# Fuzzy Logic Systems-Based Optimization and Control of Complicated High-Pressure Fuel Line with Double Fuel Injectors and One Pressure Reducing Valve

Authors: Dunqian Cao; Xue Zhang; Haoran Xue; Conglin Xu
Journal: Complexity
 (2020)

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## Abstract

High-pressure fuel line is a basic component of many modern industrial devices, and its pressure stability affects operating efficiency and safety. In this paper, the working mechanism of high-pressure fuel line with double fuel injectors and pressure reducing valve is studied as the main object, the differential equation for changes in pressure with time is established, and the rotating speed of high-pressure fuel pump cam, fuel injection time of fuel injector, and the threshold of pressure reducing valve are optimized using fuzzy logic systems to control the pressure in high-pressure fuel line within a safe range and improve the operating efficiency and safety of high-pressure fuel line. The results of numerical simulation show that the pressure reducing valve can effectively control the pressure in high-pressure fuel line.


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## Body

## 1. Introduction

Fuel-powered engines play a very important role in production and daily life. The pressure in a high-pressure fuel line will change to a certain extent when fuel oil is introduced into and ejected from it, especially during intermittent operation, in which case the amount of fuel ejected will be different from the expected amount. Such deviation will affect the operating efficiency of fuel-powered engines and may even result in high-pressure fuel line rupture when the pressure reaches a certain level.Figure1 shows a high-pressure fuel line of certain model with an internal length of l and an inner diameter of d. Fuel (oil) is supplied by a high-pressure fuel pump to the high-pressure fuel line. A high-pressure fuel pump mainly consists of plunger housing, plunger, and cam. The cam rotates to drive the plunger to move reciprocally. When the plunger reaches the bottom dead center, the plunger housing will be filled with low-pressure fuel. When the plunger moves upwards to compress the fuel, the pressure in the plunger housing will increase gradually. When the pressure in the plunger housing exceeds that in the high-pressure fuel line, the check valve connecting the plunger housing and high-pressure fuel line will open to allow fuel oil to enter the high-pressure fuel line. When the plunger reaches the top dead center, it will start to move downwards and the pressure in the plunger housing will decline gradually. When the pressure in the plunger housing is lower than that in the high-pressure fuel line, the check valve will close. And when the plunger reaches the bottom dead center again, the plunger will be refilled with low-pressure fuel. This process is repeated, as shown in Figure 2. It is obvious that the inflow velocity is determined by the angular velocity of the rotating cam [1].Figure 1
Schematic of the working mechanism of high-pressure fuel line.Figure 2
Schematic of the working mechanism of high-pressure fuel pump. (a)α = 0. (b) 0 < α < π. (c) α = π. (d) π < α < 2π.
(a)(b)(c)(d)Figure 3
Relationship between the pressure and density of a certain fuel.A high-pressure fuel line is usually fitted with one or two fuel injectors ejecting fuel oil at a fixed frequency. The amount of fuel ejected is related to various factors such as the pressure in high-pressure fuel line, the area of injector opening, and nozzle-opening time. High-pressure fuel injection systems are considered the most prospective fuel injection systems for direct injection diesel engines [2, 3]. In such systems, the pressure can be controlled flexibly and the amount of fuel injected is determined by system pressure and injection time. Therefore, the precision of pressure control is one of the important factors affecting the performance of fuel-powered engines [4].For fuel-powered engines operating under certain conditions, it is required that the pressure in high-pressure fuel line remain stable at a certain level and pressure surges be minimized, so that the amount of fuel injected can be controlled precisely [5, 6]. In [7], the multi-injection nozzle model was solved using the Monte Carlo method. This method is a random search algorithm, which is advantageous in that it does not require the first derivative of the objective function and is deficient in that it is highly random and unstable and converges slowly and that the solution found by it might be a local optimal solution. In [8], a difference equation-based model for determining cam angular velocity and a functional model-based check valve control scheme were established, and a scheme was developed for controlling cam angular velocity, injection rule, and pressure reducing valve parameters at constant pressure in several cases. In [9], a model for simulating and analyzing fuel injection systems with high-pressure fuel line was built using AVL-HYDSIM, and a series of parameters affecting the performance of such systems such as injection pressure were determined. In [10], a high-pressure fuel line pressure stabilization model based on first-order difference equation, a model for simulating pressure distribution in high-pressure fuel line, and a single-objective programming-based model for designing fuel injection systems were built using the ideas of the finite difference method, the method of characteristics, single-objective optimization, and exhaustive search, and different schemes were given for designing high-pressure fuel injection systems under different operating conditions of high-pressure fuel line. Additionally, this issue was also studied in [11, 12], in which the relationship between changes in pressure and cam angular velocity was given.In [13], the rule of changes in pressure with the volume of high-pressure fuel line and the number of fuel injectors was determined through simulation and analysis. In [14], the rule of changes in average pressure surge with length-to-diameter ratio was determined through simulation and analysis. In [15], the relationship between changes in the amount of fuel injected by the system and pressure was identified through experiments. In [16], through experiments, an ideal filter was designed to eliminate vibration induced by water hammer pressure waves in high-pressure systems. Pressure surges have a significant impact on the performance of high-pressure systems, but there are few studies aimed at analyzing the rule of pressure surges in a segmented and detailed manner. Therefore, the rule of pressure surges needs to be further studied.Excessively high internal pressure may cause instable operation and even rupture of a high-pressure fuel line. To avoid such phenomena, it is planned to install a pressure reducing check valve in the high-pressure fuel line to allow fuel oil to return to the external low-pressure fuel line when the internal pressure reaches a certain level, thus reducing the pressure in high-pressure fuel line.The focus of this study is put on how to control the cam rotating speed, fuel injection time, and the threshold of pressure reducing valve, control the pressure in high-pressure fuel line within a certain stable range, and identify the rule of changes in pressure with time. For simplicity, fuel density and pressure in the same chamber are assumed to be uniform at all times (Tables1 and 2).Table 1
Symbol description.
SymbolDescriptionPPressureρDensityEElastic modulusm,miFuel flow rate per unit timeMiFuel flow through the orificeCFlow coefficient of fuelAArea of small orificetTimeγPolar radius of camαPolar angle of camρ0Fuel density in plunger housingρ∨0Density when plunger moves to bottom dead centerρ^0Density when plunger moves to top dead centerV0Volume of plunger housingV∨0Volume when plunger moves to bottom dead centerV^0Volume when plunger moves to top dead centerd0Diameter of plunger housingd1Diameter of small orifice at fuel inletA1Area of small orifice at fuel inletωCam rotating speedlLength of high-pressure fuel linedDiameter of high-pressure fuel lineP1Internal combustion pressure in high-pressure fuel lineV1Volume of high-pressure fuel lineρ1Fuel density in high-pressure fuel lineP¯Stable pressure in high-pressure fuel lineM¯1Amount of oil into high-pressure fuel line per circle of cam rotationd2Diameter of fuel injector needle valved¯2Diameter of the lowest nozzle of fuel injector(s)θHalf-angle of fuel injector seal seathLift of needle valveA2Area of fuel injector openingd4Diameter of pressure reducing valve outletΩThreshold of pressure reducing valveTable 2
Unit.
Physical quantityUnitPressureMPaDensitymg/mm3MassmgTimemsAngular velocityrad/msLengthmmAreamm2Volumemm3


## 2. Preparations

### 2.1. Description of Symbols

Description of symbols is provided in Table1.


### 2.2. Units

The units of the physical quantity are provided in Table2.


### 2.3. Relationship between Fuel Pressure and Density

The relationship between fuel pressureP, density ρ, and elastic modulus E can be expressed as [17](1)dPdρ=Eρ.If the relationship between the elastic modulus and density of a certain fuel complies withE=aP2+bP+c, the following formulas can be obtained by solving equation (1):(2)P=Ψρ=−b+W·tan1/2·W·lnρ+Z2a,(3)ρ=ΦP=exp2Warctan2aP+bW−π−Z,where W=4ac−b2 and Z is an arbitrary constant. If fuel density is 0.85 when a known pressure of 100 is given,(4)Z=arctanb+200a1+u1u2−Wu1−u2W1+u1u2+b+200au1−u2,where u1=tan1/2ln17W,u2=tan1/2ln20W. Specifically, when(5)E=0.02893P2+3.077P+1572,then(6)P=Ψρ=226.958tan6.566lnρ−1.48−53.18,0.62<ρ<0.98,(7)ρ=ΦP=0.7765exp0.1523arctan0.0044P+0.2343,P>0.Figure3 shows the relationship between fuel pressure and density. The fuel mentioned in this paper refers to the fuel that complies with such relationship. The effect of temperature on the fuel is neglected.


### 2.4. Relationship of Fuel Flow Rate to Pressure and Density

The fuel flows from the high-pressure side to the low-pressure side through the small orifice. The amount of fuel flowing through the small orifice per unit time is(8)m=CA2ρΔP,where C is the flow coefficient, A is the area of the small orifice, ΔP is the pressure difference between two sides of the small orifice, and ρ is fuel density on the high-pressure side. When the pressure on the low-pressure side is very small, m can be approximated to(9)m=CA2ρP.


### 2.5. Analysis of Fuel Pressure in Plunger Housing

Assuming that the cam shape is defined by polar angle and polar radius asγτ, γmin=γπ/2 and γmax=γ3π/2. When the cam rotating angle is α=0, the plunger will move to the bottom dead center, the volume of plunger housing will reach its maximum, Vˇ0, and the plunger housing will be filled with low-pressure fuel (with a density ofρˇ0); when α=π, the plunger will move to the top dead center and the volume of plunger housing will reach its minimum, V^0. After the cam rotates α radians, the distance between the plunger and the top dead center is γmax−γα+π/2, and the volume of plunger housing will be(10)V0α=V^0+π4d02·hπ−hα=V^0+π4d02·γmax−γα+π2,where d0 is the diameter of plunger housing. It is obvious that(11)V0∨=V00=V^0+π4d02·γmax−γmin,and if the fuel does not enter the high-pressure fuel line, fuel density in the plunger housing will be(12)ρ0α=ρ0∨V0∨V0α,and fuel pressure therein will be(13)P0α=Ψρ0α.If the fuel can enter the high-pressure fuel line and the pressure in high-pressure fuel line remains stable atP¯1, the amount of fuel into high-pressure fuel line per circle of cam rotation will be(14)M¯1=ρ00V00−ΦP¯1V0α.


### 2.6. Area of Fuel Injector Opening

Assuming that the fuel injection nozzle has a structure shown in Figure4, the diameter of needle valve isd2, the seal seat is a cone with a half-angle of θ, and the diameter of the lowest nozzle is d¯2, when its lift is 0, the needle valve will close, when its lift is greater than 0, the needle valve will open, and the fuel will flow towards and be ejected through the injection nozzle.Figure 4
Working principle of fuel injector.Assuming that the relationship between the lift of needle valve and time within one fuel injection cycle ish = h(t), then the area of fuel injector opening will be(15)A2t=πd22+ht·tanθ2−π4d22,0≤ht≤h¯,π4d¯22,ht>h¯,where h¯=−d2+d22+d¯22/2tanθ. When the lift is relatively low, the area of fuel injector opening will be area of the clearance between the needle valve and the seal seat. When the lift is relatively high, the area of fuel injector opening will be area of the lowest injection nozzle of fuel injector(s).


### 2.7. Working Principle of Pressure Reducing Valve

The pressure reducing check valve is connected to the high-pressure fuel line via a round pipe with a diameter ofd4. When the pressure in high-pressure fuel line is higher than Ω, the pressure reducing valve will open automatically to allow the fuel in high-pressure fuel line to flow back to the external low-pressure fuel line, thus reducing the fuel pressure in high-pressure fuel line. When the pressure in high-pressure fuel line is lower than Ω, the pressure reducing valve will close automatically. It can be easily known that the amount of fuel through the pressure reducing valve per unit time is(16)m4t=C·π4d42·2ρ1tP1t,P1t>Ω,0,P1t≤Ω,=signP1t−Ω·C·π4d42·2ρ1tP1t.


## 2.1. Description of Symbols

Description of symbols is provided in Table1.


## 2.2. Units

The units of the physical quantity are provided in Table2.


## 2.3. Relationship between Fuel Pressure and Density

The relationship between fuel pressureP, density ρ, and elastic modulus E can be expressed as [17](1)dPdρ=Eρ.If the relationship between the elastic modulus and density of a certain fuel complies withE=aP2+bP+c, the following formulas can be obtained by solving equation (1):(2)P=Ψρ=−b+W·tan1/2·W·lnρ+Z2a,(3)ρ=ΦP=exp2Warctan2aP+bW−π−Z,where W=4ac−b2 and Z is an arbitrary constant. If fuel density is 0.85 when a known pressure of 100 is given,(4)Z=arctanb+200a1+u1u2−Wu1−u2W1+u1u2+b+200au1−u2,where u1=tan1/2ln17W,u2=tan1/2ln20W. Specifically, when(5)E=0.02893P2+3.077P+1572,then(6)P=Ψρ=226.958tan6.566lnρ−1.48−53.18,0.62<ρ<0.98,(7)ρ=ΦP=0.7765exp0.1523arctan0.0044P+0.2343,P>0.Figure3 shows the relationship between fuel pressure and density. The fuel mentioned in this paper refers to the fuel that complies with such relationship. The effect of temperature on the fuel is neglected.


## 2.4. Relationship of Fuel Flow Rate to Pressure and Density

The fuel flows from the high-pressure side to the low-pressure side through the small orifice. The amount of fuel flowing through the small orifice per unit time is(8)m=CA2ρΔP,where C is the flow coefficient, A is the area of the small orifice, ΔP is the pressure difference between two sides of the small orifice, and ρ is fuel density on the high-pressure side. When the pressure on the low-pressure side is very small, m can be approximated to(9)m=CA2ρP.


## 2.5. Analysis of Fuel Pressure in Plunger Housing

Assuming that the cam shape is defined by polar angle and polar radius asγτ, γmin=γπ/2 and γmax=γ3π/2. When the cam rotating angle is α=0, the plunger will move to the bottom dead center, the volume of plunger housing will reach its maximum, Vˇ0, and the plunger housing will be filled with low-pressure fuel (with a density ofρˇ0); when α=π, the plunger will move to the top dead center and the volume of plunger housing will reach its minimum, V^0. After the cam rotates α radians, the distance between the plunger and the top dead center is γmax−γα+π/2, and the volume of plunger housing will be(10)V0α=V^0+π4d02·hπ−hα=V^0+π4d02·γmax−γα+π2,where d0 is the diameter of plunger housing. It is obvious that(11)V0∨=V00=V^0+π4d02·γmax−γmin,and if the fuel does not enter the high-pressure fuel line, fuel density in the plunger housing will be(12)ρ0α=ρ0∨V0∨V0α,and fuel pressure therein will be(13)P0α=Ψρ0α.If the fuel can enter the high-pressure fuel line and the pressure in high-pressure fuel line remains stable atP¯1, the amount of fuel into high-pressure fuel line per circle of cam rotation will be(14)M¯1=ρ00V00−ΦP¯1V0α.


## 2.6. Area of Fuel Injector Opening

Assuming that the fuel injection nozzle has a structure shown in Figure4, the diameter of needle valve isd2, the seal seat is a cone with a half-angle of θ, and the diameter of the lowest nozzle is d¯2, when its lift is 0, the needle valve will close, when its lift is greater than 0, the needle valve will open, and the fuel will flow towards and be ejected through the injection nozzle.Figure 4
Working principle of fuel injector.Assuming that the relationship between the lift of needle valve and time within one fuel injection cycle ish = h(t), then the area of fuel injector opening will be(15)A2t=πd22+ht·tanθ2−π4d22,0≤ht≤h¯,π4d¯22,ht>h¯,where h¯=−d2+d22+d¯22/2tanθ. When the lift is relatively low, the area of fuel injector opening will be area of the clearance between the needle valve and the seal seat. When the lift is relatively high, the area of fuel injector opening will be area of the lowest injection nozzle of fuel injector(s).


## 2.7. Working Principle of Pressure Reducing Valve

The pressure reducing check valve is connected to the high-pressure fuel line via a round pipe with a diameter ofd4. When the pressure in high-pressure fuel line is higher than Ω, the pressure reducing valve will open automatically to allow the fuel in high-pressure fuel line to flow back to the external low-pressure fuel line, thus reducing the fuel pressure in high-pressure fuel line. When the pressure in high-pressure fuel line is lower than Ω, the pressure reducing valve will close automatically. It can be easily known that the amount of fuel through the pressure reducing valve per unit time is(16)m4t=C·π4d42·2ρ1tP1t,P1t>Ω,0,P1t≤Ω,=signP1t−Ω·C·π4d42·2ρ1tP1t.


## 3. Analysis of Fuel Density and Pressure in High-Pressure Fuel Line

Assuming thatP0t,P1t are, respectively, fuel pressure in the plunger housing and that in the high-pressure fuel line at the time of t, ρ0t,ρ1t are, respectively, fuel density in the plunger housing and that in the high-pressure fuel line at the time of t, and V1 is the volume of high-pressure fuel line, then V1=ld2/4π. The amount of fuel from the high-pressure fuel pump into the high-pressure fuel line per unit time is m1t, the amounts of fuel injected by two fuel injectors are m2t−μ1 and m2t−μ2, respectively, and the amount of fuel from the pressure reducing valve is m4t. According to the law of conservation of mass, in a small time range oft,t+Δt,(17)ρ1t+Δt−ρ1tV1=m1t−m2t−μ1−m2t−μ2−m4tΔt.LettingΔt⟶0, by rearranging the formula above, the following equation can be obtained:(18)dρ1tdt=1V1m1t−m2t−μ1−m2t−μ2−m4t,where(19)m1t=CA12ρ0tP0t−P1t,P0t>P1t,0,P0t<P1t,=signP0t−P1t·CA12ρ0tP0t−P1t,(20)m2t=C·A2t·2ρ1tP1t,(21)m4t=C·π4d42·2ρ1tP1t,P1t>Ω,0,P1t≤Ω,=signP1t−Ω·C·π4d42·2ρ1tP1t,(22)signx=0,x≤0,1,x>0.These are the equations with which fuel density in the high-pressure fuel line complies. Fuel pressure can be calculated using formula (2). If the initial conditions are given, the numerical solutions to fuel density and pressure in the high-pressure fuel line can be found by difference method.


## 4. Fuzzy Logic Systems-Based Optimization and Control

Considering the changes in fuel pressureP1t in the high-pressure fuel line in a large time range of [0, T1], mean squared error is defined as follows to measure the degree of pressure deviation P¯:(23)E=1T1∫0T1P1t−P¯2dt.The objective of optimization is to find the solutions to cam rotating speedω, fuel injection times 155 μ1,μ2, and pressure reducing valve threshold Ω and to minimize E. Therefore, the optimization 156 model can be expressed as(24)minEω,μ1,μ2,Ω=1T1∫0T1P1t−P¯2dt,where ω>0,μ1∈0,2π/ω,μ2∈μ1,2π/ω,Ω>P¯1.The optimization problem mentioned above can be solved in two cases: (1) when the high-pressure fuel line is not fitted with a pressure reducing valve and (2) when high-pressure fuel line is fitted with a pressure reducing valve.


### 4.1. High-Pressure Fuel Line without Pressure Reducing Valve

If the high-pressure fuel line is not fitted with a pressure reducing valve,m4t≡0 and problem (24) can be degenerated to(25)minEω,μ1,μ2.To minimizeEω,μ1,μ2, the amount of fuel into the high-pressure fuel line per unit time should be equal to the amount of fuel ejected per unit of time.The amount of fuel that enters the high-pressure fuel line and the amount of fuel that is injected within a period of time should be equal. Based on the working cycle of the injector, the number of cam rotations isω/2πT in one cycle, and one rotation enters the high pressure. The amount of fuel in the fuel pipe is M¯1. The amount of fuel injected by an injector in time T is M2T. Since there are two injectors in this question, then we have(26)ω2πT·M¯1=2M2T.Then, the cam rotating speed should be(27)ω=4πM2TM¯1·T,where M2T is the amount of fuel ejected by a fuel injector in one operating cycle and M2T=∫0Tm2tdt.Solving procedure is as follows:(1)
Estimate an initial rotating speed ofω0, and let i = 0, μ10=μ20=T/2,δ=0.01,ε=10−4.(2)
Letω=ωi, set search ranges μ1∈0,T,μ2∈μ1,T, perform two-dimensional search for the solution to problem (25), and find the solutions to μ1i+1,μ2i+1.(3)
Letμ1=μ1i+1,μ2=μ2i+1, set search range ω∈ωi−δ,ωi+δs, perform one-dimensional search for the solution toω, and find the solution to ωi+1.(4)
Ifδ<ε, stop searching. Otherwise, reduce the search ranges for μ1,μ2, let δ=0.5δ, i = i+1, and return to step (2).


### 4.2. High-Pressure Fuel Line with Pressure Reducing Valve

If the high-pressure fuel line is fitted with a pressure reducing valve, the operating cycle of the plunger housing may be assumed to be the same as that of fuel injector to stabilize the operation of high-pressure fuel line as far as practicable. In this case,ω=2π/T. Therefore, problem (24) can be degenerated to(28)minEμ1,μ2,Ω.The solving method for high-pressure fuel line with pressure reducing valve is similar to that for high-pressure fuel line without pressure reducing valve. The only difference is that, in the solving method for high-pressure fuel line with pressure reducing valve, the optimization of cam rotating speedω is converted to that of pressure reducing valve thresholdΩ and the initial Ω may be set to a value slightly higher than P¯1.


## 4.1. High-Pressure Fuel Line without Pressure Reducing Valve

If the high-pressure fuel line is not fitted with a pressure reducing valve,m4t≡0 and problem (24) can be degenerated to(25)minEω,μ1,μ2.To minimizeEω,μ1,μ2, the amount of fuel into the high-pressure fuel line per unit time should be equal to the amount of fuel ejected per unit of time.The amount of fuel that enters the high-pressure fuel line and the amount of fuel that is injected within a period of time should be equal. Based on the working cycle of the injector, the number of cam rotations isω/2πT in one cycle, and one rotation enters the high pressure. The amount of fuel in the fuel pipe is M¯1. The amount of fuel injected by an injector in time T is M2T. Since there are two injectors in this question, then we have(26)ω2πT·M¯1=2M2T.Then, the cam rotating speed should be(27)ω=4πM2TM¯1·T,where M2T is the amount of fuel ejected by a fuel injector in one operating cycle and M2T=∫0Tm2tdt.Solving procedure is as follows:(1)
Estimate an initial rotating speed ofω0, and let i = 0, μ10=μ20=T/2,δ=0.01,ε=10−4.(2)
Letω=ωi, set search ranges μ1∈0,T,μ2∈μ1,T, perform two-dimensional search for the solution to problem (25), and find the solutions to μ1i+1,μ2i+1.(3)
Letμ1=μ1i+1,μ2=μ2i+1, set search range ω∈ωi−δ,ωi+δs, perform one-dimensional search for the solution toω, and find the solution to ωi+1.(4)
Ifδ<ε, stop searching. Otherwise, reduce the search ranges for μ1,μ2, let δ=0.5δ, i = i+1, and return to step (2).


## 4.2. High-Pressure Fuel Line with Pressure Reducing Valve

If the high-pressure fuel line is fitted with a pressure reducing valve, the operating cycle of the plunger housing may be assumed to be the same as that of fuel injector to stabilize the operation of high-pressure fuel line as far as practicable. In this case,ω=2π/T. Therefore, problem (24) can be degenerated to(28)minEμ1,μ2,Ω.The solving method for high-pressure fuel line with pressure reducing valve is similar to that for high-pressure fuel line without pressure reducing valve. The only difference is that, in the solving method for high-pressure fuel line with pressure reducing valve, the optimization of cam rotating speedω is converted to that of pressure reducing valve thresholdΩ and the initial Ω may be set to a value slightly higher than P¯1.


## 5. Numerical Simulation

Assuming that the length of high-pressure fuel line isL = 500 and the inner diameter thereof is D = 10, then the volume thereof is V1 = 12500π. The initial fuel pressure in the high-pressure fuel line is P10=100 and fuel density therein is ρ10=0.85. The diameter of the small orifice at fuel inlet is d1=1.4. The shape of the cam is shown in Figure 5. The polar coordinate equation of its cam contour line is(29)γτ=γmin+γmax−γmin21−sinτ,where γmin=2.414,γmax=7.239.Figure 5
Shape of the cam.The inner diameter of plunger housing isd0=5, and the volume of plunger housing when the plunger moves to the bottom dead center is V^0=20. The volume of plunger housing when the plunger moves to the top dead center is(30)V0∨=V00=V^0+π4d02·γmax−γmin≈114.74.The pressure of low-pressure fuel isρ0∨=0.5 and the density thereof is ρ0=0.8044.If the pressure in high-pressure fuel line remains constant at 100, the amount of fuel into the high-pressure fuel pump per circle of cam rotation will be(31)M¯1=ρ0∨V0∨−ΦP1V^0≈ρ0∨V0∨−100V^0=75.3041.If there are no fuel injector and pressure reducing valve, the changes in the volume of plunger housing, fuel inflow, fuel pressure, and density in the high-pressure fuel line with time when the cam rotating speed isω=0.01 are shown in Figures 6 and 7, respectively.Figure 6
Changes in the volume of plunger housing and fuel inflow with time.
(a)(b)Figure 7
Changes in fuel pressure and density in the high-pressure fuel line with time.
(a)(b)Figure6 shows that when the plunger is rising and the pressure in the plunger cavity reaches a certain value, the fuel starts to enter the high-pressure fuel pipe and reaches a peak at a certain time. After that, the flow rate starts to decrease to 0.Assuming that the diameter of fuel injector needle valve isd2 = 2.5, the half-angle of seal seat is θ=π/20, and the diameter of the lowest injection nozzle is d¯2 = 1.4, then the relationship between the lift of needle valve and time within one injection cycle of T = 100 is(32)ht=1−cos15.71t2,0≤t<0.45,2,0.45≤t≤2,1−cos15.71t−2.452,2<t<2.45,0,2.46≤t≤T,According to formulas (15) and (32), the relationship between the area of fuel injector opening and time is as follows:(33)A2t=πd22+1−cos15.71t2·tanθ2−π4d22,0≤t≤0.33,π4d¯22,0.33<t<2.12,πd22+1−cos15.71t−2.452·tanθ2−π4d22,2.12≤t<2.45,0,2.45≤t≤T,.From formula (20), it can be known that the amount of fuel injected by a fuel injector in one injection cycle is(34)M2T=C·2ρ1P1∫0TA2tdt≈32.87890421.To solve equation (18), time is evenly discretized into t0,t1,t2,…,tn,…, and the interval between two adjacent time points is set to Δt. If the volume of plunger housing V0ti−1, fuel density in the plunger housing ρ0ti−1, fuel pressure in the plunger housing P0ti−1, fuel pressure in the high-pressure fuel line P1ti−1, fuel density in the high-pressure fuel line ρ1ti−1, fuel injector opening area A2ti−1, injection times (at which the two fuel injectors start to operate) μ1,μ2, and pressure reducing valve threshold Ω at the time of ti−1 are known, fuel density in the high-pressure fuel line at the time of ti will be(35)ρ1ti=ρ1ti−1+1V1m1ti−1−m2ti−1−μ1−m2ti−1−μ2−m4ti−1Δt,where(36)m1ti−1=signP0ti−1−P1ti−1·CA12ρ0ti−1P0ti−1−P1ti−1,m2ti−1−μ1=C·A2ti−1−μ1·2ρ1ti−1P1ti−1,m2ti−1−μ2=C·A2ti−1−μ2·2ρ1ti−1P1ti−1,m4ti−1=signP1ti−1−Ω·C·π4d42·2ρ1ti−1P1ti−1.Fuel pressure can be calculated using formula (6) as follows: P1ti=Ψρ1ti.


### 5.1. High-Pressure Fuel Line with Only One Fuel Injector

Assuming that the high-pressure fuel line is fitted with only one plunger house and one fuel injector, thenm2t−μ2≡0,m4t≡0,T1=1000, and the solutions found are as follows:(37)ω=0.027186,μ1=0.In this case, the average fuel pressure in the high-pressure fuel line isP¯=99.946, the mean squared error is E = 1.34898, the minimum fuel pressure is Pmin= 97.561, and the maximum fuel pressure is Pmax=102.552. The changes in fuel pressure are shown in Figure 8.Figure 8
Changes in fuel pressure in high-pressure fuel line with only one fuel injector.It can be seen from the simulation results that the pressure in the high-pressure fuel pipe is not completely cyclically changed, which will cause a slight difference in the amount of fuel injected by the injector every time in actual work, which may make the system work unstable.


### 5.2. High-Pressure Fuel Line with Two Fuel Injectors

Assuming that the high-pressure fuel line is fitted with two fuel injectors, thenm4t≡0, and the solutions found are as follows:(38)ω=0.054923,μ1=0,μ2=44.In this case, the average fuel pressure in the high-pressure fuel line isP¯=99.999, the mean squared error is E = 1.231059, the minimum fuel pressure is Pmin= 97.574, and the maximum fuel pressure is Pmax=102.472. The changes in fuel pressure are shown in Figure 9.Figure 9
Changes in fuel pressure in high-pressure fuel line with two fuel injectors.Compared with the situation where there is only one injector, the two injectors make the average pressure in the high-pressure tubing closer to the target, and the variance is also improved, but the pressure is still not periodic.


### 5.3. High-Pressure Fuel Line with Two Fuel Injectors and One Pressure-Reducing Valve

When the high-pressure fuel line is fitted with two fuel injectors and one pressure reducing valve, the fuzzy system is used to optimize.The fuzzy logic systems use IF-THEN rules to construct a mapping from an input vectorxt=x1t,x2t,…,xntT∈ℛn to an output yt∈ℛ. The rth fuzzy rule can be written as(39)ℛr:ifx1tisA1rx1tandxntisAnrxntthenytisBr,where Ar and Br are fuzzy sets with membership functions μAirxit and μBiryt, respectively. Suppose that xt belongs to some compact set. By using the product-inference rule, singleton fuzzifier, and center-average defuzzifier, the output of fuzzy logic system can be expressed as [18,19](40)yt=∑inry˜it∏j=1nμAjixjt∑i=1nr∏j=1nμAjixjt=θTtΨxt,where nr is the number of total fuzzy rules, y˜it is the point at which μBiry˜it=1,μAjixjt is the membership function of the fuzzy variable xjt characterized by Gaussian, θt=y˜1t,y˜2t,…,y˜nrtT is an adjustable parameter vector, and Ψ=Ψ1,Ψ2,…,ΨnrT is a fuzzy basis vector, where Ψi is defined as(41)Ψixt=∏j=1nμAjixjt∑i=1nr∏j=1nμAjixjt.The structure of a fuzzy system is shown in Figure10.where η,h,m are the numbers of neurons in the input, middle, and output layers, wk=wk1,…,wkhT, and ϖk=φk1∑i=1nv1isi+θ1,…,φkh∑i=1nvhisi+θhT. According to the universal approximation theorem, φx=ex−e−x/ex+e−x can be selected as function φ·. Therefore, this system can be expressed as(43)y=ϑTψx,where ϑ=w1T,…,wmTT and ψ·=ϖ1·0⋯00ϖ2·⋯0⋮⋮⋱⋮00⋯ϖm·.Figure 10
Structure of a fuzzy inference system. Its mathematical model can be expressed as [18–22](42)yks,wk=∑j=1hwkjφkj∑i=1ηvjisi+θj=wkTϖk·,The optimization is achieved using the fuzzy logic systems above and the solutions found are as follows:(44)μ1=21.4,μ2=33.7,Ω=101.14.The average fuel pressure isP¯=100.092, the minimum fuel pressure is Pmin=99.078, the maximum fuel pressure is Pmax=101.145, and the mean squared error is E = 0.151434. The changes in fuel pressure are shown in Figure 11.Figure 11
Changes in fuel pressure in high-pressure fuel line with two fuel injectors and one pressure reducing valve.From the simulation results, the pressure shows a periodic change, and the fluctuation range is very small. The results of numerical simulation show that the pressure reducing valve plays a significant role in stabilizing fuel pressure (the mean squared error for high-pressure fuel line fitted with a pressure reducing valve is only 12.3% of that for high-pressure fuel line without pressure reducing valve).The system can automatically restore the normal pressure level even if fuel pressure rises or drops abruptly due to certain reasons. The process of pressure recovery from 0.5 MPa to 100 MPa takes 6917.7 ms approximately and the process of pressure recovery from 160 MPa to 100 MPa only needs 42.41 ms (Figure12).Figure 12
Schematic of fuel pressure recovery from low or high level to normal level.
(a)(b)


## 5.1. High-Pressure Fuel Line with Only One Fuel Injector

Assuming that the high-pressure fuel line is fitted with only one plunger house and one fuel injector, thenm2t−μ2≡0,m4t≡0,T1=1000, and the solutions found are as follows:(37)ω=0.027186,μ1=0.In this case, the average fuel pressure in the high-pressure fuel line isP¯=99.946, the mean squared error is E = 1.34898, the minimum fuel pressure is Pmin= 97.561, and the maximum fuel pressure is Pmax=102.552. The changes in fuel pressure are shown in Figure 8.Figure 8
Changes in fuel pressure in high-pressure fuel line with only one fuel injector.It can be seen from the simulation results that the pressure in the high-pressure fuel pipe is not completely cyclically changed, which will cause a slight difference in the amount of fuel injected by the injector every time in actual work, which may make the system work unstable.


## 5.2. High-Pressure Fuel Line with Two Fuel Injectors

Assuming that the high-pressure fuel line is fitted with two fuel injectors, thenm4t≡0, and the solutions found are as follows:(38)ω=0.054923,μ1=0,μ2=44.In this case, the average fuel pressure in the high-pressure fuel line isP¯=99.999, the mean squared error is E = 1.231059, the minimum fuel pressure is Pmin= 97.574, and the maximum fuel pressure is Pmax=102.472. The changes in fuel pressure are shown in Figure 9.Figure 9
Changes in fuel pressure in high-pressure fuel line with two fuel injectors.Compared with the situation where there is only one injector, the two injectors make the average pressure in the high-pressure tubing closer to the target, and the variance is also improved, but the pressure is still not periodic.


## 5.3. High-Pressure Fuel Line with Two Fuel Injectors and One Pressure-Reducing Valve

When the high-pressure fuel line is fitted with two fuel injectors and one pressure reducing valve, the fuzzy system is used to optimize.The fuzzy logic systems use IF-THEN rules to construct a mapping from an input vectorxt=x1t,x2t,…,xntT∈ℛn to an output yt∈ℛ. The rth fuzzy rule can be written as(39)ℛr:ifx1tisA1rx1tandxntisAnrxntthenytisBr,where Ar and Br are fuzzy sets with membership functions μAirxit and μBiryt, respectively. Suppose that xt belongs to some compact set. By using the product-inference rule, singleton fuzzifier, and center-average defuzzifier, the output of fuzzy logic system can be expressed as [18,19](40)yt=∑inry˜it∏j=1nμAjixjt∑i=1nr∏j=1nμAjixjt=θTtΨxt,where nr is the number of total fuzzy rules, y˜it is the point at which μBiry˜it=1,μAjixjt is the membership function of the fuzzy variable xjt characterized by Gaussian, θt=y˜1t,y˜2t,…,y˜nrtT is an adjustable parameter vector, and Ψ=Ψ1,Ψ2,…,ΨnrT is a fuzzy basis vector, where Ψi is defined as(41)Ψixt=∏j=1nμAjixjt∑i=1nr∏j=1nμAjixjt.The structure of a fuzzy system is shown in Figure10.where η,h,m are the numbers of neurons in the input, middle, and output layers, wk=wk1,…,wkhT, and ϖk=φk1∑i=1nv1isi+θ1,…,φkh∑i=1nvhisi+θhT. According to the universal approximation theorem, φx=ex−e−x/ex+e−x can be selected as function φ·. Therefore, this system can be expressed as(43)y=ϑTψx,where ϑ=w1T,…,wmTT and ψ·=ϖ1·0⋯00ϖ2·⋯0⋮⋮⋱⋮00⋯ϖm·.Figure 10
Structure of a fuzzy inference system. Its mathematical model can be expressed as [18–22](42)yks,wk=∑j=1hwkjφkj∑i=1ηvjisi+θj=wkTϖk·,The optimization is achieved using the fuzzy logic systems above and the solutions found are as follows:(44)μ1=21.4,μ2=33.7,Ω=101.14.The average fuel pressure isP¯=100.092, the minimum fuel pressure is Pmin=99.078, the maximum fuel pressure is Pmax=101.145, and the mean squared error is E = 0.151434. The changes in fuel pressure are shown in Figure 11.Figure 11
Changes in fuel pressure in high-pressure fuel line with two fuel injectors and one pressure reducing valve.From the simulation results, the pressure shows a periodic change, and the fluctuation range is very small. The results of numerical simulation show that the pressure reducing valve plays a significant role in stabilizing fuel pressure (the mean squared error for high-pressure fuel line fitted with a pressure reducing valve is only 12.3% of that for high-pressure fuel line without pressure reducing valve).The system can automatically restore the normal pressure level even if fuel pressure rises or drops abruptly due to certain reasons. The process of pressure recovery from 0.5 MPa to 100 MPa takes 6917.7 ms approximately and the process of pressure recovery from 160 MPa to 100 MPa only needs 42.41 ms (Figure12).Figure 12
Schematic of fuel pressure recovery from low or high level to normal level.
(a)(b)


## 6. Conclusion

In this study, a differential equation-based model for simulating the changes in fuel density with time is built based on the analysis of changes in fuel density and pressure in the high-pressure fuel line. The calculation results show that when no pressure reducing valve is provided, the effect of double fuel injectors is equivalent to that of a single fuel injector. The only difference is that the fuel inflow and the amount of fuel injected by double fuel injectors are two times those by a single fuel injector, and there is no significant difference in pressure changes in the high-pressure fuel line. If a pressure reducing valve is provided, the operating cycle of the high-pressure fuel pump may be set to the same period as that of fuel injector(s), and the system is still able to maintain fuel pressure at a normal level. By adjusting the fuel injection time of fuel injector(s) and the threshold of pressure reducing valve, fuel pressure in the high-pressure fuel line can be controlled within a small range of fluctuations, the amount of fuel injected in each injection cycle can be maintained at almost the same level, and the operating efficiency and stability of fuel-powered engines can be improved significantly.



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*Source: 8821050-2020-11-05.xml*