Feasibility of Multiple Piston Motion Control Approaches in a Free Piston Engine Generator

Abstract

The control and design optimization of a Free Piston Engine Generator (FPEG) has been found to be difficult as each independent variable changes the piston dynamics with respect to time. These dynamics, in turn, alter the generator and engine response to other governing variables. As a result, the FPEG system requires an energy balance control algorithm such that the cumulative energy delivered by the engine is equal to the cumulative energy taken by the generator for stable operation. The main objective of this control algorithm is to match the power generated by the engine to the power demanded by the generator. In a conventional crankshaft engine, this energy balance control is similar to the use of a governor and a flywheel to control the rotational speed. In general, if the generator consumes more energy in a cycle than the engine provides, the system moves towards a stall. If the generator consumes less energy, then the effective stroke, compression ratio and maximum translator velocity must rise steadily from cycle-to-cycle until the heat transfer losses stop the increase. Moreover, when stiff springs are added to the FPEG system, the dynamics becomes more sinusoidal and more consistent with increasing spring stiffness. To understand the behavior of proposed control and cycle-to-cycle variations, a comprehensive FPEG numerical model with a 1kW target electric power was developed in MATLAB®/Simulink. An FPEG system corresponding to that numerical model has been operated in the laboratory. This MATLAB®/ Simulink numerical model has been used to examine the sensitivity of FPEG dynamics and performance parameters to the changes in design and operating inputs. A difficulty during the modeling is associated with the cycle-to-cycle energy balance, and this difficulty is also reflected in the real-world FPEG control. Therefore, the authors have devised a control strategy similar to the real world intended control methodology. In this numerical model, two different feedback control methodologies were implemented and investigated.  These control methodologies were applied to regulate the generator load with selected control or input variables, namely peak pressure, mid-stroke piston velocity, trapped compression ratio and dead center set points.   The controllers with optimized coefficients demonstrated the feasibility of energy balance management during the transient operation. Based on the simulation results, the controllers with compression ratio, peak pressure and dead center clearance set points as control variables demonstrated stable FPEG operation whereas the mid-stroke velocity failed to achieve the steady-state operation due to deviation in the piston dynamics. The simulation results from this study will be used as the pathway for improving and optimizing the experimental FPEG design.

Key words: Free piston engine, transient energy balance, piston motion, linear engine generator, two-stroke engine, control.

Introduction

A Free Piston Engine Generator (FPEG) is a novel energy conversion device that directly converts the reciprocating piston motion into electricity by using a centrally mounted electric generator [1, 2]. Unlike conventional engines with crankshaft and connecting rod mechanisms, the energy source (i.e. engine) in an FPEG device does not employ a crankshaft and flywheel combination. The piston is attached to the electric generator containing permanent magnets and windings by using a translator rod. With the elimination of crankshaft mechanism, the piston dynamics in this novel engine generator are defined by solving the system of forces acting on the piston by using Newton’s second law of motion. The various forces acting on the piston include cylinder gas forces from combustion of fuel, frictional forces from piston ring and other sources, electric generator forces from disturbance in the winding magnetic field and rebound device forces from compression of springs. The single cylinder FPEG design cannot operate without a rebound device and requires a reversal force for the compression of gas contents during the compression stroke. The rebound devices can be either gas springs or mechanical springs [3, 4]. The use of stiff mechanical springs in the final design enables stable operation and increases the operating frequency of the device. The dual cylinder FPEG device can operate without any rebound devices in the final design. However, the rebound device addition results in consistent and sinusoidal piston motion along with the improvement in operating frequency and electric power [5]. The deviations in the various forces acting on the piston alters the piston dynamics with respect to time, which, in turn, changes the engine and generator reactions to other operating variables. These dynamics deviations bring about the variable compression ratio that is beneficial for Homogenous Charge Compression Ignition (HCCI) operation of these engines, but not for other combustion modes like Spark Ignition (SI) and Compression Ignition (CI) modes [6]. Therefore, it is of extreme importance to reduce the cycle-to-cycle variations for other conventional combustion modes by exercising a proper piston dynamics control strategy [7].

In conventional internal combustion engines (ICE) with a rotating crankshaft used in a governed application, the rotating speed is controlled either by an electronic system that monitors speed with a trigger or position sensor or by a mechanical system employing centrifugal force. In either case, the responsibility of the controller is to match the power produced by the engine to the power demanded by the load. Without such a controller, the crankshaft engine system may still achieve a steady operating rotating speed, but the rotating speed will be decided by the non-linear nature of the friction or of load in the engine. This rotating speed usually will not correspond to a target operating speed. With extreme mismatch of energy demand and supply, the engine will either over speed or stall, because there will be cyclic net addition or subtraction of energy to the flywheel. The quest is for cycle to cycle constant energy, represented by a constant flywheel rotational speed.

FPEG’s are different in that and they naturally run in a limited operating frequency band [8]. If the stiff mechanical springs are added to the FPEG system, then the change in the operating frequency turns out to be even tighter when compared to the design without springs [9]. With this approach, the engine generator cannot store substantially more or less energy without changing its operating frequency. If the electric generator load consumption goes beyond the combustion output from the engine, then the stroke length, peak cylinder pressure, velocity at mid-stroke will reduce on following cycles, and the system will “ring down” to a stall [10]. If the engine combustion delivers too much energy, the excess combustion energy is accumulated by means of increased gas pressure energy, with a longer stroke and high trapped compression ratio. At mid-stroke, this same combustion energy from the engine is vested in high translator velocity, associated with an increase in stroke length. This results in the mechanical damage to the system i.e. piston crown hitting the cylinder head [11]. In the end, if more engine power is delivered than consumed by the generator, losses from in-cylinder heat transfer, increased ring friction and mass loss past the rings will consume the excess power from the engine, but this operating condition is unlikely to be optimal or efficient.

Hence, it is essential to manage the energy in the FPEG system and match the engine output to the generator demand by using an energy balance control algorithm. This transient energy balance control algorithm is similar to the use of a governor and flywheel to govern the rotational speed in a conventional crankshaft engine. In case of a conventional engine, the rotational speed changes with the constant compression ratio, whereas in an FPEG system the urge is to govern the piston dynamics at a considerably constant operating frequency. Therefore, the main aim of this research investigation is to propose and implement a basic feedback controller to monitor the energy management of the single cylinder FPEG from cycle-to-cycle. For numerical modeling purposes, it is likely to regulate the FPEG system’s energy balance by handling either engine energy output or generator demand, or both. From a MATLAB®/ Simulink modeling perspective either is desirable.  In this research investigation, the energy balance controller was implemented in the model to modify the electric generator demand for variations in different operating variables.  Currently, researchers at West Virginia University (WVU) are working on the development of a two-stroke, natural gas fueled combined heat and power (CHP) system functioning at a near-constant load of 1 kW, perhaps as an energy storage charging system with stiff mechanical springs in the final design. More detailed information regarding the FPEG prototype and structure at WVU is provided in the upcoming section. In addition to that, keeping in mind a stoichiometric natural gas experimental engine like the one considered in this investigation, the throttling of the engine creates substantial inefficiency and impacts the compression spring forces in the engine as well as the energy output.  Considering these reasons, the aim would be to modulate the electric generator load along with stable FPEG operation at a wide open throttle condition.

FPEG research using modeling is more often encountered in literature than experimental studies. Since many research investigations are in the initial phases of development of these engines, currently many studies are focused on simulation and combustion investigations rather than control architecture investigations. However, this energy balance controller design and development is beneficial in the advanced phases of research where the emissions and efficiency requirements need to be optimized for an experimental FPEG prototype. Of all the control strategies, the position motion control is most important for the FPEG system. The piston dead centers must be regulated within tight tolerances in order to ensure steady-state FPEG operation. The control of the top dead center position (TDC) for desired compression ratio avoids excessive combustion chamber pressures and ensures efficient combustion and compression process [3]. Whereas, the control of the bottom dead center (BDC) position enables proper scavenging process of fresh charge into the cylinder. Further, if the position of the piston goes beyond the prescribed TDC position, there are chances of the piston crown hitting the cylinder head. Finally, the engine optimization controls, namely ignition timing, injection timing and port timings in two stroke engines and valve timings in a four stroke free piston engine also depend upon proper position dynamics.

 Yang et al. investigated a piston dynamics controller for an FPEG system based on the Linear Quadratic Regulator (LQR) approach [12]. The authors traced the piston dead center clearance set points and regulated the amount of fuel load injected in the system. The amount of fuel load was varied based on the error calculated from the dead center set points and dynamic deviations in the FPEG system. Based on the simulation results, the proposed controller technique ensured the stable and robust FPEG operation. Similar to the piston dynamics controller proposed by Yang et al., Nemecek et al. also investigated the piston dynamics of an opposed piston configuration operated on a two-stroke mode [13]. Instead of tracking the dead center clearance set points, they provided a reference piston motion trajectory similar to a sinusoidal signal to regulate the fuel delivered to the engine. Their controller numerical modeling results demonstrated that an incorrect reference piston trajectory had a negative influence on the generator efficiency and electrical power output.

Kosaka et al. examined the influence of piston motion on the combustion and generating characteristics of a single cylinder FPEG system with gas springs [14]. They designed a controller to ensure a stable compression ratio. Based on the controller, the generator load coefficient was modified according to the deviations in the compression ratio computed by the feedback control method. The same control architecture was employed in their numerical model by Robinson et al. and Bade et al. for the dual cylinder and single cylinder FPEG designs with mechanical springs [1, 15]. Jia et al. explored a global control structure for an FPEG system with multi-layers including piston dynamics control, ignition and injection timing controls, engine starting control, actuator and supervisory controllers [16]. Mainly for the piston dynamics control, a cascade control strategy was developed for an FPEG system operation below 35 Hz. Based on the simulation results, they reported that the proposed controller was feasible to achieve stable and robust FPEG performance within 0.5s for both instantaneous 15% decrease and instantaneous 15% increase in the electric generator load.

Similar to these controller methodologies, the authors examined the stable operation and cycle-to-cycle variations of the FPEG system with and without the energy balance or piston motion controllers [17]. This investigation allowed the comparison of the FPEG behavior with and without an external controller and highlighted the importance and need for an external controller for stable operation. Based on the simulation results, for a generator load less than the engine combustion energy, the peak cylinder pressures, mid-stroke velocities and compression ratio were increased from cycle-to-cycle. Similarly, when the generator consumed higher energy than the energy generated by the engine, the peak cylinder pressures, mid-stroke velocities and compression ratio decreased from cycle to cycle and resulted in a stall. The authors also investigated the feasibility of a robust Proportional-Integral-Derivative (PID) controller and a Pseudo-Derivate Feedback (PDF) controller to regulate the compression ratio of a single cylinder FPEG system. Considering the numerical results, the PDF controller performance was found to better than the PID controller for the same controller coefficients owing to the low peak pressure attained in the cylinder.  While, the above mentioned conclusion, is true only when the same controller coefficients are considered for both the controllers. To further investigate the feasibility and select the better controller of these two, an optimization of the controller coefficients has been carried out in this investigation. This investigation is the extension of our previous publication [17]. Finally, the cycle-to-cycle variations were studied with increasing spring stiffness. The greater dominance of the stiff springs was found to reduce the cycle-to-cycle variations caused by the engine combustion stochastics.

Table 1. FPEG engine specifications and numerical model input parameters

In summary, the FPEG system necessitates a transient energy balance control algorithm with high-speed dynamic response to achieve stable operation and perhaps optimized system efficiency.  The control methodologies were applied to several control variables namely peak pressure, mid-stroke piston velocity, trapped compression ratio and dead center set points.   The influence of simulation time steps on the PID and PDF controller response characteristics was also analyzed for different target control variables. The coefficients of the PID and PDF controllers were optimized manually with respect to minimum settling time and minimal initial error peaks. The controllers with manually optimized coefficients demonstrated the feasibility of energy balance management during the transient operation. Based on the simulation results, the controllers with compression ratio, peak pressure and dead center clearance set points as control variables demonstrated stable FPEG operation whereas the mid-stroke velocity failed to achieve the steady state operation due to deviation in the piston dynamics. The simulation results presented in this study will be used as the pathway for improving and optimizing the experimental FPEG design during the transition process from starting or cranking stage to generating stage.

S. No	Specifications	Value
1	Cylinder bore	38.00 mm
2	Effective stroke	30.00 mm
3	Moving mass	2.5 kg
4	Spring number	16
5	Individual spring stiffness	30 kN/m
6	Target Compression ratio (CR)	8.25
7	Operating frequency	77 Hz ± 1 Hz

FPEG structure

Currently, researchers at WVU are investigating the feasibility of a 1 kW single cylinder, natural gas fueled, two-stroke FPEG system as a CHP system at residential level. To achieve a 1 kW electrical power output, high frequency operation of 75 Hz is desirable in order to achieve power density and efficiency. This 75 Hz operation is possible only with the specially designed mechanical springs. The single cylinder FPEG design is comprised of one internal combustion engine cylinder on the right, the electric generator in the middle, specially designed mechanical springs as rebound devices mounted inside the generator casing and a translator rod that is connected to the piston and runs through the electric generator. The piston assembly, translator rod, mechanical springs and an electric generator with permanent magnet casing are the only noteworthy reciprocating components of the FPEG system. SI combustion mode is selected as it is easier to ignite the natural gas mixture than HCCI or CI combustion mode. Exhaust and transfer ports are used for gas exchange process in this engine. These ports are at fixed positions and the opening and closing of these ports are defined by the piston displacement i.e. the scavenging process is coupled directly with the transient piston dynamics. The physics-based model descriptions for the single cylinder and dual cylinder designs can be found from the previous works of Bade et al. and Robinson et al. [15, 1]. The FPEG prototype developed at WVU is demonstrated in Figure 1 and the input parameters and prototype specifications are listed in Table 1. The various forces acting on the translator rod are shown in the free body diagram in Figure 1. The piston motion equation for a single cylinder configuration is presented in Eq.1.

$m\frac{d^{2}x}{d{t}^2}=F_{\mathit{cyl}}+F_{\mathit{spr},l}–F_{\mathit{spr},r}–F_w–F_f$

(1)

In Eq. 1, the subscripts ‘cyl’, ‘spr’, ‘f’, ‘w’, ‘l’ and ‘r’ represent engine cylinder, mechanical springs, friction, alternator work, left and right respectively. The terms ‘m’, ‘x’, ‘t’, ‘F’ represent the moving mass of the translator (including the mass of piston head, appropriate spring inertia and alternator moving magnet mass), position of the translator rod, time and force correspondingly. The dotted arrows in the free body diagram in Figure 1 represent the forces which are in favor with the translator rod motion. Whereas, the solid arrows represent the forces opposing the translator rod motion.  The corresponding mathematical equations and the procedure for the calculation of these forces can be found in our previous published works [1, 11, 15].

Energy balance controller objectives

In conventional ICE’s, the crankshaft and connecting rod mechanism is accountable for the piston dynamics control by outlining the dead centers position. In addition to the crankshaft mechanism, the flywheel is also responsible for influencing the piston dynamics. In case of a light flywheel with low inertia, the variations in the engine combustion forces influence the piston dynamics in successive cycles. In case of a heavy flywheel with high inertia, the piston dynamics are not majorly affected by the combustion differences attained in one cycle.

In contrast, without the crankshaft and connecting rod mechanism in the final design, the instantaneous summation of the forces defines the piston dynamics in the FPEG case. The cumulative nature as well as the deviations in the forces acting on the translator rod between consecutive cycles will give rise to variable compression ratio and cycle-to-cycle variations. The peak combustion pressure, maximum piston velocity, compression ratio and dead center clearance set points are control variables in an FPEG system whereas in a conventional crankshaft engine, these variables are almost fixed. If these control variables increase on consecutive cycles, the engine produces too much power and if these control variables decrease on consecutive cycles, the FPEG system will ring down to a stall. In addition to this piston dynamics control methodology, various control methodologies like ignition and injection timing controls and starting controls depend upon the piston dynamics control or energy balance control. The current exploration concentrates only on the energy management control for robust and stable FPEG operation. Therefore, the main intentions of the FPEG control system are:

1. The energy generated from the engine must be closely equivalent to the electric generator system energy demand for electricity generation every cycle.

2. The proposed control system should control the control variables like compression ratio, dead center set points, peak pressures and maximum piston velocities within tight tolerances at a substantially constant operating frequency for SI and CI combustion modes.  

3. The proposed energy balance control system should rapidly regulate the piston dynamics during the transition phase between starting and generating mode of the FPEG system.

In conventional crankshaft engines, the flywheel, crankshaft and connecting rod mechanisms take care of these objectives. In addition, the proposed control systems are devised only at the modeling level to fulfill the above-mentioned intentions. The hardware applications of the strategies mentioned in this paper are beyond the scope of the current research point. However, the simulation results from this study will be used as the pathway for improving and optimizing the experimental FPEG design.

Energy balance controller configurations

Based on the energy balance controller objectives mentioned above, the control of the target variables like compression ratio, dead center set points, peak pressures and maximum piston velocities is most important when compared to other objectives. It is well known and based on our previous investigation, a minor variation in the TDC clearance set point by +/- 1mm from the target set point results in large deviations in combustion pressures from 50 bar to 100 bar [17]. Such large variations in the cylinder pressures impact the piston dynamics and generator load characteristics in the successive cycles. Moreover, this condition results in unstable operation and cycle-to-cycle variations in the FPEG system. The numbers mentioned above are calculated from an FPEG engine with an effective stroke length of 30 mm and a target compression ratio of 8.25. As the effective stroke length is increased further say from 30 mm to 500 mm, then the TDC clearance set point deviation of +/- 1mm doesn’t attribute to major deviations in the combustion pressures.

To prevent FPEG cycle-to-cycle variations, two different controllers, i.e. PID and PDF controllers are used in the numerical model to enable the FPEG system to operate at a substantially constant control variable. The main motive behind the selection of these two controllers was due to their widespread applications in industry and their excellent performance characteristics [17]. There are many options available for the selection of the control variables. One can control any variable in the closed loop by using the measurement of the same variable or one can control a variable in the closed loop by using other control variable (i.e. generator load or fuel supply) that is sufficiently related to the primary control variable. For example, one can use compression ratio, dead center clearance set points, mid-stroke velocity and maximum cylinder pressure as an input to the energy balance controller. In this investigation, all the four different control variables mentioned above are considered. Also, the controller can manipulate either generator load or fuel supply based upon the error calculated by the controllers. In this investigation, the authors considered electric generator load as an output variable. The target values for these control variables in this investigation are selected based upon the experimental FPEG system at WVU. Moreover, the target values of these control variables are maintained constant throughout the investigation even though the increase in these control variables will increase FPEG efficiency and power output.

The PID controller in an FPEG system continuously analyses an error value, which is the difference between the measured process variable (MPV) and target set point (TSP) or target control variable and applies the gain term by using the summation of proportional, integral and derivative terms.  The control function of the PID feedback mechanism is expressed mathematically in Eq. 2.

$C\left(t\right)=K_p\mathit{ e}\left(t\right)+K_i{\int }_0^{t}e\left(t^{‘}\right)d{t}^{‘}+K_d \frac{\mathit{de}\left(t\right)}{\mathit{dt}}$

$e\left(t\right)=\mathit{TSP}–\mathit{MPV}$

(2)

In Eq. 2,

$C\left(t\right)$

is the correction applied to the generator load based on the error term,

$e\left(t\right)$

is the continuously calculated error term,

$K_p$

is the proportional term constant,

$K_i$

is the integral term constant,

$K_d$

is the differential term constant and 

$t$

is the time step in the numerical model. In the present investigation, the target trapped compression ratio is set to 8.25 which is calculated from the experimental engine cylinder (Zenoah G340RC engine) used in the experimental FPEG system. The dead center clearance set points are selected based on the trapped compression ratio and an effective stroke of 28.5 mm and the maximum cylinder pressures are maintained to be 35 bar. The effective stroke of 28.5 mm is selected because of the effective stroke of an experimental FPEG system. The MATLAB®/Simulink sub-model for the PID control function is shown in Figure 2. The above study can be expanded to any effective stroke length and compression ratio to understand how the sensitivity changes with respect to these variables. This will be the main objective of our future investigation.

On the other hand, the PDF controller generally has better load management capabilities when compared to the PID controller while the PID controller has a good target set point tracking ability. In the FPEG system, the rapid load changes pose the greater challenge due to high errors. The control function of the PDF controller is expressed mathematically in Eq. 3.

$C\left(t\right)={–K}_p*\mathit{MPV}{–K}_d*\mathit{MPV}+k_i{\int }_0^{t}e\left(t^{‘}\right)d{t}^{‘}$

 (3)

The MATLAB®/Simulink sub-model for the PDF control function is shown in Figure 3. The major advantage of the PDF controller is that it is not as much responsive to the rapid target set point changes when compared to the PID controller. Owing to this advantage, this controller makes it more suitable in the FPEG system with rapid load changes. This is due to lack of derivative or proportional terms in the forward path of the controller, thereby the response to the target set point variations is much lower than the PID controller. By employing these PID, PDF controllers and stiff springs in the system, the FPEG system behaves similarly to the conventional crankshaft engine system with substantially constant operating frequency and stable piston dynamics because of these controllers in the system.

The overall working process of the FPEG system is divided into three modes i.e. starting or cranking mode, switching or transient mode, generating or power mode. In starting or cranking mode, the centrally mounted alternator acts as a motor and provides the required motoring force to the springs to achieve mechanical resonance. Mechanical resonance is a process in which constant electromagnetic force is provided in the direction of piston velocity until desired ignition conditions are reached. The motor is managed to provide the electromagnetic force on successive strokes thereby steadily building the energy in the system. The in-cylinder pressure, the effective stroke length of the piston, and trapped compression ratio are gradually increased from cycle-to-cycle by means of mechanical resonance behavior of the springs and engine cylinders. Once the FPEG system meets the desirable ignition conditions, the fuel is injected into the cylinder and is ignited by using an ignition signal. Unlike the forces in the cranking and starting mode, where motoring pressures are repeatable and smooth, the combustion forces in the generating mode are varied from cycle-to-cycle due to changes in the combustion of fuel and air mixture. Moreover, there exists a transient period when the FPEG system switches from cranking mode to generating mode. This period is defined as the switching or transient period of an FPEG system operation. Especially, after the first combustion cycle, the stable operation of the FPEG system is prevented because of the energy mismatch between the alternator load consumption and combustion energy generation. The PID & PDF controllers in this investigation modifies the control variables like compression ratio, dead center set points, peak cylinder pressure and mid-stroke translator velocity to regulate the alternator load consumption so that the energy generated by the engine is in coordination with the energy consumed by the alternator. Without the energy balance control in the FPEG design, the ignition timing, fuel load and fuel injection parameters are not optimized for high efficiency operation and system will either ring down to a stall or operates unsteadily.

Controller simulation – Compression ratio

In this section, the response of the FPEG system during the transient mode is investigated by using PID and PDF controllers with trapped compression ratio as an input to these controllers. Unlike our previous investigation where the coefficients of these controllers are maintained same, in this investigation these coefficients are tuned manually to obtain the minimum system settling time and best performance from these controllers [17]. The results presented in this investigation are different from the results that we have concluded in the previous paper [17] because of manual optimization of the controller coefficients with an objective of reduced settling time and reduced initial error peak.

Figure 4 shows the controller behaviors in controlling the trapped compression ratio during the transient mode operation of an FPEG system. Based on the modeling results, with optimized controller coefficients, the PID and PDF controllers behaved in similar fashion to control the compression ratio. These two controllers prevented compression ratio to reach a high value (i.e. CR > 20) during the transient mode. Moreover, a slight high compression ratio is observed for the PDF controller when compared to the PID controller. The reason is mainly because of the difference in the action of proportional term in these controllers. In the PID controller, the proportional term takes the action on the error calculated from the deviation of compression ratio from target set point whereas for the PDF controller, the proportional term takes the action on the actual value of the compression ratio. This high value of the compression ratio during the initial stages of transient mode is responsible for high compression ratio in the PDF controller. Though one can prevent the overshooting of the compression ratio in the PDF controller by tweaking the controller coefficients, then the settling time would be more which is undesirable especially in a system operating at a high frequency of 78 Hz. Even though PDF controller had slightly high compression ratio, both the controllers took rapid action in controlling the compression ratio and reached the target value within 5 cycles after detecting the deviation in the compression ratio.

Figure 5 and Figure 6 show the deviation in the top dead center position set point and in-cylinder pressures for both the controllers. From Figure 4, during the first few cycles the compression ratio is below the target set point. This is also evident from the positive deviation in the position TDC set point. Once the FPEG system achieved the combustion favorable conditions, the deviation in the TDC position is reduced.  The ignition favorable condition in the numerical model is detected when the trapped compression ratio reaches the value of 6. The increasing cylinder pressures and reduction in the positive deviation in the TDC position set point explains the mechanical resonance starting mode of the FPEG system. The motoring force is managed to provide the electromagnetic force on successive strokes thereby steadily building the energy in the system. Therefore, the in-cylinder pressure, the effective stroke length of the piston, and trapped compression ratio are gradually increased from cycle-to-cycle by means of mechanical resonance behavior of the springs and engine cylinders. This is evident from the reduction in the positive deviation in the TDC position and increase in the in-cylinder pressure in Figure 5 and Figure 6.  However, if the combustion energy and alternator load is mismanaged, the energy in the FPEG system is invested into increasing the stroke length and cylinder pressures.  If the energy mismanagement is continued for few more cycles, the effective stroke, compression ratio and maximum translator velocity rises steadily from cycle-to-cycle until efficiency losses curb the increase. As the controller detected the increase in the compression ratio and cylinder pressure, the controller took action at 2.35s, as a result, the pressures and TDC deviations are reduced corresponding to the target compression ratio. To prevent that increase, the controllers scale the alternator load according to variation in the compression ratio.

  From Figure 7, it is evident that the alternator consumed a more alternator power when the compression ratio is high and reduced the alternator power so that the FPEG system can maintain the target compression ratio. Once the FPEG system reached the steady state at a target compression ratio of 8.25, the alternator power output stayed constant at 350 W. On an overall basis, both the PID and PDF controllers can control the FPEG system during the transient mode with satisfactory performance when compression ratio is used as an input to these controllers.

Controller simulation – Dead center set points

The tracking of the TDC and BDC clearance set points is the most common control method among the researchers to control the piston motion in an FPEG system [12, 16]. However, most of the control approaches mentioned in the literature are directed towards controlling the fuel load rather than the alternator load. In this investigation, the focus is upon controlling the alternator load rather than the fuel input to the engine in order to enable full throttle operation when providing the energy to an energy storage system.  The dead center clearance set points are inputted to the controller architectures devised above. The response of these controllers on the transient FPEG system behavior is explained by series of figures.

Figure 8 shows the deviation in the TDC and BCD tracking set point deviation for both PID and PDF controllers. As seen in Figure 8, both the controllers have the same response. The differences in the BDC and TDC position set point distribution for both the controllers is because of the differences in the action of multiple terms i.e. proportional, differential and integral terms on the set points and error. In case of a PID controller, the action of the controller depends upon the deviation in the set points. The integral, differential and proportional terms take action in the alternator load based on the error calculated. However, in a PDF controller, the integral action depends upon the error calculated whereas the differential and proportional term actions depend upon the actual process control variable. These differences caused the differences in the trends shown in Figure 8.

Figure 9 shows the variation in the trapped compression ratio for both controllers when dead center clearance set points are provided as an input to these controllers. Similar to the compression ratio in Figure 4, the compression ratio also attained the high initial peak. It is clear now that the PID and PDF controllers are unable to prevent the high initial peak of compression ratio. The main reason for that high peak is because of the delay in the detection of error and the action of the controller. However, once the controller detects the error, the action is much rapid. It is clear from the sharp decline in the compression ratio from 13 to 8 in the span of three consecutive cycles as evident from Figure 4 and Figure 9. From Figure 9, it is seen that the PDF controller has significantly smoother response than the PID controller.  This is because of the better load handling capabilities of the PDF controller taking rapid action on the deviations in the set points. While this is on explanation for this behavior, the manual selection of the controller response might also be reason for the deviation. The cycle-to-cycle variations in the in-cylinder pressures for both the controllers are presented in Figure 10. Similar to cylinder pressure in Figure 6, the in-cylinder pressures are within reasonable limits. In compression ratio control case, the pressure rose to 60 bar because of high initial peak and reduced to 30 bar because of controller action. In case of a dead center set point control, the cylinder pressure rose to 40 bar because of high initial peak and reduced to 30 bar within 5 cycles because of controllers taking action on the deviations. Similar to the compression ratio control, both the PID and PDF controllers can control the FPEG system during the transient mode with satisfactory performance when dead center clearance set points are used as an input to these controllers.

Controller simulation – Peak cylinder pressure

Though many researchers investigated the control of the FPEG systems using compression ratio and dead center clearance set points, the control of these systems by using peak in-cylinder pressure and mid-stroke velocity has not been explored until now. The upcoming sections in this research investigation builds upon the feasibility of peak cylinder pressure and mid-stroke velocity control approaches. In this section, the peak in-cylinder pressure is provided as an input to both the controllers. The target value of peak in-cylinder pressure is set at 35 bar for both the controllers. The results presented in this and the upcoming section are fundamental numerical results with manually tuned controller coefficients.

Figure 11 shows the variation in the compression ratio of an FPEG system with peak cylinder pressure as an input to the controller. The compression ratio in this case has a much initial peak of 60 when compared to 15-20 in the previous cases like compression ratio and dead center clearance set point inputs. As said previously, the basic feedback controllers have no control in avoiding the initial error peaks. So, this high initial compression ratio of 60 has nothing to do with the selection of the controller. However, after detecting the peaks, the PID controller has smoother variations in the compression ratio when compared to the PDF controller. One reason for such smoother variations in the PID controller is the selection of the controller coefficients. It is well known that the slight increase or decrease in the compression ratio and dead center deviations cause rapid variations in the cylinder pressures as evident from Figure 12 and Figure 13. From Figure 11, 12 and 13, for a compression ratio of 60 and TDC deviation of 2 mm, the in-cylinder pressure rose to 250 bar unlike the previous case where the pressures are around 60 bar. With such high cylinder pressures, the selection of controller coefficients and the number of significant digits in the controller coefficients becomes more critical. The variations are more clearly seen in the TDC position deviation in Figure 13. Other reason for the rapid variations in the PDF controller is because of the lack of proportional and differential terms in the forward path of the controllers i.e. the proportional and differential actions are calculated from actual pressure rather than error in pressures. Though the compression ratio is high initially, the controllers took the corrective action to control the in-cylinder pressures. The only difference is the gradual control action rather than the fast action in the previous cases. Once the system reached the steady state, both the controllers maintained same cylinder pressures as seen in Figure 14. With the simulation results and cylinder pressure from Figure 14, it is evident that the maximum cylinder pressure can be used as an input to the controllers in order to maintain the steady state operation of the FPEG system. However, considering the gradual control, a more detailed parametric investigation of controller coefficients will be performed in future with an objective of rapid corrective action and less settling time of the controllers in this case.

Controller simulation – Mid-stroke piston velocity

Based on the dominance of forces acting on the translator rod, the FPEG systems are classified into three types. The first type refers to the spring dominant FPEG system. In such a spring dominant FPEG system, the spring forces acting on the translator rod are high when compared to the forces generated by the combustion of fuel. As spring forces dominate the combustion forces, the springs are mainly responsible for the dynamics of the piston. The nuances in the combustion process have no impact on the dynamics of the piston. The advantage of such a spring dominant system is consistent and sinusoidal motion and also reduction in the cycle-to-cycle variations. The operating frequency and mid-stroke velocities increase with increasing the spring stiffness. However, there are also some disadvantages associated with such a spring dominant system. The main disadvantage in such a system is the slow expansion process in the beginning of expansion stroke which might result in the increased heat transfer losses and emissions than the FPEG system without springs.

The second type of classification refers to the equally dominant system. As the name implies, the equally dominant FPEG system is the system in which the peak combustion forces and peak spring forces are almost same. The behavior of this system lies in between the spring dominant system and the FPEG system without springs. In this system, both springs and combustion forces impact the dynamics of the system. The third classification refers to the combustion driven FPEG or no-spring FPEG system. In this system, the spring forces are negligible in nature when compared to the combustion forces acting on the translator rod. In this system, only combustion forces define the dynamics and operating frequency. This FPEG is most widely investigated around the world because of its unique characteristics. The advantages of this system is fast expansion stroke and reduced heat transfer losses when compared to the spring dominant system. However, the cycle-to-cycle variations are high because of the nuances in the combustion process.

The current investigation is carried out on an FPEG system with stiff mechanical springs. The dynamics are almost sinusoidal in nature and they are mainly defined by stiff springs in the system. Moreover, the maximum translator velocity happens exactly at the mid-stroke especially for such a spring dominant system due to the nature of stiff spring forces. The system of forces acting on the piston in such a system is shown in Figure 15. From Figure 15, it is evident that the peak spring forces are of magnitude of 11 kN whereas the peak cylinder force is around 5 kN. The spring force magnitude is more than twice the peak cylinder force magnitude in the current system. In this section, the mid-stroke piston velocity is provided as an input to the controllers. The target mid-stroke velocity is set to 7 m/s based on the mean piston speed calculated for the experimental FPEG system. Figure 16 shows the mid-stroke velocity plotted against cycle number for both PID & PDF controller. Based on Figure 16, the mid-stroke velocities rose to the value of 10 m/s because of high initial peak and then gradually reduced to the set point value of 7m/s. However, unlike the previous cases where the set point is tracked smoothly, the PID & PDF controllers failed to maintain the mid-stroke velocity to 7 m/s. It is evident from cycle-to-cycle variations in the mid-stroke velocities from 200 to 250 cycles in Figure 16. The main reason for cycle-to-cycle variations in this case is because of the lack of proper information in the control variable regarding the dead center positions. In the previous three cases, the dead center positions are defined either by using the compression ratio, clearance set points and maximum cylinder pressures. However, in the case of mid-stroke velocity input only mid-stroke velocity is defined without any information on the location of dead centers. As a result, the FPEG system found its own dead center locations based on the non-linear nature of the forces acting on the piston. These dead center locations are not ideal and are deviated from the target dead center set points. This is evident from the velocity versus position distribution of an FPEG system at steady state when compression ratio and mid-stroke velocity is used an input to the controllers. Based on Figure 17, the dead center positions are shifted towards right when compared to the compression ratio input condition. Owing to this deviation in the dead centers, the port locations, opening and closing of ports and combustion attributed are affected thereby causing the cycle-to-cycle variations. From Figure 17, it is evident that the mid-stroke velocity is same for an FPEG system when mid-stroke velocity and trapped compression ratios are provided as an input to these controllers. The only difference is that the dynamics are shifted towards right in case of a mid-stroke velocity approach. The trapped compression ratio and cylinder pressures are reduced in this case when compared to their respective values in the previous sections. The variations in the trapped compression ratio during transient and steady state operation are shown in Figure 18 and Figure 19. Moreover, both the PID & PDF controllers behaved in the same way in this operation highlighting the dominance of the stiff springs in the system.

From Figure 17 and Figure 19, the deviation in the FPEG system dynamics changed the operational characteristics of the system. The trapped compression ratio during steady state operation in this case was found to be 6 when compared to the target value of 8.25. With reduced compression ratio, the in-cylinder pressures are also reduced to 10 – 15 bar thereby reducing the indicated and electrical power outputs. In addition to the compression ratio, the expansion ratio is also reduced as the exhaust port opening location is moved closer to the cylinder TDC position. Moreover, the amount of exhaust gas retained is increased with the reduction in the pressure difference between the combustion chamber and exhaust port. With the simulation results presented above, it is clear that the mid-stroke velocity itself is not enough to operate the FPEG system steadily at an optimum performance. However, if mid-stroke velocity is used in conjunction with other control variables peak cylinder pressure, compression ratio and dead center set points, then the FPEG system can be operated steadily at the optimized performance. Moreover, one can also use the reference trajectory profile like a sinusoidal signal while maintaining the same mid-stroke velocity as an input to these controllers for stable FPEG operation. Our future investigations will focus upon using the reference trajectory signal with same mid-stroke velocity for controlling the FPEG system.

Effect of numerical model time step on controller behavior

In a PID and a PDF controller, the integral and differential term action is calculated based on the time based integral and time based differentiation calculated in the error calculated by the controller. An investigation was carried to investigate the effect of the numerical model time step on the behavior of the controller during the transient mode. In this case, the maximum in-cylinder pressure is provided as an input to the controller. Four different time steps are considered in this investigation i.e. 0.5μs, 5μs, 7.5μs and 10μs. The change in the trapped compression ratio with different time steps during the transient mode operation is shown in Figure 20.  From Figure 20, it is clear that the numerical model time step yields substantially similar conclusions without any major deviations in the behavior of the controller. The controllers with different numerical time steps reached the steady state operation within the same time. Moreover, the system attained same dynamics for all the time steps during the steady state operation as evident from Figure 21. Based on this, it can be concluded that only the controller coefficients have the major impact on the controller set point ability tracking and control rather than the numerical model time steps.

Cycle-to-cycle variations

In the previous investigation, the authors claimed that the addition of mechanical springs reduces the cycle-to-cycle variations experienced by the FPEG system [17]. In this investigation, the FPEG system cycle-to-cycle variations were investigated with different control algorithms with same number of springs.  The cycle-to-cycle variations are evaluated by using the coefficient of variation (COV) in compression ratio and peak in-cylinder pressures. According to the standard definition of COV, the COV’s are calculated by using the standard deviation of the corresponding variable divided by the mean of that variable when the FPEG system is operating at a steady state. Figure 22 shows the cycle-to-cycle variations in the compression ratio and peak cylinder pressures with different control variables provided to the input. The combustion duration and combustion efficiency were changed from cycle-to-cycle with a random number generator within a particular range i.e. 1.5 to 2.5 ms for combustion duration and 85% to 95% for combustion efficiency.

From Figure 22, it is evident that the COV of peak pressure and the compression ratio are maintained below 5% for the controllers when compression ratio, dead center clearance set points and maximum cylinder pressures are provided as an input or control variables. When the mid-stroke velocity is selected as a control variable to the controller, the COV of peak pressure and compression ratio rose to 8 % and 20 % respectively. So, based on the simulation results presented in this investigation, the FPEG system can be controller steadily by using compression ratio controller, dead center clearance set points controller and maximum pressure controller. The mid-stroke velocity controller considering only the maximum piston velocity of the piston is not sufficient enough to control the FPEG system. However, the mid-stroke velocity controller when used in conjunction with other controller will reduce cycle-to-cycle variations. In addition to that, the reference trajectory signal (similar to a sinusoidal signal) with the same mid-stroke velocity can reduce the cycle-to-cycle variations even further. Our future investigations will focus on the evaluation of the PID & PDF controller with reference velocity trajectory signal as an input to the controller. Moreover, the controller investigations will also be carried for these inputs by regulating engine input fuel load rather than the regulating the alternator load which is considered in this investigation. Finally, the behavior of the above-mentioned controllers will be considered for rapid changes in the engine operating conditions during generating mode by introducing the stochastics in the numerical model. All these controller studies will then be used as the pathway for improving and optimizing the experimental FPEG design.

Conclusions

Unlike the conventional engines with crankshaft and connecting rod mechanism, the energy source i.e. engine in an FPEG device does not employ a crankshaft and flywheel combination as a result the dead center set points are not fixed. Depending upon the magnitude of forces acting on the piston, the piston dynamics are changed from cycle-to-cycle thereby resulting in variable compression ratio and variable stroke length. These variable compression ratio and variable stroke length are not attractive for an FPEG system operating at wide open throttle conditions and at a near constant load. To overcome these variations, a transient energy balance and piston motion controller is necessitated by an FPEG system for stable operation. Using a detailed MATLAB®/Simulink numerical model developed by the authors, an investigation was carried out to manage the energy generation and consumption in an FPEG system during the transient operating mode.  Based on that, two different basic feedback controllers (i.e. PID and PDF controllers) were implemented and investigated for the energy management purpose. Considering the design space of an FPEG system, there are many options available for the selection of the control variables. In this investigation, the trapped compression ratio, dead center clearance set points, maximum cylinder pressure and mid-stroke piston velocity were selected as control variables to the controller. Depending upon on the deviations in the control variables, these controllers regulated the alternator load.

Based on the simulation results, it is clear that both the PID and PDF controllers with manually tuned and optimized coefficients behaved in a similar fashion for all the control variables considered in the system.  Both of these controllers with optimized coefficients demonstrated the feasibility of energy balance management during transient operation. The controllers with compression ratio, peak cylinder pressure and dead center clearance set points as control variables demonstrated stable FPEG operation whereas the mid-stroke velocity failed to achieve steady operation. The main reason for the failure of mid-stroke velocity approach is because of defining only mid-stroke velocity without any information on the location of dead centers. In the former three cases, the dead center positions are defined either by using the compression ratio, clearance set points and maximum cylinder pressures. As a result of this, the FPEG system with mid-stroke velocity control found its own natural dead center set points based on the non-linear system of forces in the system and also because of the nature of the stiff spring forces. These dead center locations are not ideal and deviated from the target dead center set points resulting in cycle-to-cycle variations and non-optimized performance. Finally, the COV’s of compression ratio and peak pressures was calculated for these control approaches. The COV was found to be below 5% in all the cases except the mid-stroke velocity approach.

Considering the significant control challenges imposed by an FPEG system, more detailed analysis of these control architectures is required to overcome these challenges. Our future investigations will focus on the evaluation of these controller with reference velocity trajectory signal as an input to the controller and by regulating engine input fuel load rather than the regulating the alternator load which is considered in this investigation. Moreover, the behavior of the above-mentioned controllers will also be considered for rapid changes in the engine operating conditions during generating mode by introducing the stochastics in the numerical model.

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