Design and Performance Analysis for the Low-Power Holding Mechanism of the All-Electric Subsea Gate Valve Actuator

: The all-electric subsea gate valve actuator is one of the critical components of the all-electric subsea production control system. To bridge the gap of the low-power holding mechanism in the all-electric subsea gate valve actuator of the subsea production system, minimize the power consumption and cable number for control and improve the open-position keeping performance of all-electric subsea gate valve actuator, this paper proposed a novel low-power holding mechanism for the all-electric subsea gate valve actuator which can be applied to all-electric subsea gate valve actuators with various valve sizes and process pressure ratings. The proposed low-power holding mechanism uses an electromagnet as a driving element, combines the spiral transmission and the cam-like transmission, and only requires a holding force of approximately 2–7% of the maximum load of the closing spring to keep the valve open. The proposed low-power holding mechanism converts the axial force of the closing spring into the circumferential force, which substantially reduces the output force required for the driving element of the low-power holding mechanism and the number of the actuator’s control cables. Analytic models are created for the lockable maximum load of the closing spring and the permissible stroke of the locking tab with regard to the design variables. The parameter e ﬀ ects and the corresponding sensitivities are discussed by numerical analysis. The design parameters and the lockable maximum load of the closing spring of the low-power holding mechanism are obtained.


Introduction
Late in the last century, in order to meet the needs of deep-water gas and oil exploitation, subsea equipment suppliers began to develop and improve the all-electric subsea control technology [1]. The all-electric Subsea Production Control System (AE), a leading-edge and breakthrough technology, which didn't need hydraulic fluid and was driven completely by electric power, was proposed and developed [2]. The key benefits of the AE system compared to the conventional Multiple Electric-Hydraulic Control System (MEH) are as follows [3][4][5][6][7][8][9][10]: the simplified system structure, enhanced control performance, improved system reliability and availability, increased the oil recovery factor and environmental friendliness. Furthermore, the advantages of AE system also include: the decreased power consumption [11,12], reduced Capital expenditure(CAPEX) and Operating number for control and improve the open-position keeping performance of an all-electric subsea gate valve actuator, only requiring a holding force of approximately 2%~7% of the maximum load of the closing spring to keep the valve open and with the ability to be applied to all-electric subsea gate valve actuators with various valve sizes and process pressure ratings, to bridge the gap of the low-power holding mechanism in the all-electric subsea gate valve actuator of the subsea production system.
The rest of the paper is organized as follows: Section 2 briefly introduces the working process of the all-electric subsea gate valve actuator proposed, followed by a full presentation of the configuration, work principle and advantage of the proposed failsafe closing and low-power holding mechanism. Section 3 presents the mechanical and kinematic models. In Section 4, the detailed discussion of the parameter effects and sensitivity analysis is performed based on the established model and results. Finally, conclusions are given in Section 5.

Research on the Configuration of Failsafe Closing and Low-Power Holding Mechanism
Based on the function of the all-electric subsea gate valve actuator, a configuration of the all-electric subsea gate valve actuator is proposed in this paper, as shown in Figure 1, wherein the structure mainly includes the subsea gate valve assembly, failsafe closing and low-power holding mechanism, subsea ambient pressure compensating mechanism, screw drive mechanism, multi-motor redundant driving mechanism, valve position indicating mechanism and ROV interface. During the valve opening process, multi-motor redundant driving mechanism or ROV drives the valve stem and gate downward to open the subsea gate valve through the screw drive mechanism. After the subsea gate valve is fully opened, the low-power holding mechanism locks the failsafe closing mechanism to ensure that the valve remains open with the minimum power consumption. When the all-electric subsea production control system breaks down, the low-power holding mechanism and multi-motor redundant driving mechanism are power-off, the failsafe closing mechanism drives the valve stem and gate upward to realize the emergency failure close.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 17 gate valve actuator, only requiring a holding force of approximately 2%~7% of the maximum load of the closing spring to keep the valve open and with the ability to be applied to all-electric subsea gate valve actuators with various valve sizes and process pressure ratings, to bridge the gap of the low-power holding mechanism in the all-electric subsea gate valve actuator of the subsea production system. The rest of the paper is organized as follows: Section 2 briefly introduces the working process of the all-electric subsea gate valve actuator proposed, followed by a full presentation of the configuration, work principle and advantage of the proposed failsafe closing and low-power holding mechanism. Section 3 presents the mechanical and kinematic models. In Section 4, the detailed discussion of the parameter effects and sensitivity analysis is performed based on the established model and results. Finally, conclusions are given in Section 5.

Research on the Configuration of Failsafe Closing and Low-Power Holding Mechanism
Based on the function of the all-electric subsea gate valve actuator, a configuration of the all-electric subsea gate valve actuator is proposed in this paper, as shown in Figure 1, wherein the structure mainly includes the subsea gate valve assembly, failsafe closing and low-power holding mechanism, subsea ambient pressure compensating mechanism, screw drive mechanism, multi-motor redundant driving mechanism, valve position indicating mechanism and ROV interface. During the valve opening process, multi-motor redundant driving mechanism or ROV drives the valve stem and gate downward to open the subsea gate valve through the screw drive mechanism. After the subsea gate valve is fully opened, the low-power holding mechanism locks the failsafe closing mechanism to ensure that the valve remains open with the minimum power consumption. When the all-electric subsea production control system breaks down, the low-power holding mechanism and multi-motor redundant driving mechanism are power-off, the failsafe closing mechanism drives the valve stem and gate upward to realize the emergency failure close. Failsafe closing mechanisms of the subsea valve include the mechanical failsafe closing mechanism and Uninterrupted Power System (UPS) failsafe closing mechanism [29,30]. As the service life cycle of the subsea Xmas tree or manifold is usually around 20 years, long term continuous power supply makes the UPS failsafe closing mechanism unpractical. The mechanical failsafe closing mechanism is most commonly used for this purpose. In the application of the mechanical failsafe closing mechanism for a subsea valve actuator, the spring-type configuration is Failsafe closing mechanisms of the subsea valve include the mechanical failsafe closing mechanism and Uninterrupted Power System (UPS) failsafe closing mechanism [29,30]. As the service life cycle of the subsea Xmas tree or manifold is usually around 20 years, long term continuous power supply makes the UPS failsafe closing mechanism unpractical. The mechanical failsafe closing mechanism is most commonly used for this purpose. In the application of the mechanical failsafe closing mechanism for a subsea valve actuator, the spring-type configuration is reliable and widely used [4,25]. However, the force of the closing spring needs to be big enough to oppose the high-pressure fluid in the valve when a failure close occurs. So when the valve is opened, the power system needs to resist the large spring forces, which consume a lot of energy [25]. Thus, the low-power holding mechanism also needed to be designed in the actuator of this paper. In addition, a single Electric Subsea Control Module (ESCM) is claimed to control up to 32 functional groups [6]. However, a single ESCM can control up to 16 functional groups including sensors in an engineering application [12], which puts forward requirements for the number of the actuator's control cables.
Based on the above analysis, the low-power holding mechanism is proposed in this paper, as shown in Figure 2, in which a spiral groove is machined inside the locking ring sleeve; a thread and four cam-like notches are machined outside the locking ring and inside the bottom, respectively. Four locking tabs are adopted due to the higher symmetry and stability of this configuration, compared with the design of two locking tabs. Furthermore, the requirements of this configuration for the size of the locking ring are moderate, because each locking tab needs a cam-like notch of the locking ring. The higher the number of locking tabs, the higher the number of cam-like notches necessary, which leads the size of the locking ring being easily unable to meet the requirements. Thus, the configuration of four locking tabs has advantages in realizing product function, improving product performance and meeting design constraints.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 17 reliable and widely used [4,25]. However, the force of the closing spring needs to be big enough to oppose the high-pressure fluid in the valve when a failure close occurs. So when the valve is opened, the power system needs to resist the large spring forces, which consume a lot of energy [25]. Thus, the low-power holding mechanism also needed to be designed in the actuator of this paper. In addition, a single Electric Subsea Control Module (ESCM) is claimed to control up to 32 functional groups [6]. However, a single ESCM can control up to 16 functional groups including sensors in an engineering application [12], which puts forward requirements for the number of the actuator's control cables. Based on the above analysis, the low-power holding mechanism is proposed in this paper, as shown in Figure 2, in which a spiral groove is machined inside the locking ring sleeve; a thread and four cam-like notches are machined outside the locking ring and inside the bottom, respectively. Four locking tabs are adopted due to the higher symmetry and stability of this configuration, compared with the design of two locking tabs. Furthermore, the requirements of this configuration for the size of the locking ring are moderate, because each locking tab needs a cam-like notch of the locking ring. The higher the number of locking tabs, the higher the number of cam-like notches necessary, which leads the size of the locking ring being easily unable to meet the requirements. Thus, the configuration of four locking tabs has advantages in realizing product function, improving product performance and meeting design constraints. The locking and holding process of the low-power holding mechanism is shown in Figure 2. The driving mechanism pushes the spring support cylinder against the closing spring until the valve is fully opened, at which point the closing spring is at the limit, and the electromagnets are energized and push the locking ring sleeve upward. The locking ring sleeve drives the locking ring to rotate forward through the spiral groove. Meanwhile, the locking ring drives locking tabs to move inward through the cam-like notch. The locking process of the low-power holding mechanism is completed when the locking tabs fully enter the actuator working cavity and make contact with the slope of the spring support cylinder to block the cylinder moving back. After the locking process, the holding process of the low-power holding mechanism is started. In the holding process, the locking tab and the spring support cylinder constrain each other, and the position of the locking tab is restricted by the locking ring, to ensure that the valve remains open with the minimum power consumption. The locking and holding process of the low-power holding mechanism is shown in Figure 2. The driving mechanism pushes the spring support cylinder against the closing spring until the valve is fully opened, at which point the closing spring is at the limit, and the electromagnets are energized and push the locking ring sleeve upward. The locking ring sleeve drives the locking ring to rotate forward through the spiral groove. Meanwhile, the locking ring drives locking tabs to move inward through the cam-like notch. The locking process of the low-power holding mechanism is completed when the locking tabs fully enter the actuator working cavity and make contact with the slope of the spring support cylinder to block the cylinder moving back. After the locking process, the holding process of the low-power holding mechanism is started. In the holding process, the locking tab and the spring support cylinder constrain each other, and the position of the locking tab is restricted by the locking ring, to ensure that the valve remains open with the minimum power consumption.
When a valve fail-close occurs, the electromagnets are de-energized, thereby the position of the locking tab is not restricted by the locking ring. The unlocking process is opposite to the locking process, and the closing spring force is the driving force for the unlocking process. The closing spring drives the Appl. Sci. 2020, 10, 6119 5 of 17 locking tabs to move outward through the slope of the spring support cylinder. Meanwhile, the locking tabs drive the locking ring to rotate backward through the cam-like notch surface, and the locking ring pushes the locking ring sleeve downward through the spiral groove. The unlocking process is completed when the locking tab and the spring support cylinder are separated. After the unlocking process, the closing spring continues to release rapidly and drives the spring support cylinder and the valve stem upward until the valve is fully closed.
The proposed low-power holding mechanism converts the axial force of the closing spring into the circumferential force by the spiral groove and the cam-like notches, which substantially reduces the output force required for the driving element of the low-power holding mechanism and increases the feasibility and reliability of the holding mechanism. In addition, different from the form of Cameron's "clutch" motor, the proposed low-power holding mechanism is driven by the electromagnet, which effectively reduces the number of the actuator's control cables.
When a fail-close occurs, the maximum load produced by the high-pressure fluid which the all-electric subsea gate valve actuator can oppose depends on the force of the closing spring. According to the configuration of the low-power holding mechanism, the maximum load of the closing spring is related to the size of the low-power holding mechanism. However, the permissible stroke of the locking tab also depends on the size of the low-power holding mechanism and needs to be larger than the given working stroke, which will form a constraint for the optimization of the locking and holding ability of the low-power holding mechanism. Thus, it is necessary to analyze the mechanical and kinematic models of the low-power holding mechanism in detail.

Mechanical Model of the Low-Power Holding Mechanism
The low-power holding mechanism adopts four locking tabs to share the maximum load of the closing spring. Figure 3 shows the force analysis of the spring support cylinder and single locking tab in the holding process, in which F ml /4 is the maximum load of the closing spring shared by the single locking tab, R 16 and R 61 are the interactive force between the spring support cylinder and the single locking tab, R is the radial component force acting on the spring support cylinder by the single locking tab in the symmetrical direction, R 63 and R 36 are the interactive force between the single locking tab and the actuator working cavity, F r and F r are the interactive radial thrust between the single locking tab and the locking ring.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 17 When a valve fail-close occurs, the electromagnets are de-energized, thereby the position of the locking tab is not restricted by the locking ring. The unlocking process is opposite to the locking process, and the closing spring force is the driving force for the unlocking process. The closing spring drives the locking tabs to move outward through the slope of the spring support cylinder. Meanwhile, the locking tabs drive the locking ring to rotate backward through the cam-like notch surface, and the locking ring pushes the locking ring sleeve downward through the spiral groove. The unlocking process is completed when the locking tab and the spring support cylinder are separated. After the unlocking process, the closing spring continues to release rapidly and drives the spring support cylinder and the valve stem upward until the valve is fully closed.
The proposed low-power holding mechanism converts the axial force of the closing spring into the circumferential force by the spiral groove and the cam-like notches, which substantially reduces the output force required for the driving element of the low-power holding mechanism and increases the feasibility and reliability of the holding mechanism. In addition, different from the form of Cameron's "clutch" motor, the proposed low-power holding mechanism is driven by the electromagnet, which effectively reduces the number of the actuator's control cables.
When a fail-close occurs, the maximum load produced by the high-pressure fluid which the all-electric subsea gate valve actuator can oppose depends on the force of the closing spring. According to the configuration of the low-power holding mechanism, the maximum load of the closing spring is related to the size of the low-power holding mechanism. However, the permissible stroke of the locking tab also depends on the size of the low-power holding mechanism and needs to be larger than the given working stroke, which will form a constraint for the optimization of the locking and holding ability of the low-power holding mechanism. Thus, it is necessary to analyze the mechanical and kinematic models of the low-power holding mechanism in detail.

Mechanical Model of the Low-Power Holding Mechanism
The low-power holding mechanism adopts four locking tabs to share the maximum load of the closing spring. Figure 3 shows the force analysis of the spring support cylinder and single locking tab in the holding process, in which Fml/4 is the maximum load of the closing spring shared by the single locking tab, R16 and R61 are the interactive force between the spring support cylinder and the single locking tab, R is the radial component force acting on the spring support cylinder by the single locking tab in the symmetrical direction, R63 and R36 are the interactive force between the single locking tab and the actuator working cavity, Fr and F′r are the interactive radial thrust between the single locking tab and the locking ring.  In the holding process, the spring support cylinder and the single locking tab are in a state of mechanical equilibrium. Figure 4 shows the force diagram of the spring support cylinder and the single locking table as shown in Figure 4a, the relation between R 61 and F ml can be expressed as: where λ is the angle of the slope of the single locking tab; ϕ is the friction angle; F ml is the maximum load of the closing spring.
mechanical equilibrium. Figure 4 shows the force diagram of the spring support cylinder and the single locking table as shown in Figure 4a, the relation between R61 and Fml can be expressed as: where λ is the angle of the slope of the single locking tab; φ is the friction angle; Fml is the maximum load of the closing spring. As shown in Figure 4b, the relation between R61 and Fr can be given as: Combining Equations (1) and (2), the relation between Fr and Fml can be expressed as: (a) (b) Figure 4. The force diagram of (a) the spring support cylinder and (b) the single locking tab. Figure 5 shows the force diagram of the contact between the single locking tab and the locking ring in the holding process, in which R67 and R76 are the interactive forces between the single locking tab and the locking ring. To facilitate the analysis, the force diagram in Figure 5 is simplified, as shown in Figure 6. This assumes that O1 is the center of the actuator working cavity; O2 is the center of the notch of the locking ring; O3 is the center of the curved surface of the locking tab; C is the contact point between the locking tab and locking ring; the straight line, MN, is the normal direction of the contact point C; S is the starting point of the notch of the locking ring, which is the contact point between the notch and the actuator working cavity; E is the ending point of the notch of the locking ring, which is the intersection of the notch and the line O1O2; δ is the angle of the notch of the locking ring. The angle between R76 and MN is φ since R76 is the resultant force of normal pressure and friction; the direction of Fr is parallel to O1O3. Because the notch of the locking ring and the actuator working cavity are not concentric, a circumferential force Ft is introduced at the contact point C, which is not only the resistance to overcome by the low-power retaining mechanism in the holding process, but also the driving force of the backward rotation of the locking ring in the unlocking process. The locking ring rotates around the center O1 in the locking or unlocking process, so Ft is perpendicular to the segment O1C. Draw line segments begin in the direction of Fr and R76, and end at points A and B, respectively, where the segment AB∥Ft and passes through the center O1. As shown in Figure 4b, the relation between R 61 and F r can be given as: Combining Equations (1) and (2), the relation between F r and F ml can be expressed as: Figure 5 shows the force diagram of the contact between the single locking tab and the locking ring in the holding process, in which R 67 and R 76 are the interactive forces between the single locking tab and the locking ring. To facilitate the analysis, the force diagram in Figure 5 is simplified, as shown in Figure 6. This assumes that O 1 is the center of the actuator working cavity; O 2 is the center of the notch of the locking ring; O 3 is the center of the curved surface of the locking tab; C is the contact point between the locking tab and locking ring; the straight line, MN, is the normal direction of the contact point C; S is the starting point of the notch of the locking ring, which is the contact point between the notch and the actuator working cavity; E is the ending point of the notch of the locking ring, which is the intersection of the notch and the line O 1 O 2 ; δ is the angle of the notch of the locking ring. The angle between R 76 and MN is ϕ since R 76 is the resultant force of normal pressure and friction; the direction of F r is parallel to O 1 O 3 . Because the notch of the locking ring and the actuator working cavity are not concentric, a circumferential force F t is introduced at the contact point C, which is not only the resistance to overcome by the low-power retaining mechanism in the holding process, but also the driving force of the backward rotation of the locking ring in the unlocking process. The locking ring rotates around the center O 1 in the locking or unlocking process, so F t is perpendicular to the segment O 1 C. Draw line segments begin in the direction of F r and R 76 , and end at points A and B, respectively, where the segment AB F t and passes through the center O 1 . Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 17  As shown in Figure 6, the relationship between Ft and Fr can be obtained by calculating the relationship between AB and AC. The relevant design parameters of the low-power holding mechanism are listed in Table 1, and the intermediate parameters in the calculation process are listed in Table 2.

Parameter Description
Symbol Radius of the actuator working cavity r1 Radius of the notch of the locking ring r2 Radius of the locking tab r3 Center distance between the actuator working cavity and the locking ring notch (|O1O2|) l Radial deflection angle of the center of the locking ring notch (∠O2O1O3) α

Parameter Description Symbol
Center distance between the actuator working cavity and the locking tab (|O1O3|) h Distance between the center of the actuator working cavity and the contact point C (|O1C|) r  As shown in Figure 6, the relationship between Ft and Fr can be obtained by calculating the relationship between AB and AC. The relevant design parameters of the low-power holding mechanism are listed in Table 1, and the intermediate parameters in the calculation process are listed in Table 2.

Parameter Description
Symbol Radius of the actuator working cavity r1 Radius of the notch of the locking ring r2 Radius of the locking tab r3 Center distance between the actuator working cavity and the locking ring notch (|O1O2|) l Radial deflection angle of the center of the locking ring notch (∠O2O1O3) α

Parameter Description Symbol
Center distance between the actuator working cavity and the locking tab (|O1O3|) h Distance between the center of the actuator working cavity and the contact point C (|O1C|) r Figure 6. The force analysis of the contact between the single locking tab and the locking ring in the holding process.
As shown in Figure 6, the relationship between F t and F r can be obtained by calculating the relationship between AB and AC. The relevant design parameters of the low-power holding mechanism are listed in Table 1, and the intermediate parameters in the calculation process are listed in Table 2.

Parameter Description Symbol
Radius of the actuator working cavity r 1 Radius of the notch of the locking ring r 2 Radius of the locking tab r 3 Center distance between the actuator working cavity and the locking ring notch (|O 1 O 2 |) l Radial deflection angle of the center of the locking ring notch (∠O 2 O 1 O 3 ) α r 2 = r 2 3 + h 2 + 2r 3 h cos arcsin l sin α r 2 − r 3 and (5) ).
As F r O 1 O 3 and CO 1 ⊥ AB, ∠CAB can be given as: ).
In O 1 O 2 C, ∠O 1 CO 2 can be expressed as: And ∠ACB = ∠O 2 CA − ϕ = ∠CO 1 O 3 + ∠O 1 CO 2 − ϕ, so ∠ACB can be given as: Combining Equations (3), (7) and (9), the relation between F t and F ml can be expressed as: Since four locking tabs are adopted to share the maximum load of the closing spring, the total circumferential force is 4F t . Therefore, the total resistance torque between the locking ring and the locking ring sleeve can be expressed as: Figure 7 shows the force analysis of the locking ring sleeve and the locking ring in the holding process. Assuming that the driving force of the locking ring sleeve is fully sustained by the spiral groove of the locking ring, the actuator working cavity applies no force on the locking ring sleeve. In the holding process, the locking ring tends to rotate backward to drive the locking ring sleeve to move upward. In Figure 7, F D is the output force of the electromagnet, R 85 is the resultant force of the housing acting on the locking ring sleeve, R 75 and R 57 are the interactive force between the locking ring sleeve and the locking ring, R 37 is the resultant force of the actuator working cavity acting on the locking ring, β is the spiral angle of the locking ring.
As shown in Figure 7, the force to resist the backward rotation of the locking ring is the circumferential resultant force of R 57 and R 37 , and satisfies: where R is the spiral radius of the locking ring; β = arctan( S 2πR ), and S is the pitch. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 17 As shown in Figure 7, the force to resist the backward rotation of the locking ring is the circumferential resultant force of R57 and R37, and satisfies: where R is the spiral radius of the locking ring; arctan( ) 2 S R β π = , and S is the pitch. The relation between R57 and FD can be expressed as: By substituting Equations (10) and (13) into Equation (12), the lockable maximum load of the closing spring can be obtained, that is: where ( ) ( )

Kinematic Analysis of the Low-power Holding Mechanism
In the locking or unlocking process, the locking tab moves linearly in the cam-like notch of the locking ring, and the locking ring rotates around the actuator working cavity. Figure 8 shows the relative position between the locking tab and the locking ring in the locking state and unlocking The relation between R 57 and F D can be expressed as: By substituting Equations (10) and (13) into Equation (12), the lockable maximum load of the closing spring can be obtained, that is: where h = (r 2 − r 3 ) 2 + l 2 + 2l(r 2 − r 3 ) cos α + arcsin l sin α r 2 − r 3 1 2 ,

Kinematic Analysis of the Low-Power Holding Mechanism
In the locking or unlocking process, the locking tab moves linearly in the cam-like notch of the locking ring, and the locking ring rotates around the actuator working cavity. Figure 8 shows the relative position between the locking tab and the locking ring in the locking state and unlocking state, respectively. Thus, the constraint condition for the locking tab to complete the locking and unlocking actions can be expressed as: where [L] is the permissible stroke of the locking tab; L is the working stroke of the locking tab. The permissible stroke of the locking tab depends on the size of the low-power holding mechanism. To facilitate the analysis, the movement of the locking ring and the locking tab is equivalent so that the locking ring remains fixed, while the locking tab not only moves linearly in the notch but also rotates around the actuator working cavity during the locking and unlocking processes. Figure 9 shows the kinematic analysis of the unlocking process of the low-power holding mechanism. Assuming that O 32 is the center of the curved surface of the locking tab in an unlocking state; O 31 is the intersection of the segment O 1 O 32 and the arc formed by the rotation of O 3 around O 1 , m is the length of the segment O 1 O 32 . The other symbols have the same values as depicted in Figure 6. The permissible stroke of the locking tab, [L], can be expressed as: Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 17 state, respectively. Thus, the constraint condition for the locking tab to complete the locking and unlocking actions can be expressed as: where [L] is the permissible stroke of the locking tab; L is the working stroke of the locking tab.
(a) (b) The permissible stroke of the locking tab depends on the size of the low-power holding mechanism. To facilitate the analysis, the movement of the locking ring and the locking tab is equivalent so that the locking ring remains fixed, while the locking tab not only moves linearly in the notch but also rotates around the actuator working cavity during the locking and unlocking processes. Figure 9 shows the kinematic analysis of the unlocking process of the low-power holding mechanism. Assuming that O32 is the center of the curved surface of the locking tab in an unlocking state; O31 is the intersection of the segment O1O32 and the arc formed by the rotation of O3 around O1, m is the length of the segment O1O32. The other symbols have the same values as depicted in Figure  6. The permissible stroke of the locking tab, [L], can be expressed as:   The permissible stroke of the locking tab depends on the size of the low-power holding mechanism. To facilitate the analysis, the movement of the locking ring and the locking tab is equivalent so that the locking ring remains fixed, while the locking tab not only moves linearly in the notch but also rotates around the actuator working cavity during the locking and unlocking processes. Figure 9 shows the kinematic analysis of the unlocking process of the low-power holding mechanism. Assuming that O32 is the center of the curved surface of the locking tab in an unlocking state; O31 is the intersection of the segment O1O32 and the arc formed by the rotation of O3 around O1, m is the length of the segment O1O32. The other symbols have the same values as depicted in Figure  6. The permissible stroke of the locking tab, [L], can be expressed as: Figure 9. The kinematic analysis of the unlocking process of the low-power holding mechanism. Figure 9. The kinematic analysis of the unlocking process of the low-power holding mechanism.
As shown in Figure 9, the rotation angle of the locking ring in the locking or unlocking process θ can be given as: So ∠O 1 O 32 O 2 and m can be expressed as follows, respectively: By substituting Equations (15) and (21) into Equation (18), the permissible stroke of the locking tab [L] can be rewritten as: (22)

Parameter Effects
The effects of the design parameters in Table 1 on the lockable maximum load of the closing spring F ml and the permissible stroke of the locking tab [L], respectively, are analyzed in this section. Assume that the parameters of the low-power holding mechanism are: λ = 45 • , ϕ = 5.7 • , S = 850 mm, R = 183 mm, F D = 1600 N, the required working stroke of the locking tab L = 8 mm, and the design variable ranges are listed in Table 3. Table 3. The design variable ranges.

Design Variable Description Variable Variable Range
Radius of the actuator working cavity r 1 130~150 mm Radius of the notch of the locking ring r 2 130~150 mm Radius of the locking tab r 3 10~20 mm Center distance between the actuator working cavity and the locking ring notch l 30~40 mm Radial deflection angle of the center of the locking ring notch α 35~45 • Figure 10 shows the variable effects on the lockable maximum load of the closing spring F ml . F ml can be increased by increasing r 2 and/or reducing α. According to Equations (5), (10) and (11), in the holding process, the resistance arm, r, increases with the increase in r 2 or decrease in α, while the resistance force, F t , decreases with the increase in r 2 or decrease in α, and the reduction in F t is greater than the increment in r, resulting in the decrease in the total resistance torque T. Thus, with a given maximum lockable load, F ml , the required force, F D , of the electromagnet can be reduced by increasing r 2 and/or decreasing α; in other words, with a given output force, F D , of the electromagnet, the maximum lockable load, F ml , can be increased by increasing r 2 and/or decreasing α. As shown in Figure 10, Fml decreases with the increase in r3 and l. According to Equations (5), (10) and (11), both r and Ft increase with the increase in r3 and l, resulting in the increase in the total resistance torque T. Thus, with a given output force, FD, of the electromagnet, the maximum lockable load, Fml, can be decreased by increasing r3 and l. Figure 11 shows the variable effects on the permissible stroke of the locking tab [L].
[L] increases with the increase in r1 and α or decrease in r3. The reason is that the angle of the notch of the locking ring, δ, increases with the increase in r1 and α or decrease in r3, resulting in the rise of the rotation angle, θ, of the locking ring in the locking or unlocking process. Thus [L] increases with the increase in r1 and α or decrease in r3. As shown in Figure 10, F ml decreases with the increase in r 3 and l. According to Equations (5), (10) and (11), both r and F t increase with the increase in r 3 and l, resulting in the increase in the total resistance torque T. Thus, with a given output force, F D , of the electromagnet, the maximum lockable load, F ml , can be decreased by increasing r 3 and l. Figure 11 shows the variable effects on the permissible stroke of the locking tab [L].
[L] increases with the increase in r 1 and α or decrease in r 3 . The reason is that the angle of the notch of the locking ring, δ, increases with the increase in r 1 and α or decrease in r 3 , resulting in the rise of the rotation angle, θ, of the locking ring in the locking or unlocking process. Thus [L] increases with the increase in r 1 and α or decrease in r 3 .
(a) (b) Figure 10. The lockable maximum load of the closing spring with (a) effects of r2 and l; (b) effects of r3 and α.
As shown in Figure 10, Fml decreases with the increase in r3 and l. According to Equations (5), (10) and (11), both r and Ft increase with the increase in r3 and l, resulting in the increase in the total resistance torque T. Thus, with a given output force, FD, of the electromagnet, the maximum lockable load, Fml, can be decreased by increasing r3 and l. Figure 11 shows the variable effects on the permissible stroke of the locking tab [L].
[L] increases with the increase in r1 and α or decrease in r3. The reason is that the angle of the notch of the locking ring, δ, increases with the increase in r1 and α or decrease in r3, resulting in the rise of the rotation angle, θ, of the locking ring in the locking or unlocking process. Thus [L] increases with the increase in r1 and α or decrease in r3. As shown in Figure 11, [L] can be decreased by increasing r2 and/or decreasing l. The reason is that as shown in Figure 9, the position of the contact point of the locking tab in the unlocking state, is not affected by the variation of r2 and l, while the position of the contact point of the locking tab in Figure 11. The permissible stroke of the locking tab with (a) the effects of r 1 and r 3 ; (b) the effects of r 2 and r 3 ; (c) the effects of l and α.
As shown in Figure 11, [L] can be decreased by increasing r 2 and/or decreasing l. The reason is that as shown in Figure 9, the position of the contact point of the locking tab in the unlocking state, is not affected by the variation of r 2 and l, while the position of the contact point of the locking tab in the locking state show upward tendency with the increase in r 2 or decrease in l, resulting in the reduction in the relative position distance between the contact point in the locking state and that in the unlocking state. Thus [L] decreases with the increase in r 2 or decrease in l.

Sensitivity Analysis
Sensitivity is often used to evaluate the change of performance caused by a unit variable increment. The variable effects in Section 4.1 on the objective function can be quantified by the sensitivity. The sensitivity analysis of the lockable maximum load of the closing spring F ml and the permissible stroke of the locking tab [L] is carried out in this section. Figures 12 and 13 show the sensitivities of F ml and [L] with regard to the change rate of variables in the design space, respectively, with a sampling interval of 0.01% of the full range.
Sensitivity is often used to evaluate the change of performance caused by a unit variable increment. The variable effects in Section 4.1 on the objective function can be quantified by the sensitivity. The sensitivity analysis of the lockable maximum load of the closing spring Fml and the permissible stroke of the locking tab [L] is carried out in this section. Figures 12 and 13 show the sensitivities of Fml and [L] with regard to the change rate of variables in the design space, respectively, with a sampling interval of 0.01% of the full range.   increment. The variable effects in Section 4.1 on the objective function can be quantified by the sensitivity. The sensitivity analysis of the lockable maximum load of the closing spring Fml and the permissible stroke of the locking tab [L] is carried out in this section. Figures 12 and 13 show the sensitivities of Fml and [L] with regard to the change rate of variables in the design space, respectively, with a sampling interval of 0.01% of the full range.   As shown in Figure 12, the absolute value of the sensitivities of F ml increases with the increase in r 2 , and decreases with the increase in l, α and r 3 . When the change rate ranges from 0% to 54.65%, l is the most sensitive variable and after that α, r 2 , r 3 ; when the change rate ranges from 54.65% to 90.48%, l is still the most sensitive one and after that r 2 , α, r 3 ; when the change rate ranges from 90.48% to 100%, the most sensitive variable is r 2 and after that l, α, r 3 . The average variable sensitivities of F ml are listed in Table 4. In the design space, the most sensitive variable of F ml is l and after that α, r 2 and r 3 . As shown in Figure 13, the absolute value of the sensitivities of [L] increases with the increase in r 3 , and decreases with the increase in r 2 l, and α; the sensitivity of [L] to r 1 remains on the lowest level in the full range; α is the most sensitive variable in the full range and after that l; when the change rate ranges from 0% to 30.63%, r 2 is the third most sensitive variable and r 3 is the fourth. The average variable sensitivities of [L] are listed in Table 5. In the design space, the most sensitive variable of [L] is α and after that l, r 3 , r 2 and r 1 .

Analysis
According to the numerical analysis in Section 4.1, when r 1 = 150 mm, r 2 = 150 mm, r 3 = 10 mm, l = 30 mm and α = 35 • , the lockable maximum load of the closing spring is F ml = 109.414 kN and the permissible stroke of the locking tab is [L] = 6.4059 mm. However, the required working stroke of the locking tab L = 8 mm > [L], which means the above variable set cannot satisfy the requirement of the low-power holding mechanism. Thus, further optimization design is needed.
According to the sensitivity analysis in Section 4.2, in the design space of the low-power holding mechanism, [L] is more sensitive to l and α rather than the other variables, r 1 , r 2 and r 3 . However, F ml is also sensitive to l. So α is the better option to be adjusted comparing with l to maximize the effect on the lockable maximum load of the closing spring. Through further numerical simulation, the improved parameter set of the low-power holding mechanism is obtained, as listed in Table 6, with the permissible stroke of the locking tab [L] = 8.0031 mm, which satisfies [L] > L = 8 mm, and the lockable maximum load of the closing spring is F ml = 71.44 kN after the parameter adjustment.

Conclusions
To bridge the gap of the low-power holding mechanism in the all-electric subsea gate valve actuator of the subsea production system, minimize the power consumption and cable number for control and to improve the open-position keeping performance of all-electric subsea gate valve actuator, this paper proposed a novel low-power holding mechanism for the all-electric subsea gate valve actuator which can be applied to all-electric subsea gate valve actuators with various valve sizes and process pressure ratings. The paper also created mechanical and kinematic models, respectively, with regard to design variables, including the radius of the actuator working cavity r 1 , the radius of the notch of the locking ring r 2 , the radius of the locking tab r 3 , the center distance between the actuator working cavity and the locking ring notch l, and the radial deflection angle of the center of the locking ring notch α. Based on these analytic models, the variable effects on the lockable maximum load of the closing spring and the permissible stroke of the locking tab were analyzed. The parameter adjustment method was illustrated to satisfy the design criterion that the working stroke of the locking tab L should not exceed the permissible stroke of the locking tab [L]. The detailed conclusions are as follows: • The proposed low-power holding mechanism is driven by the electromagnet and combines spiral transmission and cam-like transmission to convert the axial force of the closing spring into circumferential force to minimize the output force required as well as the number of the actuator's control cables, to improve the reliability and feasibility of the holding mechanism. The proposed low-power holding mechanism only requires a holding force of approximately 2%~7% of the maximum load of the closing spring to keep the valve open.

•
With a given output force of the electromagnet, the lockable maximum load the low-power holding mechanism can be increased by increasing r 2 or decreasing r 3 , l and α. In the design space, the variable sensitivities of the lockable maximum load increases with the increase in r 2 or the decrease in r 3 , l and α; the lockable maximum load is most sensitive to l and after that come α, r 2 and r 3 with regard to the average sensitivities in the design space.

•
The permissible stroke of the locking tab increases with the increase in r 1 , l and α or the decrease in r 2 and r 3 . In the design space, the variable sensitivities of the permissible stroke increase with the increase in r 3 or the decrease in r 2 , l and α; the sensitivity of the permissible stroke to r 1 remains on the lowest level in the full range; with regard to the average sensitivities of the permissible stroke, the most sensitive variable is α and after that l, r 3 , r 2 and r 1 .

•
The design criterion is that the working stroke of the locking tab L should not exceed the permissible stroke of the locking tab [L]. The parameters can be adjusted to satisfy the criterion considering the variable sensitivities meanwhile minimizing the effect on the lockable maximum load. In the given case, the final parameter set of the low-power holding mechanism is: r 1 = 150 mm, r 2 = 150 mm, r 3 = 10 mm, l = 30 mm and α = 39.31 • , with a lockable maximum load of the closing spring of 71.44 kN.
Further research will focus on the analysis of the influence of contact deformation and surface morphology on the modeling and the open-position keeping performance of the low-power holding mechanism. In addition, the future scope of the work can be directed toward the reliability assessment of the all-electric subsea gate valve actuator considering the subsea environment and the production fluid factors to ensure the long term application of the actuator in the subsea environment. The procedure in the paper can be applied to the design and research of other key components of the all-electric subsea production control system, such as the all-electric subsea ball valve actuator, which is required for the failsafe close ability and the open-position keeping function.

Conflicts of Interest:
The authors declare no conflict of interest.