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

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.)e results of numerical simulation show that the pressure reducing valve can effectively control the pressure in high-pressure fuel line.


Introduction
Fuel-powered engines play a very important role in production and daily life. e 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. Figure 1 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 highpressure fuel line. A high-pressure fuel pump mainly consists of plunger housing, plunger, and cam. e 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 highpressure 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. is 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].
A high-pressure fuel line is usually fitted with one or two fuel injectors ejecting fuel oil at a fixed frequency. e 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.
erefore, 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.
is 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, singleobjective 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-todiameter 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. erefore, 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. e 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 highpressure 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 (Tables 1 and 2).

Description of Symbols.
Description of symbols is provided in Table 1.

Units.
e units of the physical quantity are provided in Table 2.
(7) Figure 3 shows the relationship between fuel pressure and density. e fuel mentioned in this paper refers to the fuel that complies with such relationship. e effect of temperature on the fuel is neglected.

Relationship of Fuel Flow Rate to Pressure and Density.
e fuel flows from the high-pressure side to the lowpressure side through the small orifice. e amount of fuel flowing through the small orifice per unit time is 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

Analysis of Fuel Pressure in Plunger Housing.
Assuming that the cam shape is defined by polar angle and polar radius as c(τ), c min � c(π/2) and c max � c(3π/2). Area of fuel injector opening d 4 Diameter of pressure reducing valve outlet Ω reshold of pressure reducing valve 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, � V 0 , and the plunger housing will be filled with low-pressure fuel (with a density of ( � ρ 0 ); 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 c max − c(α + π/2), and the volume of plunger housing will be where d 0 is the diameter of plunger housing. It is obvious that and if the fuel does not enter the high-pressure fuel line, fuel density in the plunger housing will be and fuel pressure therein will be If the fuel can enter the high-pressure fuel line and the pressure in high-pressure fuel line remains stable at P 1 , the amount of fuel into high-pressure fuel line per circle of cam rotation will be 2.6. Area of Fuel Injector Opening. Assuming that the fuel injection nozzle has a structure shown in Figure 4, the diameter of needle valve is d 2 , the seal seat is a cone with a halfangle 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. Assuming that the relationship between the lift of needle valve and time within one fuel injection cycle is h � h(t), then the area of fuel injector opening will be 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).

Working Principle of Pressure Reducing Valve.
e pressure reducing check valve is connected to the highpressure fuel line via a round pipe with a diameter of d 4 . 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

Analysis of Fuel Density and Pressure in High-Pressure Fuel Line
Assuming that P 0 (t), P 1 (t) are, respectively, fuel pressure in the plunger housing and that in the high-pressure fuel line at the time of t, ρ 0 (t), ρ 1 (t) are, respectively, fuel density in the plunger housing and that in the high-pressure fuel line at the time of t, and V 1 is the volume of high-pressure fuel line, then V 1 � ld 2 /4π. e amount of fuel from the high-pressure fuel pump into the high-pressure fuel line per unit time is m 1 (t), the amounts of fuel injected by two fuel injectors are m 2 (t − μ 1 ) and m 2 (t − μ 2 ) , respectively, and the amount of fuel from the pressure reducing valve is m 4 (t). According to the law of conservation of mass, in a small time range of[t, t + Δt], Letting Δt ⟶ 0, by rearranging the formula above, the following equation can be obtained: where h Figure 4: Working principle of fuel injector. 4 Complexity ese 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.

Fuzzy Logic Systems-Based Optimization and Control
Considering the changes in fuel pressure P 1 (t) in the highpressure fuel line in a large time range of [0, T 1 ], mean squared error is defined as follows to measure the degree of pressure deviation P: e objective of optimization is to find the solutions to cam rotating speed ω, fuel injection times 155 μ 1 , μ 2 , and pressure reducing valve threshold Ω and to minimize E. erefore, the optimization 156 model can be expressed as where ω > 0, μ 1 ∈ [0, 2π/ω], μ 2 ∈ [μ 1 , 2π/ω], Ω > P 1 . e 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.

High-Pressure Fuel Line without Pressure Reducing Valve.
If the high-pressure fuel line is not fitted with a pressure reducing valve, m 4 (t) ≡ 0 and problem (24) can be degenerated to To minimize E(ω, μ 1 , μ 2 ), 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.
e 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 is ω/2πT in one cycle, and one rotation enters the high pressure. e amount of fuel in the fuel pipe is M 1 . e amount of fuel injected by an injector in time T is M 2 (T). Since there are two injectors in this question, then we have ω 2π en, the cam rotating speed should be where M 2 (T) is the amount of fuel ejected by a fuel injector in one operating cycle and M 2 (T) � T 0 m 2 (t)dt. Solving procedure is as follows: (1) Estimate an initial rotating speed of ω (0) , and let i � 0, , perform two-dimensional search for the solution to problem (25), and find the solutions to , set search range ω ∈ [ω (i) − δ, ω (i) + δs], perform one-dimensional search for the solution to ω, and find the solution to ω (i+1) . (4) If δ < ε, stop searching. Otherwise, reduce the search ranges for μ 1 , μ 2 , let δ � 0.5δ, i � i+1, and return to step (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. erefore, problem (24) can be degenerated to e solving method for high-pressure fuel line with pressure reducing valve is similar to that for high-pressure fuel line without pressure reducing valve. e only difference is that, in the solving method for high-pressure fuel line with pressure reducing valve, the optimization of cam rotating speed ω is converted to that of pressure reducing valve threshold Ω and the initial Ω may be set to a value slightly higher than P 1 .

Numerical Simulation
Assuming that the length of high-pressure fuel line is L � 500 and the inner diameter thereof is D � 10, then the volume thereof is V 1 � 12500π. e initial fuel pressure in the high-Complexity 5 pressure fuel line is P 1 (0) � 100 and fuel density therein is ρ 1 (0) � 0.85. e diameter of the small orifice at fuel inlet is d 1 � 1.4. e shape of the cam is shown in Figure 5. e polar coordinate equation of its cam contour line is where c min � 2.414, c max � 7.239. e inner diameter of plunger housing is d 0 � 5, and the volume of plunger housing when the plunger moves to the bottom dead center is V 0 � 20. e volume of plunger housing when the plunger moves to the top dead center is e pressure of low-pressure fuel is ρ 0 ∨ � 0.5 and the density thereof is ρ 0 � 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 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 is ω � 0.01 are shown in Figures 6 and 7, respectively. Figure 6 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 is d 2 � 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 According to formulas (15) and (32), the relationship between the area of fuel injector opening and time is as follows: From formula (20), it can be known that the amount of fuel injected by a fuel injector in one injection cycle is To solve equation (18), time is evenly discretized into t 0 , t 1 , t 2 , . . . , t n , . . . , and the interval between two adjacent time points is set to Δt. If the volume of plunger housing V 0 (t i−1 ), fuel density in the plunger housing ρ 0 (t i−1 ), fuel pressure in the plunger housing P 0 (t i−1 ), fuel pressure in the high-pressure fuel line P 1 (t i−1 ), fuel density in the highpressure fuel line ρ 1 (t i−1 ), fuel injector opening area A 2 (t i−1 ), injection times (at which the two fuel injectors start to operate) μ 1 , μ 2 , and pressure reducing valve threshold Ω at the time of t i−1 are known, fuel density in the high-pressure fuel line at the time of t i will be  where

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, then m 2 (t − μ 2 ) ≡ 0, m 4 (t) ≡ 0, T 1 � 1000, and the solutions found are as follows: In this case, the average fuel pressure in the highpressure fuel line is P �99.946, the mean squared error is E � 1.34898, the minimum fuel pressure is P min � 97.561, and the maximum fuel pressure is P max �102.552. e changes in fuel pressure are shown in Figure 8.
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.

High-Pressure Fuel Line with Two Fuel Injectors.
Assuming that the high-pressure fuel line is fitted with two fuel injectors, then m 4 (t) ≡ 0, and the solutions found are as follows: In this case, the average fuel pressure in the highpressure fuel line is P �99.999, the mean squared error is E � 1.231059, the minimum fuel pressure is P min � 97.574, and the maximum fuel pressure is P max �102.472. e changes in fuel pressure are shown in Figure 9.
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.

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. e fuzzy logic systems use IF-THEN rules to construct a mapping from an input vector x(t) � [x 1 (t), x 2 (t), . . . , x n (t)] T ∈ R n to an output y(t) ∈ R.
e r th fuzzy rule can be written as where A r and B r are fuzzy sets with membership functions μ A r i (x i (t)) and μ B r i (y(t)), respectively. Suppose that x(t) 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] where n r is the number of total fuzzy rules, y i (t) is the point is the membership function of the fuzzy variable x j (t) characterized by Gaussian, θ(t) � [y 1 (t), y 2 (t), . . . , y n r (t)] T is an adjustable parameter vector, and Ψ � [Ψ 1 , Ψ 2 , . . . , Ψ n r ] T is a fuzzy basis vector, where Ψ i is defined as e structure of a fuzzy system is shown in Figure 10.where η, h, m are the numbers of neurons in the input, middle, and output layers, w k � [w k1 , . . . , w kh ] T , and   8 Complexity to the universal approximation theorem, φ(x) � e x − e − x / e x + e −x can be selected as function φ(·). erefore, this system can be expressed as where ϑ � [w T 1 , . . . , w T m ] T and ψ(·) � ϖ 1 (·) 0 · · · 0 0 ϖ 2 (·) · · · 0 ⋮ ⋮ ⋱ ⋮ 0 0 · · · ϖ m (·) e optimization is achieved using the fuzzy logic systems above and the solutions found are as follows:  Complexity e average fuel pressure is P �100.092, the minimum fuel pressure is P min �99.078, the maximum fuel pressure is P max �101.145, and the mean squared error is E � 0.151434. e changes in fuel pressure are shown in Figure 11. From the simulation results, the pressure shows a periodic change, and the fluctuation range is very small. e 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).
e system can automatically restore the normal pressure level even if fuel pressure rises or drops abruptly due to certain reasons. e 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 (Figure 12).

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. e 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. e 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.

Data Availability
e data used to support the findings of this study are included within the article.