible for certain high-pressure units, it exceeds the nominal working pressure of many surface control systems. It should be noted that the higher pressure was selected as an additional safety margin to prevent unintended opening of the preventer stack due to dynamic pressure fluctuations or partial volumetric efficiency loss. This indicates that a constrained optimization with P ≤ 207 bar could further enhance practical applicability, while still maintaining a safety margin as a priority under real operating conditions. Such an extension is therefore recommended for future investigation.

Overall, the combined use of experimental, statistical, and heuristic methods has proven highly effective in achieving robust system optimization with minimal experimental effort. The methodology presented here can serve as a foundation for further development of intelligent, energy-aware control systems for critical hydraulic components in high-risk industrial environments.

Beyond the numerical optimization results, the reduction in power demand carries direct industrial implications. Lower energy requirements enable the use of smaller accumulator capacities and more compact power supply units, which directly decreases both capital investment and maintenance costs. Additionally, reduced power peaks translate into greater operational autonomy, particularly in remote or offshore drilling sites where energy supply is constrained and reliability is paramount. From a sustainability perspective, minimizing power consumption also reduces fuel usage for diesel-driven generators, indirectly lowering emissions and improving the environmental footprint of drilling operations. Therefore, the proposed optimization does not only advance the technical efficiency of BOP control systems but also enhances their economic viability and environmental compatibility in real-world oil applications

The developed model has direct applicability in field operations of hydraulic BOP control systems. By providing a quantitative link between pump operating parameters (pressure, flow rate, and leakage) and the required activation power, the model enables engineers to determine optimal control settings that minimize energy consumption while ensuring reliable well closure. In practice, this framework can be used to:

I.

guide the design and adjustment of surface control units,

II.

support predictive maintenance by monitoring deviations of operating parameters from the optimized configuration, and

III.

reduce energy demand, thereby allowing smaller accumulator capacity and improving system autonomy in remote drilling locations. These functions highlight the potential of the proposed methodology to enhance both the efficiency and reliability of blowout preventer operation under real industrial conditions.

5. Conclusions

This research demonstrated that optimizing the power required to activate a jaw-type blowout preventer (BOP) using an axial piston pump can be successfully achieved through the application of combined statistical and metaheuristic methods. Taguchi analysis, regression modeling, and the Grey Wolf Optimizer (GWO) algorithm enabled the identification of key factors, quantification of their influence, and determination of an optimal configuration with minimal energy consumption.

The following key conclusions can be drawn:

• Key Influencing Factors: Flow rate and pressure were consistently identified as the most influential parameters across all analytical approaches. Volumetric efficiency was also shown to have a significant impact, with higher efficiency directly contributing to reduced power demand. Clearance leakage had a lesser, though statistically relevant, effect.
• Optimization Results: The optimization process identified the operating point at a pressure of 257.94 bar and a flow rate of 33.20 L/min, with a volumetric efficiency of 0.95 and a leakage of 1.997 L/min, resulting in a minimized activation power of 5.0001 kW. This corresponds to a reduction of more than 90% compared with the highest measured power value in the experimental design, thereby confirming the effectiveness of the proposed methodology. These values not only quantify the optimal configuration but also clearly demonstrate that flow rate is the dominant factor influencing power demand, followed by pressure, volumetric efficiency, and leakage. The fact that the optimum flow rate aligns with the requirements of API Standard 53 underscores the industrial relevance of the results, whereas the higher pressure emphasizes the need for further refinement and verification under field operating conditions. Moreover, the reliability level of approximately 93% indicates that the optimized parameters not only reduce energy consumption but also enhance the operational safety and dependability of the BOP system. Overall, the findings suggest that the combined Taguchi–GWO methodology is not only effective for minimizing power consumption but also representative of a generalizable framework for improving the energy efficiency, safety, and reliability of hydraulically driven safety devices.
• Model Validity and Reliability: The developed regression model achieved a coefficient of determination of R² = 99.94%, while ANOVA confirmed the statistical significance of all included factors. Residual analysis and VIF values confirmed the model’s robustness and absence of multicollinearity.
• System Reliability Assessment: Based on deviation analysis from the optimal parameter configuration, a reliability model was introduced using a weighted Euclidean distance and an exponential reliability function. Under optimized conditions, the BOP system achieved a reliability index of approximately 93%, qualifying it as a highly reliable safety component.
• Practical Engineering Implications: The findings underscore the potential for energy savings and performance improvements in critical hydraulic systems through integrated modeling and optimization techniques. This is particularly relevant for safety-critical applications in the oil industry, where system autonomy and rapid response are essential.
• Future Directions: While the proposed model demonstrated excellent performance under controlled laboratory conditions, future research should include experimental validation in real-world environments, where factors such as temperature variation, formation pressure dynamics, and fluid composition may affect system behavior. Additional research may explore dynamic simulations, repeated activation cycles, and nonlinear modeling techniques such as neural networks or adaptive neuro-fuzzy inference systems (ANFISs). Furthermore, multi-criteria optimization—including cost, safety margin, and response time—could enhance system adaptability and applicability in field operations. Laboratory and field conditions differ in terms of temperature fluctuations, formation-induced dynamic pressures, and variable fluid properties, which can significantly affect the performance of hydraulic ram-type BOP systems. Laboratory experiments allow precise measurement of hydraulic parameters in a controlled environment; however, they may not fully capture transient behaviors and operational uncertainties encountered under actual field conditions. To address these limitations, future research should incorporate real-time sensor data from both surface and downhole monitoring systems. This approach would enable dynamic adjustment of hydraulic parameters based on actual operating conditions, enhancing model predictive accuracy and improving the reliability and safety of BOP operations in the field. Future work should also include a systematic sensitivity analysis of the GWO hyperparameters, particularly the number of agents and iterations. In this study, the choice of 30 agents and 100 iterations was pragmatic and ensured a stable balance between exploration and exploitation at low computational cost. However, a detailed analysis could provide additional insights into robustness and support the development of adaptive optimization strategies for more complex hydraulic systems.

In summary, the methodological framework presented in this study provides a valuable foundation for the design of energy-efficient, robust, and intelligent control systems for hydraulic safety equipment. Its adaptability to future enhancements makes it suitable for application in a wide range of industrial conditions, particularly in the context of next-generation blowout prevention and well control systems. Modern industrial systems, such as those used for monitoring and controlling safety equipment in the oil industry (e.g., BOPs), heavily rely on accurate and continuous data acquisition. This task is performed by numerous sensors that must operate reliably and without interruption, often in environments lacking access to the electrical grid. Consequently, there is a growing need for energy-efficient and autonomous solutions. The utilization of renewable energy sources, particularly solar power, emerges as an optimal solution for powering such sensor stations. Moreover, advanced optimization methods, such as the Taguchi method and GWO, which have already demonstrated effectiveness in reducing energy consumption during BOP operation, can be successfully adapted for optimizing the parameters of solar-powered systems for similar Internet of Things (IoT) nodes.