Intake Valve Profile Optimization for a Piston-Type Expander Based on Load

Abstract

Intake valve parameters significantly affect the performance of the piston-type expander (PTE). To improve compressed energy utilization efficiency, intake valve parameters need to be regulated according to load. In this paper, an electro-pneumatic variable valve actuation (EPVVA) system was proposed for independent control distributing valve parameters. The trajectory planning for the intake valve was proposed to obtain good mechanical properties. Then, the intake valve duration angle was optimized, and the optimum intake valve lift curves were obtained at different rotational speeds. Results show that the energy efficiency decreased with the intake valve duration angle increasing. The output power ascended sharply with increasing intake valve duration angle, but the amplitude of power growth decreased. The output power had a maximum value at a specific intake valve duration angle. The gray relation analysis (GRA) method was applied to obtain the optimum intake duration angle based on output power and energy efficiency. Finally, the optimum intake valve trajectories were presented under different rotational speeds. Results are helpful for the future control of the piston-type expander.

1. Introduction

In recent years, energy conservation and carbon emission reduction, crucial global concerns, have drawn considerable attention. Many countries present carbon peak and carbon-neutral policies and put them into practice. As the world’s largest developing country globally, China has committed to being carbon peak by 2030 and carbon neutral by 2060 [1]. All the policies show that renewable energy will develop rapidly in the next few years. Nevertheless, renewable energy such as wind and solar power is characterized by intermittency and volatility, causing severe negative impacts on the power grid [2]. The energy storage system (ESS) can effectively cope with this challenging issue due to its flexible charging and discharging capabilities [3].

Compressed air energy storage (CAES) is a concept for thermo-mechanical energy storage with the potential to offer low-cost, large-scale, and fossil-fuel-free operations [4]. With the development of CAES, small-scale systems of CAES are also explored in both academic studies and industrial projects [5,6,7]. In a CAES system, the expander is a critical component that decides the power output and the overall energy conversion efficiency. The selection of expanders highly depends on both the system operations and the discharge power capacity of the energy storage system [8]. Generally, two main types of expanders can be selected in the market: the positive displacement (volume) type, such as piston-type expanders (PTE), screw and scroll expanders, and the dynamic (velocity) types, such as radial and axial turbines. The PTEs are suitable for scaled-down CAES systems owing to the low flow rate, high-pressure ratio, and low rotational speed [9,10]. The PTE has been used in marine and stationary power plants and attempts to apply it in the automotive field [11].

PTE as a compressed air engine has been investigated. Mathematical models of compressed air engines were established. The operational characteristics of the engine under both single-stage and two-stage expansion were analyzed by some researchers [12,13]. Liu et al. used a multi-objective optimization method to obtain maximum cycle efficiency under the engine intake pressure ratio at 32 [14]. A virtual prototype with a newly developed valve system was designed to study the compressed air engine’s dynamic performance and the optimized valve-controlling strategies [15]. These applications and studies on PTEs have demonstrated the technical viability of this type of expander to be used in the CAES system. However, due to the uncertainty of renewable energy characteristics and load requirements in the CAES system, the expander often works under off-design conditions. The off-design conditions affect the energy efficiency of the expander significantly. Therefore, the PTE usually runs under off-design conditions, which is difficult to run efficiently in a wide range [16].

Based on previous studies, the opening and closing of the inlet and exhaust valves significantly affect the performance of the PTE. However, adjusting the opening and closing of the inlet and exhaust valve is challenging to meet load demand by a cam mechanism. To meet the load demand, Yu et al. proposed a pneumatic circuit that consists of a regulator, air-operated regulator, electronic pressure control unit, and pressure sensor to obtain the requirement load [17] in high pressure and significant flow conditions. However, a large quantity of energy is lost in the throttling process. It is necessary to find a new approach to solve the above problem.

The PTE often works in high-pressure and extensive flow conditions to improve the output power, making airflow regulation difficult. Few variable valve actuation (VVA) technologies for PTE were investigated. In fact, fast-switching pneumatic solenoids can be applied in some high-precision control fields. For example, Shi et al. proposed a pneumatic vibration isolation system based on fast-switching pneumatic solenoids and used pressure difference to improve the system control precision [18,19]. This study aims to provide an electro-pneumatic variable valve actuation system for PTE, and the intake valve duration angle is optimized to obtain high energy efficiency and output torque. In the VVA system, double-acting pneumatic actuators govern the forward and backward motions of the engine valves. Fast-switching pneumatic solenoids control the pressurized air through the actuator. The desired intake and exhaust valve motion profiles are achieved by energizing and de-energizing the solenoids.

The rest of the paper is organized as follows: the system structure, working principle, mathematical model, and trajectory planning are described in detail in Section 1. Simulation and optimization are carried out in Section 2. Section 3 is devoted to the summary and conclusions.

2. Methodology

2.1. EPVVA System of Piston Expander Configuration

The schematic diagram of the EPVVA system of the piston expander is illustrated in Figure 1. The major components include the intake valve actuator, exhaust valve actuator, intake valve, exhaust valve, accumulator, regulator, connectors, preload spring, engine head, piston, link, and crank. The engine intake valve (IEV) and exhaust valve (EEV) are connected to double-acting pneumatic actuators using connectors. The double-acting pneumatic actuator has upper and lower chambers. In this paper, the upper and lower chambers of the intake valve actuator are renamed the a and b chambers. The exhaust valve actuator’s upper and lower chambers are called the c and d chambers.

Pressurized airflow into the actuator is controlled by the solenoids V₁, V₂, V₃, V₄, V₅, V₆, V₇, and V₈. They are commercially available high-frequency solenoid valves, in which the solenoid’s plunger directly controls the fluid flow. V₁ and V₂ are connected to chamber a, and V₅ and V₆ are connected to chamber b. Each solenoid valve is connected in pairs to achieve the desired mass flow rate controlled according to the crankshaft angle to accomplish the expander intake, expansion, and exhaust processes shown in Figure 2. The working process can be described as follows:

Intake process: when the crankshaft’s angle is equal to 0 degrees, the V₂ solenoid coil is energized, the V₁ solenoid coil is de-energized, the V₅ solenoid coil is de-energized, and the V₆ solenoid coil is energized. The pressurized air from the accumulator flows into chamber a. The air inside chamber b flows out, and the pressurized air pushes the intake actuator piston connected with the intake valve down to open the IEV. Meanwhile, the pressurized air from the supply-side flows into the expander cylinder through the intake port and pushes the expander piston downward. After the crankshaft’s angle reaches point C₁, the V₂ solenoid coil is de-energized, the V₁ solenoid coil is energized, the V₅ solenoid coil is energized, and the V₆ solenoid coil is de-energized. The pressurized air from the accumulator flows into chamber b, and the residual air inside chamber a flows out until the IEV closes.

Expansion process: the V₂ solenoid coil is de-energized, the V₁ solenoid coil is energized, the V₅ solenoid coil is energized, the V₆ solenoid coil is de-energized, and the IEV is closed. The pressurized air entering the expander cylinder expands and pushes the piston downward to the bottom dead center (BDC).

Exhaust process: when the expander piston reaches BDC, the V₃ solenoid coil is energized, the V₄ solenoid coil is de-energized, the V₇ solenoid coil is de-energized, and the V8 solenoid coil is energized. The pressurized air from the constant pressure air tank flows into chamber c and pushes the exhaust actuator piston to the exhaust valve to open the EEV. The residual air inside the expander cylinder flows out through the EEV. After the crankshaft’s angle reaches point C₂, the V₃ solenoid coil is de-energized, the V₄ solenoid coil is energized, the V₇ solenoid coil is energized, the V₈ solenoid coil is de-energized. The pressurized air from the accumulator flows into chamber d, and the residual air inside chamber c flows out until the EEV closes.

2.2. Expander Mathematical Model

The following assumptions were made:

(1)

The working air of the system follows all ideal gas laws.

(2)

There is no leakage in the working process, and the area of the piston rod end is too small to be considered.

(3)

Supply temperature is equal to atmospheric temperature.

(4)

The flow of air moving into and out of the cylinder is a stable one-dimensional flow.

Because there is no leakage in the working process, the charge and discharge of air does not simultaneously happen. The heat transfer can be neglected during the active process, owing to the high operation speed. The following equation can illustrate the energy equation for the expander:

$$c_v{m}_{cy}\frac{d{T}_{cy}}{dt}=\left(c_p{T}_{atm}-c_v{T}_{cy}\right)G_{ic}-R{T}_{cy}{G}_{oc}-p_{cy}\frac{d{V}_{cy}}{dt}$$

(1)

where c_v is the specific heat at constant volume, c_p is the specific heat at constant pressure, m_cy is the air mass of cylinder, T_cy is the air temperature of cylinder, T_atm is the atmosphere temperature, G_ic is the intake air mass flow, G_oc is the exhaust air mass flow, p_cy is the air pressure of cylinder, V_cy is the air volume of the cylinder, and R is the gas constant.

The intake air mass flow G_ic and the exhaust air mass flow G_oc can be calculated by:

$$G_i=\left\{\begin{array}{ll}{A}_{ei}{p}_H\sqrt{\frac{k}{R{T}_H}{\left(\frac{2}{k+1}\right)}^{\frac{k+1}{k-1}}}\hfill & \raisebox{1ex}{$p_L$}\!\left/ \!\raisebox{-1ex}{$p_H$}\right.\le 0.528\hfill \\ A_{ei}{p}_H\sqrt{\frac{2k}{\left(k-1\right)R{T}_H}{\left[{\left(\raisebox{1ex}{$p_L$}\!\left/ \!\raisebox{-1ex}{$p_H$}\right.\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}-{\left(\raisebox{1ex}{$p_L$}\!\left/ \!\raisebox{-1ex}{$p_H$}\right.\right)}^{\raisebox{1ex}{$k+1$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\right]}^{\frac{k+1}{k}}}\hfill & \raisebox{1ex}{$p_L$}\!\left/ \!\raisebox{-1ex}{$p_H$}\right.>0.528\hfill \end{array}\right.$$

(2)

where the subscripts H and L denote the upstream and downstream compressed air. The subscript i indicates the intake air or exhaust air, A_e is the effective area, and k is the adiabatic exponent, equal to 1.4.

The valve area of the frustum-shaped site is calculated by:

$$A_{ei}=N_v\pi l_v\cos a\left(D_s+l_v\cos a\sin a\right)$$

(3)

where a is the valve face angle, N_v is the number of the intake valve and exhaust valve, D_s is the diameter of the valve small head, and l_v is the valve displacement.

Using the ideal gas law, the engine cylinder pressure is determined by:

$$p_{cy}=\frac{m_{cy}R{T}_{cy}}{V_{cy}}$$

(4)

$$V_{cy}=A_{cy}\left(x+x_0\right)$$

(5)

where A_cy, x, and x₀ are the area of the piston, the piston displacement, and cylinder clearance, respectively.

In the above equation, x is the piston displacement, which is calculated by:

$$x=r_{cs}+L_{rod}-r_{cs}\cos \left({\theta }_{cs}\right)-\sqrt{L_{rod}^2-r_{cs}^2{\sin }^2\left({\theta }_{cs}\right)}$$

(6)

where r_cs, L_rod, and θ_cs are the crank radius (half stroke), piston rod length, and crank angle, respectively.

2.3. Evaluation Indicator

If the friction torque is neglected, in a steady state, the output torque can be expressed as:

$$T_{or}=T_i-T_r$$

(7)

where T_i is the indicated torque, and T_r is the reciprocating torque.

The relationship between the in-cylinder pressure p_cy and the indicated torque T_i can be expressed as:

$$T_i=\left(p_{cy}-p_{atm}\right)A_{cy}{r}_{cs}G\left({\theta }_{cs}\right)$$

(8)

where

$$G\left({\theta }_{cs}\right)=\raisebox{1ex}{$\sin \left({\theta }_{cs}+\beta \right)$}\!\left/ \!\raisebox{-1ex}{$\cos \beta $}\right.$$

(9)

$$\beta ={\sin }^{-1}\frac{r_{cs}\sin {\theta }_{cs}}{L_{rod}}$$

(10)

The reciprocating torque is given by:

$$T_r=M{r}_{cs}^{2}G\left({\theta }_{cs}\right)G_1\left({\theta }_{cs}\right){\dot{\theta }}_{cs}^2$$

(11)

where

$$G_1\left({\theta }_{cs}\right)=\cos \left({\theta }_{cs}\right)+\frac{\lambda \sin \left({\theta }_{cs}\right)\cos \left({\theta }_{cs}\right)}{{\left(1-{\lambda }^2{\sin }^2\left({\theta }_{cs}\right)\right)}^{3/2}}-\frac{\lambda {\sin }^2\left({\theta }_{cs}\right)}{{\left(1-{\lambda }^2{\sin }^2\left({\theta }_{cs}\right)\right)}^{1/2}}$$

(12)

$$\lambda =\frac{r_{cs}}{L_{rod}}$$

(13)

where M is the piston, rings, pin, and small end connecting rod mass.

The output power of the piston expander is expressed as follows:

$$P=\frac{\raisebox{1ex}{${\displaystyle \int T_i}$}\!\left/ \!\raisebox{-1ex}{$t$}\right.\times n}{9550}$$

(14)

where t is time, and n is the speed of the crankshaft (rpm).

Energy utilization efficiency can be expressed as:

$$\eta =\frac{P}{E_{air}}\times 100\%$$

(15)

where E_air is the air power which can be calculated.

$$E_{air}=G_{ic}R{T}_{ic}\ln \left(\raisebox{1ex}{$p_{ic}$}\!\left/ \!\raisebox{-1ex}{$p_0$}\right.\right)$$

(16)

where T_ic is the temperature of the supply air. p_ic is the pressure of the supply air.

2.4. Trajectory Planning for Engine Valve

A typical valve position and the corresponding valve event performance parameters are shown in Figure 2. Here, l_iv is the IEV maximum lift, l_ev is the EEV maximum lift, t_il is the intake valve up to the maximum lift’s corresponding time, t_id is the intake valve beginning to the close’s corresponding time, and t_ic is the intake valve’s fully closed corresponding time. Similarly, t_el, t_ed, and t_ec are the exhaust valve’s up to maximum lift corresponding time, the exhaust valve’s beginning to close corresponding time, and the exhaust valve’s fully closing corresponding time.

To minimize energy loss during the valve opening and closing process, t_il, t_ic − t_id, t_ed − t_el, and t_ec − t_ed should reduce as much as possible. The valve velocity should be low when approaching the closing position to avoid the mechanical components’ noise, wear, and tear. The typical third-order trajectory planning for the point–point motion is shown in Figure 3, where T₀ and l_max represent the movement time from closing to opening and the lift of the engine valve, respectively. The movement distance l can be given as:

$$l=a+bt+c{t}^2+d{t}^3$$

(17)

According to boundary conditions, when t = 0, l = 0; when t = 0, v = 0; when t = T₀, v = 0; when t = T₀, l = l_max. Based on the above analysis, the date of the trajectory planning is calculated by digital integration, as shown in Equation (18).

$$\left\{\begin{array}{l}a=0\hfill \\ b=0\hfill \\ b\cdot T_0+c\cdot T_0^2+d\cdot T_0^3=l_{\max }\hfill \\ b+2c\cdot T_0+3d\cdot T_0^2=0\hfill \end{array}\right.$$

(18)

A, b, c, and d can be calculated by Equation (25).

$$\left\{\begin{array}{l}a=0\hfill \\ b=0\hfill \\ c=\raisebox{1ex}{$3l_{\max }$}\!\left/ \!\raisebox{-1ex}{$T_0^2$}\right.\hfill \\ d=-\raisebox{1ex}{$2l_{\max }$}\!\left/ \!\raisebox{-1ex}{$T_0^3$}\right.\hfill \end{array}\right.$$

(19)

The opening profile of the valve can be described as:

$$l=\raisebox{1ex}{$3l_{\max }$}\!\left/ \!\raisebox{-1ex}{$T_0^2$}\right.t^2-\raisebox{1ex}{$2l_{\max }$}\!\left/ \!\raisebox{-1ex}{$T_0^3$}\right.t^3$$

(20)

The closing profile of the valve can be obtained by using the same method.

Based on the above analysis, the lift curves of IEV and EEV can be described by:

$$l_{iv}=\left\{\begin{array}{ll}{l}_{iv}\left(t\right)\hfill & t_{up}\le t<t_{il}\hfill \\ l_{i\max }\hfill & t_{il}\le t<t_{id}\hfill \\ l_{iv}\left(t_{ic}-t\right)\hfill & t_{id}\le t<t_{ic}\hfill \end{array}\right.$$

(21)

$$l_{ev}=\left\{\begin{array}{ll}{l}_{ev}\left(t\right)\hfill & t_{down}\le t<t_{el}\hfill \\ l_{e\max }\hfill & t_{el}\le t<t_{ed}\hfill \\ l_{ev}\left(t_{ec}-t\right)\hfill & t_{ed}\le t<t_{ec}\hfill \end{array}\right.$$

(22)

$$t_{id}=t_{il}+\raisebox{1ex}{${\theta }_{id}$}\!\left/ \!\raisebox{-1ex}{$6\times n$}\right.$$

(23)

where t_up is the corresponding time when the piston reaches to top dead center, and t_down is the corresponding time when the piston comes to down dead center, θ_id is the intake duration angle, n is the rotation speed of the engine, l_imax is the maximum lift of intake valve, l_emax is the maximum lift of exhaust valve. Additionally, t_il − t_up, t_ic − t_id, t_el − t_down, and t_ec − t_ed are equal to T₀.

3. Result and Discussion

3.1. Valve Open and Close Time

As mentioned before, the fast-switching valves are responsible for opening and closing the flow connections between the pneumatic cylinder and the constant pressure tank. The response time and flow rate of the fast-switching valve will influence the pressure of the pneumatic cylinder and then affect the engine valve’s kinematic character. The fast switching valve should have a high frequency, high-speed response, and high flow rate to obtain a fast response to the engine valve low rate. Two kinds of fast switching valves can be selected in this work; one is the two 2/2-ways fast switching valve with series number MHJ10-S-2,5-QS-6-HF produced by FESTO company, and the other is the two 2/2-ways fast switching valve of series number SX12-JJ produced by the company SMC. The specifications of the two kinds of switching valves are shown in

Table 1. Specification of high-speed on–off valve.

Company	Effective Range (bar)	Response Time of Power on (ms)	Response Time of Power off (ms)	C (dm3/(s/bar))	b
FESTO	0.5–6	1.3	0.6	0.66	0.36
SMC	1.5–7	0.6	0.75	0.7	0.21

.

In Table 1, C is the sonic conductance, and b is the critical pressure ratio. According to the response time and flow rate parameters comparison, the fast-switching valve SX12-JJ was selected. Based on the specification of SX12-JJ, the response time and flow rate are shown in Figure 4. To simplify the calculation process, the flow opening area of the on/off valve was linearized. From Figure 4, the opening time is about 1 ms and the closing time is about 0.5 ms.

The opening time and closing time are variable in the different external loads. The valve force diagram is shown in Figure 5. Newton’s second law allows taking the balance of the forces influencing the movement of the valve:

$$F_a-F_b-F_{sp}-b_m\dot{y}-F_{cy}=M\ddot{y}$$

(24)

where

$$F_a=p_a\frac{\pi D_a^2}{4}$$

(25)

$$F_b=p_b\frac{\pi \left(D_a^2-D_b^2\right)}{4}$$

(26)

$$F_{sp}=k_{sp}\left(y_{sp0}+y\right)$$

(27)

$$F_{cy}=p_{cy}\frac{\pi D_e^2}{4}$$

(28)

where p_a and p_b are the air pressure in the chamber a and chamber b, D_a and D_b are diameters of piston and piston rod, k_sp is the stiffness of the spring, y_sp0 is the pre-compression length, D_e is the diameter of the valve head, M is the mass of moving parts including piston, piston rod, connector, valve, b_m is the piston viscosity coefficient. The computational procedure for air pressure in chamber a and chamber b is similar to the calculation process of cylinder pressure.

The initial parameters are shown in

Table 2. Initial structure parameters.

Parameter	Symbol	Value
Diameter of the valve head (mm)	De	20
Diameter of the piston (mm)	Da	80
Diameter of the piston rod (mm)	Db	40
stiffness of spring (N/m)	ksp	200
Diameter of force balance block (mm)	Dc	16
Stroke of cylinder (mm)	L	4
Ambient pressure (bar)	patm	1
Air pressure in the intake port (bar)	ps	50
Air pressure supply to flow into actuator (bar)	pp	8
Pre-compression length (mm)	ysp0	1
Mass of moving parts (kg)	M	0.5
piston viscosity coefficient (N·s/m)	bm	80

.

The maximum force acting on the valve can be calculated by:

$$F_{\max }=p_s\times \frac{\pi }{4}{D}_e^2=50\times {10}^5\times \frac{3.14}{4}\times 20\times 20\times {10}^{-6}=1570N$$

(29)

When the engine opens, the pressure in the engine cylinder has not immediately reached maximum pressure. Thus, the external force was assumed to be 0 N, 392.5 N, 785 N, 1177.5 N, and 1570 N. Based on these external forces, the movement characteristics of the intake engine valve are shown in Figure 6.

It is evident that the time corresponding to the valve to maximum lift increases when the external force increases from 0 N to 1570 N. When the external force is equal to 0 N, the time corresponding is equal to 2.9 ms; when the external force is equal to 1570 N, the time corresponding is equal to 6.1 ms. In the actual process, the external force varies with air pressure in cylinder, and the air pressure in the cylinder is lower than the supply air pressure in the intake port, owing to the throttling effect. Under normal conditions, it is easier to close the engine valve than to open it based on valve force. As described in Section 4, t_il, t_ic − t_id, t_ed − t_el, and t_ec − t_ed should reduce as much as possible. In this paper, t_il, t_ic − t_id, t_ed − t_el, and t_ec − t_ed are set to 5 ms.

3.2. Boundary Conditions

The system was simulated in the Matlab/Simulink software using the developed mathematical model for the preliminary investigation of the expander performance under different valve trajectories. The simulation was initially run using arbitrary values for the model parameters. Some of the model parameters that are design dependents, such as valve actuator moving mass (m) and friction (F_fr), are presumed to be known values in the simulation.

Table 3. Basic structure parameters.

Parameter	Symbol	Value
Cylinder bore (mm)	Dcy	85
Cylinder stroke (mm)	Scy	88
Radius of the crankshaft (mm)	r	44
Ratio of crankshaft and link rod	λ	0.3
Expander cylinder clearance (mm)	x0	4
Supply pressure (bar)	ps	50
Number of the intake valve and exhaust valve	Nv	1
Diameter of valve small head (mm)	Ds	12
Maximum lift of intake valve (mm)	limax	4
Maximum lift of exhaust valve (mm)	lemax	5
Specific heat at constant pressure (J/(kg·K))	Cp	1005
Specific heat at constant volume (J/(kg·K))	Cv	718
Gas constant (J/(kg·K))	R	287

shows the final values selected for the system parameters. According to the mathematical model above, each parameter can be changed for comparison while all other parameters are kept constant.

The intake valve and exhaust valve trajectories depend on l_iv, t_il, t_id, t_ic, l_ev, t_el, t_ed, and t_ec. The pneumatic actuators are used to drive the intake valve and exhaust valve. In this paper, T₀ is set as 5 ms, which is explained in Section 3.1. Thus, t_id − t_il, defined as intake valve duration time can be regulated according to different rotation speeds. The exhaust valve keeps opening during the exhaust process to reduce exhaust resistance.

3.3. Intake Valve Duration Angle Optimization

This section optimizes the intake valve duration angle according to different rotation speeds. The average output power and energy efficiency are the optimization objectives shown in Equations (14) and (15). Figure 7 shows the average output power and energy efficiency obtained in different intake valve duration angles based on given conditions shown in Table 3 when the rotation speed is equal to 800 rpm.

Under most conditions, the PTE system has an inherent tradeoff between energy efficiency and output power. As shown in Figure 7, the energy efficiency decreases as the intake valve duration angle increases. The energy efficiency reaches a maximum value when the intake valve duration angle is equal to 10°. Compared with the 85° intake valve duration angle, the energy efficiency increases by 30.05%. Initially, the output power ascends sharply with the increasing intake valve duration angle, but the amplitude of power growth decreases. Additionally, the output power peaks at approximately 19.57 kW. After this peak, the output power starts to descend slowly.

The primary reason is that the residual pressure increases with the intake valve duration angle, as shown in Figure 8.

The primary reason is that the residual pressure increases with the intake valve duration angle, as shown in Figure 8. When the intake valve duration angle is equal to 10°, the residual pressure (RP) value is equal to about 7 bar, which is the air pressure in the expander cylinder when the piston reaches BDC. The residual pressure increases from 7 bar to 33.4 bar when the intake valve duration angle varies from 10° to 80°. The higher residual pressure leads to more energy loss. However, the output power increases with the pressure in the expander cylinder increasing. However, with the residual pressure reaching a specific value, more energy from pressurized air will be consumed during the exhaust process. Thus, the output power has an extreme value at an exact rotational speed.

Figure 9 shows an intake valve position, velocity trace, and the corresponding valve event performance parameters when the intake valve duration angle is equal to 10°.

The expander’s energy efficiency and output power cannot reach optimum at the same time. An effective, comprehensive evaluation method should be proposed to obtain the optimum intake valve duration angle based on energy efficiency and output power. This paper uses the gray relation analysis (GRA) method [15,20] to obtain the expected indicator. The evaluation set E = (P, η) = (E₁, E₂) is formed by the indicators in Equations (14) and (15). The calculative process of the GRA method is shown as follows:

Step 1. Calculate the decision matrix D. D_ij is the calculated value of E_j through the simulation model’s running with different intake valve duration angles. Then, D = (D_ij)_N×2 is called the gray relational decision matrix. i is No. i intake valve duration angle, and j is the indicator number.

Step 2. According to the decision matrix D, the maximum and minimum values corresponding to evaluated indicators can be obtained, expressed as D_min = (D_0min, D_1min) and D_max = (D_0max, D_1max).

Step 3. Normalization of the decision matrix. The decision matrix can be normalized by using the D_min and D_max, described by the following equation:

$$f_{ij}=\frac{D_{ij}-D_{oj\min }}{D_{oj\max }-D_{oj\min }}$$

(30)

Step 4. Calculated the relation degree. The relational degree between the calculated indicators and the leading indicators can be calculated by:

$$\begin{array}{cc}{R}_i={\displaystyle \sum _{j=1}^2{f}_{ij}{W}_j},& i=1,2,\dots ,N\end{array}$$

(31)

where W_j is the vector of weight, W_j = (0.6, 0.4)^T.

The value of R_i can determine the optimal intake valve duration angle. The bigger R_i is, the more the calculated indicators fit with the leading indicators.

The optimal performance indicators and corresponding intake valve duration angles are obtained by simulating calculations in

Table 4. Performance indicators and optimum intake valve duration angle.

Optimum Intake Valve Duration Angle (deg)	Output Power (kW)	Energy Efficiency (%)	Speed (rpm)
15	8.91	56.8	600
20	12.1	50.9	700
25	15.02	44.86	800
30	17.27	38.9	900
30	18.06	35.1	1000
30	18.23	31.2	1100

.

In different rotational speed conditions, optimal trajectories of the intake valve can be obtained, which are shown in Figure 10.

When the rotational speed is lower than 900 rpm, the intake valve duration angle is equal to 30°. When the rotation speed is higher than 900 rpm, the intake valve duration angle is equal to 30°. Due to the different rotational speeds, the opening and closing angles of the intake valve are different during the same opening and closing times. Figure 10 gives the reference intake valve trajectories at different rotational speeds, which is the basement to control the intake valve actuator.

4. Conclusions

In this paper, an EPVVA system was proposed for independent control distributing valve parameters. Based on the EPVVA system, the working principle was introduced, and the mathematical model of the system was developed. Meanwhile, the working characteristics were obtained by numerical analysis. Trajectory planning for the intake valve and the effect of intake valve duration angle were carried out. The optimum intake valve duration angles at different rotational speeds were obtained by using the GRA method. Some conclusions can be drawn:

(1)

The energy efficiency decreases with the intake valve duration angle increasing.

(2)

Initially, the output power ascends sharply with increasing intake valve duration angle, but the amplitude of power growth decreases. When the rotational speed is equal to 800 rpm, the output power peaks at approximately 19.57 kW when the intake valve duration angle is equal to 75°, after this peak, the output power starts to descend slowly.

(3)

When the rotational speed varies from 600 rpm to 900 rpm, the corresponding intake valve duration angle increases from 15° to 30°. When the rotation speed is higher than 900 rpm, the intake valve duration angle is equal to 30°.

The future work will focus on the precise control of this VVA system and the experimental study in the actual PTE to comprehensively study the performance of the proposed system.