Research on optimization design method of actuator parameters with stepless capacity control system for reciprocating compressor ①

Zhou Chao(周 超), Zhang Jinjie , Sun Xu, Wang Yao

(*Compressor Health and Intelligent Monitoring Center of National Key Laboratory of Compressor Technology, Beijing University of Chemical Technology, Beijing 100029, P.R.China) (**State Key Laboratory of Compressor Technology, Anhui Provincial Laboratory of Compressor Technology, Hefei 230031, P.R.China) (***Beijing Key Laboratory of Health Monitoring Control and Fault Self-Recovery for High-End Machinery, Beijing University of Chemical Technology,Beijing 100029, P.R.China)

Abstract

Key words: reciprocating compressor, stepless capacity control system, non-dominated sorting genetic algorithm II (NSGA-II), fuzzy analytic hierarchy process (FAHP)

0 Introduction

In petroleum and chemical enterprises, many chemical or physical processes require high-pressure gases, and reciprocating compressors play an irreplaceable role because of their outstanding advantages in compression efficiency and compression ratio. Reciprocating compressor belongs to positive displacement compressor, the discharge is constant under normal working conditions, but the fluctuation of technological process makes the device not run at full load. Reciprocating compressor usually runs below the designed discharge, so it is usually equipped with capacity control system to meet the operating requirements. The capacity regulation by pressing-off suction valve during partial stroke can realize 0-100% stepless flow regulation theoretically by controlling the opening time of the unloading device in the compression process. At the same time, it also reduces energy consumption[1]. The capacity control system is installed on reciprocating compressor in a petrochemical refinery. When the load is 40%-60%, the power can be saved by 300 kW/h. Calculated at 0.5 yuan/kWh, running for one year (that is 8 000 h) can save about 1.2 million yuan of electricity, with obvious energy-saving effect.

When the reciprocating compressor is installed with the stepless capacity control system[2-5], the actuator, control hardware and hydraulic system often fail because the design or control parameters are not optimized according to the actual situation in the field.The failure of hydraulic system is involved in Ref.[6]. The failure of control system is involved in Ref.[7]. The failure of actuator is involved in Refs[8-12]. Due to application problems, researchers begin to avoid problems in structure and parameter design. Wu et al.[13]analyzed the influence of inlet oil pressure on valve plate movement based on the mathematical model of valve plate movement and designed the optimal inlet oil pressure. Meanwhile, the speed buffer structure is designed to greatly reduce the impact of valve plate on valve seat and extend the life of valve plate. Li et al.[14]established a streamlined valve model and studied the relationship between inlet oil pressure and tilt angle and valve plate life, which provided theoretical basis for optimal design of actuator. Cao et al.[15]studied characteristics of hydraulic actuator and system modeling based on AMESim, and proposed an improved design. Li et al.[16]deduced an accurate calculation formula of discharge, and designed the GUI, which can quickly complete the design of the force of ejection. In conclusion, the researchers set single optimization goal and didn’t consider the diversity of parameters which have complex relationship of mutual dependence and contradiction, so, the design results have limitations, which can not make the system in an efficient operation state.

The objectives are mutually restricted, and there are multiple groups of non-dominant solutions. Traditional multi-objective optimization methods can not be used to solve the above problems. The evolutionary multi-objective optimization method has achieved good results in practical application, and researchers have achieved rich research results. The non-dominated sorting genetic algorithm II (NSGA-II)[17,18]is currently one of the most popular multi-objective genetic algorithm (GA), which reduces the complexity of non-inferior sort genetic algorithm, and has the advantages of fast running speed and good convergence of solution set, and becomes the benchmark of other multi-objective optimization algorithms.

In this paper, mathematical models of actuators and compressors are established. According to the mutual influence relationship and importance of parameters, the indicated power deviation, impact velocity of ejection, inlet oil pressure and spring stiffness are selected as objective functions. The Pareto frontier is obtained by NSGA-II. Finally, the optimal solution is selected by fuzzy analytic hierarchy process and compared with the traditional design value. The result obtained is significantly better than that obtained by the traditional design method, which verifies the feasibility and effectiveness of the method.

1 Stepless capacity control system and mathematical model

The stepless capacity control system principle of reciprocating compressor is shown in Fig.1. Under normal conditions, when the piston moves to the position of inner dead point (point A), it starts to reverse movement, the suction valve begins to close, and then the compressor compresses and exhausts. The area of the P-V diagram contains regions I and II. When the piston moves to the position of inner dead point (point A), it starts to move in the opposite direction. The hydraulic system provides a large hydraulic driving force to the unloader. The suction valve is forced to be pushed open by the unloader. The valve plate is at the lower limit and the gas returns to the suction chamber. When the piston moves to point B, the volume meets the production requirements. The hydraulic system provides the hydraulic driving force of the unloader to reduce, the spring force in the unloader overcomes the hydraulic driving force and friction, the unloader retracts, and the suction valve begins to close. The area of the P-V diagram contains region II. According to the area in the P-V diagram, the capacity control system can not only make the reciprocating compressor meet the volume requirements, but also save energy in region I. Nomenclature is listed in Table 1.

Fig.1 Schematic diagram of valve and chamber

1.1 Mathematical model of actuator

The actuator is mainly composed of hydraulic cylinder, unloader including reset spring, executing fork, mandril and other components.The hydraulic system includes hydraulic stations and pipelines to provide the driving force for the actuator to overcome the spring force, gas force, friction force, etc.The suction valve is delayed in closing from open state. Part of the gas returns to the suction chamber in compression stroke to realize the purpose of capacity control.

Table 1 Nomenclature

In order to analyze the motion characteristics of the actuator with stepless capacity control system, several hypotheses are provided.

(1) Don’t consider the rebound of actuator during impact.

(2) Don’t consider time delay of hydraulic system oil supply or unloading, that is, the start time of ejection or withdrawal of the loader is consistent with the start time of oil supply or unloading in the hydraulic oil unit.

i) Hydraulic driving force

(1)

where,Fhrepresents the hydraulic driving force,p1represents the oil pressure in the ejection process of the actuator,p2represents oil pressure in the withdrawal process of the actuator,Aunloaderrepresents transversal area of hydraulic piston,θ1represents start angle of ejection,θ2represents end angle of ejection,θ3represents start angle of withdrawal,θ4represents end angle of withdrawal.

ii) Spring force of actuator

Fs=kunloader(x0+x)

(2)

where,Fsrepresents spring force of actuator,kunloaderrepresents spring stiffness of actuator,x0represents pre-compression,xrepresents actuator displacement.

iii) Differential equations of motion of ejection and withdrawal

(3)

where,mrepresents actuatormass,Firepresents gasforce of suction,αrepresents installation angle of actuator,frepresents total friction of actuator,Fcyrepresents gas force of cylinder,γrepresents gas force coefficient of valve plate, when executing fork contacts the valve plate,γ=1, instead,γ=0.

iv) Initial condition

where,x′(0) represents initial velocity in ejection or withdrawal of actuator,va(0) represents initial displacement in ejection or withdrawal of actuator,Lrepresents actuatortrip.

1.2 Mathematical model of compressor with stepless capacity control system

As shown in Fig.2 and Table 2, on account of changing opening and closing states by actuator under capacity control condition, many new processes are added compared with the compressor model under normal working conditions. As shown in Fig.2, the suction valve is forced open during compression stroke (crank angle isθs3-θ3). The movement state of valve plate is changed. In the closing process of suction valve, if the acceleration of the unloader is less than the suction valve plate acceleration (crank angle isθs5-θ4), the motion state of the valve plate closing process will also change.

Table 2 Description of unloader and suction valve

Fig.2 Displacement diagram of unloader and suction valve

Based on the above analysis, a compressor mathematical model based on capacity control system is established.Before establishing the mathematical model, the following hypothesis is proposed.

(1) The suction valve is an automatic valve, which

is not affected by the actuator during the opening process.

(2) The motion of the exhaust valve and suction valve plates is one-dimensional.

(3) The flow of gas through the valve gap is a one-dimensional flow of ideal gas and an adiabatic process.

(4) The cylinder transfers heat with the cooling water in outer wall, which is simulated as an inter-wall heat exchanger, and its heat transfer coefficient isB(J/(m2·s)).

1.2.1 Expansion, suction and compression processes

Under capacity control condition, the motion of the actuator doesn’t affect the expansion, suction and compression processes[19]. The motion differential equations of the suction valve plate in different processes are respectively:

(4)

(5)

where,hrepresents valve plate displacement,θrepresents crank angle,αsvAsvrepresents instantaneous effective valve gap area of suction valve,Aprepresents area of valve plate,krepresents spring stiffness of valve plate,krepresents ratio of specific heat of gas,Vcyrepresents cylinder volume,V0represents relative clearance volume,βrepresents coefficient of heat transfer,C=B2πrcyrcrk,rcrkrepresents crank radius,trepresents time,H0represents precompression of valve plate spring,Mvrepresents the valve quality,Rrepresents gas constant,Tsrepresents suction temperature,βrepresents coefficient of applied force of gas,Zrepresents number of the spring,psrepresents inlet pressure,pcyrepresents cylinder pressure,Vhrepresents stroke volume,λrepresents ratio between the crankshaft radius and connecting rod length,rcyrepresents radius of the cylinder.

Eq.(4) is the motion differential equations of the suction valve plate in the expansion and compression process, and Eq.(5) is the motion differential equations of the suction valve plate in the suction process.

1.2.2 Backflow and suction valve closing process

(1) Backflow process

(2) Suction valve closing process

θ3≤θ≤θ4(7)

When the withdrawal speed of valve plate is less than the unloader, the equation is consistent with Eq.(5).

2 Multi-objective optimization mathematical model of actuator based on NSGA-II

2.1 Multi-objective optimization mathematical model

The hydraulic pressure and the reset spring stiffness have great influence on the safety, reliability and performance of the capacity control system. The reduction of hydraulic pressure can decrease the design cost of hydraulic system, decrease the impact of ejection speed and increase the safety of the system, but the actuator can not meet the requirements of ejecting when the hydraulic pressure is small. The decrease of spring stiffness can reduce the design value of hydraulic pressure, but when the spring stiffness decreases, the impact speed of ejection will increase, which is not conducive to system reliability. On the other hand, on account of the valve plate retracts with the actuator, the reduction of spring stiffness will lead to the reduction of the actuator’s retract acceleration, so the valve plate closing time increases, the quantity of reflux increases, and the load deviation increases. Therefore, it is necessary to consider the mutual inhibition and contradictory relationship among multiple parameters and objectives, such as hydraulic pressure, spring stiffness, impact velocity of ejection, and regulating effect, as shown in Fig.3. Based on the mathematical model of actuator and compressor in Section 2, the multi-objective optimization study of load deviation, impact velocity of ejection, hydraulic pressure and spring stiffness is carried out. The multi-objective mathematical model of the unloader is shown in Eqs(8,9):

Fig.3 Parameter relational graph

minf(X)=[f1f2…fm]

(8)

subX=(x1x2…xn)

xlower≤xi≤xupper(i=1,2,…,n)

(9)

where,f(X) is the objective equation ofX;m,nis the number of objective function and decision variables respectively;xloweris the lower limit of the decision variable;xupperis the upper limit of the decision variable.

Therefore, the objective function is to minimize the spring stiffness, oil inlet pressure, impact velocity of ejection and load adjustment deviation. The objective function is described as follows.

(1)Spring stiffness and oil inlet pressure

The design of reducing the system hydraulic pressure and the spring stiffness of unloader can reduce the processing requirements and costs of the actuator and hydraulic system, and increase the safety coefficient of the system.

f1=k

f2=p1

(2)Load adjustment deviation

Under normal condition, the suction valve plate retracts automatically. Through numerical calculation of Eqs(4) and (5), the withdrawal time of the suction valve plate is 1.56 ms. Actuator parameters are shown in Table 3. Reciprocating compressor parameters are shown in Table 4. Ultimate withdrawal time of unloader is 4 ms with capacity control system. Since the mass of the valve plate is small relative to the unloader, the acceleration of the valve plate is large relative to the unloader. Therefore, under the capacity control condition, the valve plate is withdrawn together with the unloading device, namely, Eq.(7) is adopted as the differential equations of valve plate withdrawal process.

Table 3 Actuator parameters

Table 4 Reciprocating compressor parameters

Through numerical calculation, the indicated power of compressor with different stiffness is obtained. The load adjustment deviation can be obtained by subtracting the indicated power of the compressor with different stiffness under the normal condition from the indicated power of the compressor under the capacity control condition, that isΔη. The relationship betweenΔηandkis obtained as shown in Fig.4, and through rational fitting, when the numerator degree is 1, denominator degree is 2, fitting degree is high, the relationship betweenΔηandkis shown in Eq.(10).

Fig.4 Fitted curve and simulation curve

(10)

(3) Impact velocity of ejection

When the angle isθ1≤θ≤θ2, it is the ejection process of the actuator, the initial conditions are substituted into the ejection motion differential equation of the actuator, and the velocity at the lower limit of the actuator namely the impact velocity of ejection, can be obtained:

(11)

2.2 Analysis of optimization results based on NSGA-II

Fig.5 shows the flow of multi-objective optimization algorithm, which is divided into 3 parts. The first part constructs mathematical equations according to the structure and characteristics of the actuator and compressor, the second part analyzes the mutual influence of various parameters and obtains the relational expression according to the mathematical model, and the third part completes the multi-objective optimization calculation with NSGA-II.

The setting parameters of non-dominant sorting multi-objective optimization is shown in Table 5. Figs 6-7 show the feasible solution set graph among the 4 targets. As can be seen from Fig.6, with the decrease of inlet oil pressure or spring stiffness, the impact velocity decreases, but the deviation of indicated power increases. Fig.6 shows the Pareto front of spring stiffness and oil inlet pressure when Gen=50, 500, 1 000 and 2 000. As the number of iterations increases, the region of feasible solution gradually decreases and is close to the Pareto front.

Every solution in the NSGA-II solution set is a nondominant solution, and every solution cannot dominate the others. For practical engineering problem of application, the solution of multi-objective problem is not only an optimization problem. When Pareto front is found, the final optimal solution needs to be selected according to the relative importance of the optimization target. Although the deviation of indicated power affects the adjustment accuracy, it can be compensated by the control method. Therefore, the weight of the inlet oil pressure, spring stiffness, impact speed and deviation of indicated power is 0.3: 0.3: 0.3: 0.1. The relationship betweenrijand the weight of factorswiandwjisrij=0.5+(wi-wj)β, 0<β≤0.5, takingβ=0.3, therefore precedence relation matrix is:

Table 5 Parameter of NSGA-II

ParameterValuePopulation size100Maximum generation1000Mutation fraction0.7Crossover fraction0.4Variation ratio0.02Crossover ratio0.02

Fig.5 Multi-objective optimization design algorithm flow chart

Fig.6 Spring stiffness and hydraulic pressure Pareto front

Fig.7 k-p1-v-Δη Pareto front

Fuzzy consistency matrix is

Therefore,

=(0.2533, 0.2533, 0.2533, 0.24)

According to the weight coefficient, the optimal solution can be obtained:

Take the first 3 groups of solutions, as shown in Table 6.

The optimized result of NSGA-II is compared with the traditional design parameters. The traditional design parameters are selected and calculated based on feasibility without considering the optimized design. Therefore, it can be seen from Table 7, the parameters decrease after the optimized design that could reach at least 17% and the maximum could reach 201%.

3 Conclusion

In this work, the key parameters of reciprocating compressor actuator and hydraulic system are optimized under capacity control system. Due to the mutual influence of spring stiffness, inlet oil pressure, impact velocity of ejection and deviation of indicated power, the improvement of any parameter will lead to the deterioration of other parameters. In order to solve the multi-parameter optimal design effectively, NSGA-II is used to solve the problem and compared with the traditional results.

Table 6 Optimal solution of actuator parameters

Table 7 Optimal solution of actuator parameters

(1) The differential equation of actuator motion is established to analyze the relationship between spring stiffness, inlet oil pressure, impact velocity of ejection,which lays a theoretical foundation for multi-objective optimization.

(2) The mathematical model of compressor under the capacity control system is established. On normal condition, the automatic withdrawal time of the valve plate is about 1.56 ms, while the minimum withdrawal time of the unloader on capacity control condition is 4 ms, so the valve plate and the unloader are withdrawn to the top limit together.The motion equation of the valve plate on capacity control system is related to the motion of the actuator. Therefore, the influence of spring stiffness on the indicated power and the displacement of the valve plate is analyzed by combining the motion differential equation of the actuator with the mathematical model of the compressor. The influence of spring stiffness on indicated power and displacement of valve plate on capacity control system is analyzed. It can be seen from the results that with the increase of spring stiffness, the closing time of valve plate decreases and the backflow decreases. Therefore, the deviation of indicated power of capacity control system decreases, but the increase of stiffness will lead to the increase of inlet oil pressure, and the relationship curve between spring stiffness and indicating power deviation is obtained through rational fitting.The above research lays a theoretical foundation for multi-objective optimization.

(3) The optimal design is carried out by using NSGA-II to get Pareto front, and the optimal parameters of actuator and hydraulic system are obtained by adopting fuzzy analytic hierarchy process. The spring stiffness is 33 214 N/m, the inlet oil force is 310.63 N, the impact velocity of ejection is 0.2372 m/s, and the deviation of indicated power is 7.2523 kW. Compared with the traditional design parameters, the optimized design can reduce the spring stiffness or inlet oil pressure by at least 17% and the maximum by 201%.