The free-piston electromechanical hybrid power system (FEHS) affords the advantages of a simple construction and high thermal efficiency due to the removal of the crankshaft. However, the unrestricted trajectory of the piston linkage assembly (PLA) also gives rise to challenges for stable operation during the process of startup or operation. In order to realize the stable operation of free-piston electromechanical hybrid power system, this paper proposed a load control strategy. First, a dynamic model is established through thermodynamic and electromagnetic theory, and its effectiveness is verified by experiment and simulation. On this basis, a coupling load control model based on linear active disturbance rejection control (ADRC) is developed. The reliability of the proposed load control strategy is validated under different interference fluctuations. The simulation results demonstrate that no matter whether the interference occurs during the startup process or the operation process, the proposed control strategy exerts effective limiting function over the piston linkage assembly and maintain its stable operation. Moreover, compared with the proportion integral differential (PID) control strategy, the proposed strategy exhibits faster response times and a smoother startup process. The compression ratio fluctuation range was reduced from 0.1 to 0.001, and the control accuracy has been greatly improved.
The free-piston electromechanical hybrid power system (FEHS) is a novel energy conversion device that directly couples the primary power machine (internal combustion engine) with the secondary power machine (generator) [1–3]. Due to the small friction loss and energy transfer loss, the energy efficiency of FEHS can reach about 40% . Moreover, the absence of the crankshaft enables the PLA to move freely between the top dead center (TDC) and bottom dead center, unrestricted by mechanical limitations. The “free” nature of FEHS renders it advantageous for variable compression ratio and robust fuel adaptability [5–8]. However, it also gives rise to a challenge in maintaining the constant dead center position of the PLA, thereby impeding the operational stability of FEHS [9–11]. To address this challenge, researchers have developed various control strategies for the stable operation of FEHS.
To achieve reliable operation and eliminate potential interference factors, various control strategies have been proposed to address all aspects of FEHS operation. Mikalsen et al. [12–14] proposed a piston motion controller for FEHS, which uses the predicted TDC of the next cycle as the proportion integral differential (PID) controller input, thereby reducing the time delay of signal transmission and achieving faster response time. Furthermore, Jia et al. [15,16] proposed a cascade control strategy, which utilizes velocity error and displacement error as PID controller input respectively to ensure the stable operation of FEHS by adjusting the amount of fuel injection.
Moreover, a concept of virtual crankshaft was proposed by the University of Minnesota for a hydraulic free-piston engine prototype, replacing the function of the crankshaft with an active motion controller. Two feed-forward control schemes were then designed by Li et al. [17–19] to improve the tracking performance of the controller. Additionally, a transient motion control strategy consisting of a detection and shifting algorithm was proposed. The application of these control schemes greatly improved the stability and robustness of the FEHS. In addition, Liu et al.  proposed an independent pressure and flowrate control for the prototype. The controller takes the load pressure, flowrate, and piston trajectory as input signals and realizes the control of pressure and flowrate by adjusting the fuel amount and digital valve.
The motion process of a free piston engine under homogeneous charge compression ignition combustion condition was studied by Alrbai et al. . The stability between cycles is realized by adjusting the generator thrust constantly according to the position of bottom dead center by PD controller. Zhang et al.  proposed a real-time motion control strategy for FEHS based on the position domain from a dynamic perspective. The motion trajectory of the piston tracks the reference trajectory accurately with errors of less than 0.2 mm through the coordinated regulation of the electromagnetic force of the generator and the gas force of the engine. Focusing on the process control of free-piston engines, Xu et al.  presented a hierarchical hybrid control system for a four-stroke FEHS based on the principle of energy balance. The current of the generator is adjusted by a PID controller to change the load force and achieve the objective of stable piston motion.
In conclusion, the current control strategy for FEHS is mostly to use PID controller to achieve control goals. Although PID controller have the advantages of simple implementation and strong versatility, it falls short in meeting the high standards of control accuracy required by complex, high-dimensional, multivariate, nonlinear, and strongly-coupled FEHS systems. The use of PID controller in FEHS has the disadvantages of insufficient accuracy, slow response time, and serious overshoot. The proposed linear active disturbance rejection load control strategy aims to address the limitations of PID controllers in FEHS. In this paper, a coupled load control model of FEHS dynamics and LADRC controller is established. Taking the combustion fluctuation as the embodiment of interference, it is verified that the proposed control strategy can realize the stable operation of the PLA under different interference degrees by adjusting the electromagnetic force. The simulation results show that the LADRC controller is superior to the PID controller in terms of control accuracy, response time, and overshoot.
2 Free-Piston Electromechanical Hybrid Power System Configurations and Features
The FEHS studied in this paper is directly coupled by a linear electric machine and free piston engines. The fundamental structure is depicted in Fig. 1. The operational process of the FEHS markedly differs from that of conventional power systems due to the absence of a crankshaft mechanism. During the startup phase, the generator serves as the power output device of the system, propelling the PLA to execute linear motion and facilitating the combustion of the fuel-air mixture within the cylinder. Following ignition within the cylinder, the engines supply energy to propel the PLA. At this point, the generator acts as energy-consuming component of the system to consume the chemical energy generated by combustion. When the PLA reaches the TDC of the opposing side, the mixture meets the combustion conditions and starts to burn. The gas force engendered by combustion propels the PLA in a reverse motion, realizing alternating combustion between cylinders on both sides and the continuous operation of the system.
3 Mathematical Model
3.1 Dynamic Model
3.1.1 Dynamic Equation.
where t denotes time and m, x are the mass and displacement of PLA, A is the area of PLA top surface, pl and pr represent the gas pressure of the left and right cylinders, respectively, psl and psr denote the scavenging back pressure of the left and right pistons, Ff is the sum of the friction on the PLA, and Fe represents the electromagnetic force produced by the generator.
3.1.2 Thermodynamic Model.
where Q represents the heat release from combustion, p, V are pressure and volume of gas in cylinder, k denotes adiabatic index which is taken as 1.4 in this paper.
where Qin is the energy released by fuel.
3.1.3 Generator and Friction Model.
where cf denotes friction coefficient and v is PLA velocity.
3.1.4 Model Validation.
The matlab/simulink (Chongqing University, Chongqing, China) is used to simulate the established model, the simulation results and the experimental results are shown in Fig. 3. It is clear from the figure that the simulation results can be well matched with the experimental results, the PLA trajectory motion ranges of both are −0.047 m, 0.047 m, and the peak velocity are −7.02 m/s, 7.02 m/s. However, the peak velocity of the experimental results is slightly larger than the simulation results. The reason for this phenomenon is that there is a pressure shock effect in the experiment, which causes the velocity curve of the PLA to fluctuate slightly. The experimental results and simulation results verify the correctness of the model established in this paper.
3.2 Control Model
3.2.1 Linear Active Disturbance Rejection Control Description.
where wc is the controller bandwidth, ξ is the damping ratio of the system which is taken as ξ = 1 in this paper. At this point, the problem of parameter tuning of LADRC turns to the problem of choosing appropriate wc and w0. The optimized final controller parameters are w0 = 9, wc = 7, and b0 = 18.
3.2.2 Control Design for Free-Piston Electromechanical Hybrid Power System.
In order to ensure the superiority brought by the unique structure of the FEHS, a reliable control strategy is indispensable. The LADRC is suitable for applying in FEHS, on the one hand, it does not rely on specific model of controlled plant; on the other hand, it possesses the ability of estimating the interference and enters the compensation factor into the system in a feedforward way. Figure 4 shows the control flowchart of FEHS with LADRC. In this paper, the combustion fluctuation caused by abnormal fuel injection or abnormal fuel combustion is taken as the embodiment of the interference, and the electromagnetic force is regulated by LADRC to realize that the PLA's motion track conforms to the desired trajectory, so as to achieve the purpose of continuous and stable operation of FEHS.
4 Results and Discussion
4.1 Interference Occurs During Startup.
The status information of FEHS when interference occurs under different situation are shown in Fig. 5. It can be seen from Fig. 5(a) that FEHS does not have the adjustment ability when the controller is not used. The compression ratio curves change with the input energy, and the system directly stops operation when the input energy falls below a certain value. Figure 5(c) shows that under the control of LADRC, the compression ratio quickly reached stability after a small fluctuation in the startup process. In contrast, the system under the control of PID changes dramatically in the compression ratio due to overshoot as shown in Fig. 5(b), which greatly exceeds the stable compression ratio and takes a long time to adjust to achieve stability. This indicates that the system startup process vibrates large and the startup performance is poor when using the PID controller. Form Fig. 5(d), it can see that the dead center of LADRC after system stabilization are closer to the target value (0.098, 0), and the error of dead center is smaller; while the distribution of the dead center of PID is more scattered, and the jumping is greater.
The phase diagram of LADRC controller and PID controller is shown in Fig. 6. It can be seen that the starting points both are (0.06, −10). However, the phase curve of LADRC controller changes smoothly and evenly diffuses outwards, while PID controller changes dramatically, with phase curves disordered shrink inward. LADRC controller has high control accuracy, and plays a good control role in the process of system operation, including the conversion process of generator-motor mode. As a result, the phase curves transition smoothly, and the dead center of PLA is virtually unchanged. On the contrary, due to the lack of control accuracy, PID controller has overshoot phenomenon in the process of starting, there is obvious fluctuation in the curve span and PLA dead center, a flat start process of the system cannot be achieved.
4.2 Interference Occurs During Operation.
Figure 7 shows the motion information of FEHS when interference occurs during operation. Figures 7(a) and 7(b) display the impact of interference on the displacement curves under operation conditions. Under the control of the LADRC controller, the position of dead center is nearly invariant to the presence of interference, whereas with PID controller, despite its ability to maintain relative stability of dead center position in the face of interference, there still exists a notable fluctuation phenomenon in comparison to the stability offered by the LADRC controller. Figures 7(c) and 7(d) show the appearance of interference at 0.2 s causes fluctuations in the compression ratio curve, with a fluctuation range of 0.001 under LADRC and 0.1 under PID. It is illustrated that LADRC has more precise control accuracy, and FEHS with LADRC controller possesses better anti-interference ability when interference occurs.
Owing to the elimination of the crankshaft, the mechanical structure no longer imposes constraints on the PLA, thus presenting a challenge to the steady operation of the FEHS. To address the issue of easy instability of FEHS operation, a load control strategy is proposed. Initially, a dynamic model of the FEHS is established, and its feasibility is confirmed through simulation and experimentation. Building upon this foundation, a coupling control model based on LADRC is developed. Taking the abnormal input energy caused by combustion fluctuation as the interference factor, the reliability of the proposed load control strategy under different disturbances is verified. Simulation results demonstrate that the suggested load control strategy ensures the stable operation of the PLA, irrespective of the occurrence of interferences during both the startup or operation processes. Furthermore, in comparison to PID controller, the utilization of LADRC controller yields a swifter control response and a smoother startup process, and the control accuracy is greatly improved.
National Natural Science Foundation of China (Grant No. 51805056; Funder ID: 10.13039/501100001809).
Natural Science Foundation of Chongqing, China (Grant No. cstc2019jcyj-msxmX0407; Funder ID: 10.13039/501100005230).
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.