Software Enabled Variable Displacement Pumps: Experimental Studies
Proceedings of IMECE2006 2006 ASME International Mechanical Engineering Congress and Exposition November 5-10, 2006, Chicago, Illinois, USA
IMECE2006-14973 SOFTWARE ENABLED VARIABLE DISPLACEMENT PUMPS - EXPERIMENTAL STUDIES ∗
Michael B. Rannow, Haink C. Tu, Perry Y. Li†and Thomas R. Chase Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis, MN 55455 Email: {rann0018,tuxxx021,pli,trchase}@me.umn.edu
ABSTRACT The majority of hydraulic systems are controlled using a metering valve or the use of variable displacement pumps. Metering valve control is compact and has a high control bandwidth, but it is energy inefficient due to throttling losses. Variable displacement pumps are far more efficient as the pump only produces the required flow, but comes with the cost of additional bulk, sluggish response, and added cost. In a previous paper [1], a hydromechanical analog of an electronic switch-mode power supply was proposed to create the functional equivalent of a variable displacement pump. This approach combines a fixed displacement pump with a pulse-width-modulated (PWM) on/off valve, a check valve, and an accumulator. The effective pump displacement can be varied by adjusting the PWM duty ratio. Since on/off valves exhibit low loss when fully open or fully closed, the proposed system is potentially more energy efficient than metering valve control, while achieving this efficiency without many of the shortcomings of traditional variable displacement pumps. The system also allows for a host of programmable features that can be implemented via control of the PWM duty ratio. This paper presents initial experimental validation of the concept as well as an investigation of the system efficiency. The experimental apparatus was built using available off-the-shelf components and uses a linear proportional spindle valve as the PWM valve. Experimental results confirm that the proposed approach can achieve variable control function more efficiently than a valve controlled system, and that by increasing the PWM frequency and adding closed-loop control can decrease system
response times and of the output ripple magnitude. Sources of inefficiency and their contributions are also investigated via modeling, simulation and are validated by experiments. These indicate design parameters for improving inefficiency.
1
Introduction All hydraulic systems require a power source, which usually consists of a prime-mover driving a pump. In most cases, the prime-mover runs at a constant speed, so variation of the flow sent to the load can only be achieved by varying the displacement of the pump or diverting a portion of the flow away from the load. Variable displacement is typically accomplished mechanically by varying the swash plate angle or eccentricity of the pump [7]. These techniques require moving a significant mass, which leads to a low control bandwidth, as well as bulky and expensive equipment [1]. Flow can also be varied using metering valve control, which usually consists of diverting excess flow across a relief valve or a restriction in a bleed-off configuration [7]. This involves little moving mass, so a high bandwidth can be achieved. However, the diverted flow is bled off at a high pressure, which results in a large amount of wasted energy. The goal of our research is to create a device that has the functionality and efficiency of a variable displacement pump but without the penalties in bandwidth, bulk, or cost. This will be achieved by applying the switch-mode power supply [6] from power electronics to a hydraulic system by mating a fixed displacement pump with a high-speed on/off valve, check valve, and 1
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an accumulator. Since on/off valves exhibit low loss when fully open or fully closed, the concept should be more energy efficient than metering valve-based control. Additionally, since the output flow is varied by changing the duty ratio of the PWM valve, it is simpler and more compact than traditional variable displacement pumps which rely on mechanical mechanisms to alter the flow rate. It is also expected to be more responsive. Proposals to use on/off valves to control fluid power systems have been around for awhile. For example, Luo et al. [10] proposed an apportioning system based on synchronized pilot operated poppet valves in the late 1980’s. In recent years, there has been a resurgence of activities in this area, partly due to the wide spread use of electrohydraulic concepts. For example, Gu et al. [3] uses the switch-mode converter concept to develop hydraulic transformers; and Barth et al. [4] substituted a PWM valve in place of a proportional valve to control a pneumatic load. Our current focus is on creating an integrated design that combines the various hydraulic elements (fixed displacement pump, flywheel, on/off valves, check valves and accumulator) as well as embedded closed-loop control to create a standalone system with multiple user-selectable modes of operation. Since these modes can be achieved via different control algorithms, they are implemented flexibly in software. Some examples of these software-enabled features include pressure control, flow/load control, the ability to run the prime-mover at its optimal speed, and the ability to compensate for non-ideal engine behavior. For further details please refer to [1]. In addition to the variable displacement pump, other applications of the hydraulic switch-mode concept are also currently being pursued. In this paper, the feasibility of a software enabled PWM variable displacement pump is experimentally validated. The paper explores the ripple size, response time, and the effect of closed loop feedback on their trade offs as discussed in [1], as well as the overall efficiency of the system when compared to metering valve control. A model is developed to describe several non-ideal effects of the system, and is verified through experiment and a multi-physic modeling software, Dymola. This study highlights the effects of various design parameters on efficiency and performance. The remainder of this paper includes the following: Section 2 gives an overview of the proposed concept, as well as a set of ideal and non-ideal models for the system, and for the prediction of energy loss. Section 3 presents experimental results concerning system efficiency and performance, and lastly, Section 4 contains some concluding remarks.
I−out I−in L Inductor
C PWM
Load Vout
Vin
Boost converter (a) DC-DC boost converter
(b) PWM variable displacement pump Figure 1.
Electrical DC-DC boost converter and its hydraulic analog
pump (and possibly an additional flywheel) is the functional equivalent of the inductor, and serves to smooth the load on the engine driving the pump. The check valve acts as a diode and ensures that there is no back flow from the load when the PWM valve is open. The accumulator, the analog of a capacitor, provides smoothing of the output flow. And finally, the high speed on/off valve has the role of a transistor. The PWM pump thus has two basic modes of operation. When the on/off valve is closed, the check valve opens, and the full flow of the pump is directed to the load and accumulator. Excess flow is stored in the accumulator. Once the on/off valve opens, the full flow of the pump is dumped to tank. This creates a drop in pressure on the inlet side which causes the check valve to close. The accumulator then provides the flow to the load until the valve closes again. 2.1
Idealized Model The system is modeled as a two state system, with the energy storage elements being the pump inertia and the accumulator. The idealized model contains several simplifying assumptions: the gas in the accumulator obeys the ideal gas law and operates adiabatically, the PWM and check valves are ideal (i.e. no time
2 Concept Overview and System Modeling The concept of a PWM based variable displacement pump comes directly from the DC-DC boost converter found in power electronics and is essentially its hydro-mechanical equivalent. Figure 1 shows a comparison between the two systems. In the hydraulic case, the inertia of the fixed displacement 2
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In steady-state, the equations become:
delay or pressure drop), and fluid friction and compressibility effects are negligible. Under these assumptions, the system can be modeled by the following equations:
Qout = D ˙ = −u(t) Pout + Γ(t) Jf ω 2π D Pout ˙ u(t) ω − Qout (t) Pout = γ V 2π | {z }
(1) (2)
(3)
if s(t) ≥ (t/T )mod1 if s(t) < (t/T )mod1
(4)
β(Pin ) Qadd (t) P˙in = Vin
where T is the period of the PWM cycle. In the analysis of switch-mode systems, the state-space averaging technique of using the duty ratio to average the system dynamics in the on and off states is often used. While there is evidence that this technique may introduce some inaccuracy and in some cases leading to instability [2], the design rule-of-thumb is that if the switching frequency is “fast enough”(10 times faster than the control bandwidth), the method is accurate. Using this method, the system dynamics can be written as: D Pout + Γ(t) 2π (1+1/γ) D Pout s(t) ω − Qout (t) = γ 1/γ 2π P0 V0 | {z }
(7)
Non-Ideal Model The model developed above demonstrates the feasibility of using PWM to create a software-enabled variable displacement pump. However, if an investigation of system performance and efficiency is to be undertaken, a more detailed model must be obtained that eliminates some of the simplifying assumptions made in the previous section. As described in [3], many effects must be considered in the analysis of switch-mode hydraulic systems. In a real system, efficiency and performance can be degraded by transition time in the PWM valve, fluid compressibility in the inlet oil volume, loss across a fully open PWM valve, closing time for the check valve, hysteresis in the accumulator bladder, fluid friction, leakage, and according to [5], variation in the adiabatic assumption of the gas in the accumulator during charging and discharging. In this paper, three of these effects are examined: the loss across the fully open PWM valve, the transition time of the PWM valve and the dynamics of the inlet oil volume. These three effects are dependent on the inlet pressure, which was not included in the idealized model, Eqs. (5) and (6). These effects can be included by adding the inlet pressure as a state with the following dynamics:
where Po and Vo are the initial pressure and volume of the gas in the accumulator. The input, u(t), is a pulse-width-modulation signal for the on/off valve, where u = 0 corresponds to the valve being fully open, and u = 1 corresponds to the valve being fully closed. u(t) is generated by comparing the desired duty ratio, s(t) ∈ [0, 1], to a periodic sawtooth wave: ( 1 u(t) = 0
s(t)D Pout 2π
2.2
where Pout is the pressure at the accumulator, ω is the rotational velocity of the pump, D is the pump displacement, J f is the rotary moment of inertia of the pump/flywheel, Γ(t) is the engine torque, Qout (t) is the output flow rate, γ is the ratio of specific heat at constant pressure and constant temperature of the gas in the accumulator, and V is the volume of compressed gas in the accumulator which satisfies: γ
Γ=
These equations show that the flow out of the system and torque on the engine are modified by the duty ratio s(t), representing a pump with an effective displacement of s(t)D.
−V˙
Pout V γ = P0V0
s(t)D ω; 2π
˙ = −s(t) Jf ω
(5)
P˙out
(6)
(8)
where β(Pin ) is the pressure dependent bulk modulus of the fluid, Vin is the volume of the oil which extends from the pump to the check valve and the PWM valve (see Fig. 1(b)), and Qadd (t) is the flow required to fill the void in the inlet volume once the oil is compressed. This is equal to the difference between the input flow from the pump and the output flows through the PWM and check valves. From Eq. (8), it is clear that the compressibility of the fluid causes the inlet volume to act like an accumulator with a volumetric stiffness, dP/dV , of β(Pin )/Vin . The inverse of the stiffness has a linear effect on the energy stored in the pressurized fluid, and thus, if Vin is increased, the energy stored in the fluid is also increased in a linear fashion. Once the PWM valve opens, this energy is lost and must be replaced in the next cycle. As the PWM frequency increases, this energy loss will occur more frequently, leading to a power loss that is increases linearly with frequency.
−V˙
3
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Energy is also lost due to throttling that occurs across the PWM valve as it transitions between fully open and fully closed. These losses are taken into account by modeling the valve as an adjustable orifice with area a(t). With this addition, the non-ideal model becomes:
to consist of two primary components: the energy lost due to throttling while the PWM valve is transitioning between fully open and fully closed, and the energy lost due to fluid compressibility. This gives the general form of a1 :
√ β(Pin ) D ω − a(t)K Pin − Qcheck (t) P˙in = Vin 2π D ˙ = − Pin + Γ(t) Jf ω 2π
a1 = Ethrottle + Ecomp
In calculating a1 it was assumed that the bulk modulus of the hydraulic oil in the system was pressure dependent, and the orifice of the PWM valve was circular. The energy lost due to throttling during valve transition was calculated using:
(10)
(1+1/γ)
P (Qcheck (t) − Qout (t)) P˙out = γ out 1/γ Po Vo
(11)
Z
where K is the discharge coefficient for the PWM valve, and the PWM valve orifice area, a(t), is the control input to the system. If the valve dynamics are known, a(t) can be determined from the valve model given some input signal. Qcheck (t) is the flow through the check valve, and, assuming that the check valve is ideal and that the inlet pressure is essentially static in the high pressure state, it is defined by:
Qcheck (t) =
( 0
√ D 2π ω − a(t)K Pin
if Pin < Pout if Pin ≥ Pout
Ethrottle =
Z
Pin Q pwm dt +
tclose
topen
Pin Q pwm dt
(15)
where tclose and topen are the times required for the system to transition from Plow → Phigh and Phigh → Plow respectively. Plow corresponds to the pressure drop across the fully open PWM valve. Phigh is simply the load pressure of the system. The inlet pressure of the system (Pin ) and the flow through the PWM valve (Q pwm ) are related to the open area of the PWM valve by the orifice equation:
(12)
( Pi Pin (t) = Phigh
2.3
Power Loss Model In order to investigate the sources of energy loss in a PWM system, the non-ideal model was extended to describe several of the predicted effects in detail: 1) the loss due to throttling during transition, 2) the loss due to compressibility in the fluid volume, and 3) the loss due to the pressure drop across the fully open PWM and check valves. The losses due to throttling during transition and compressibility in the inlet volume occur every PWM cycle and were predicted to be linear with frequency. The loss across the PWM and check valves is dependent on the amount of flow that passes over each valve and thus are dependent on the duty ratio. The overall power loss in the system modeled to be affine in both duty ratio and frequency: Powerlost = a0 + a1 f + a2 s
(14)
(9)
Q pwm (t) =
(
if Pi < Phigh if Pi ≥ Phigh
Dω 2π
if Pi < Phigh if Pi ≥ Phigh
p Ka(t) Phigh
(16)
(17)
Pi (t) is the predicted pressure drop across the on/off valve if with full flow across it given by
Pi (t) =
(13)
Dω 2πa(t)K
2
(18)
where K is the orifice discharge coefficient and a(t) is the open orifice area, determined using the geometric relationship derived by Richer and Hurmuzlu [9].
where a0 , a1 and a2 are coefficients for baseline power loss, frequency dependency, and duty ratio dependency respectively. f is the PWM frequency, and s is the duty ratio. The following sections describe a method for predicting the coefficients in Eq. (13).
r 2 a(xe ) = nh 2Rh arctan
xe 2Rh − xe
p − (Rh − xe ) xe (2Rh − xe )
(19) Rh is the radius of the circular orifice, and xe is the effective spool displacement. Eq. (19) was applied assuming no spool/orifice
2.3.1 Frequency Dependent Loss The frequency dependent coefficient a1 (the energy lost per cycle) was assumed 4
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overlap. xe as a function of time is determined from the dynamics of the PWM valve. Energy loss due to fluid compressibility, the other component of the frequency dependent energy loss, was determined by:
8
Z Vf
P dV
(20)
Vi
Ecomp = −V
Plow
2
0
1
2
3 4 Load Pressure (Pa)
5
6
7 6
x 10
Energy lost due to fluid compressibility
(21) P dP β(P)
(22) and the desired duty ratio component a2 .
where V is the total inlet volume of the system. β(P), the pressure dependent bulk modulus of the fluid, was determined using the model derived by Yu et al. [8]:
Dω Dω + (∆Pcheck − ∆Ppwm ) s 2π 2π Dω a2 = (∆Pcheck − ∆Ppwm ) 2π
Plost = ∆Ppwm
β(1 + 10−5 P)(1+1/γ) βe (P) = (1 + 10−5 P)(1+1/γ) + 10−5 R(1 − c1 P)(β/γ − 105 − P) (23) β is the bulk modulus of air free oil, P is the inlet pressure in Pa, γ is the ratio of specific heats for air, R is the ratio of entrained air volume to total volume, c1 is the variation of air bubble volume due to changing amounts of dissolved air, and βe is the effective bulk modulus of oil with air. Figure 2 depicts the results from numerically integrating Eq. (22) for various values of R and load pressure (Phigh ).
Dω Dω + ∆Pcheck s 2π 2π
(25) (26)
Eq. (26) reveals that pressure drops encountered by the load flow have a positive effect on a2 , while those encountered by the diverted flow have a negative effect. a0 , the contstant loss coefficient consists both of the constant term from Eq. (25) and any loss due to constant effects (i.e. fluid friction and leakage). These effects are difficult to predict and we did not attempt to model them in this paper. To verify the predictive models described in this section, the multi-physic modeling software package Dymola was used to create a complete system model. In addition to the three effects mentioned, Dymola was also used to model an idealized accumulator and load orifice, as well as the fluid volumes along the load branch of the circuit. The Yu bulk modulus model was also used, and air was included in the system. A schematic of the model used is shown in Fig. 3. The PWM valve in the Dymola was modeled with a circular orifice and second order spool dynamics, and it contains the open-valve throttling effect and the transition time effect. The Dymola model does not include any check valve dynamics, accumulator hysteresis, or fluid friction effects. The pressure drop across the accumulator orifice was also assumed to be negligible. The results of these simulations, as well as results obtained from an experimental setup, are presented in the following section.
2.3.2 Duty Ratio Dependent Loss The duty ratio dependent energy loss coefficient a2 was calculated by finding the power loss due to pressure drops encountered by the diverted flow, as well as non-load pressure drops encountered by the load flow. It was assumed that the dominant pressure drops would be due to the fully open PWM valve and the check valve. This leads to the power loss equation: Plost = ∆Ppwm (1 − s)
0
Figure 2.
dP dV
Z P high
6
4
Eq. (20) represents the energy required to compress the fluid in the inlet volume of the system from low pressure to high pressure. Using the definition of bulk modulus and performing a coordinate transformation on Eq. (20), the final equation for calculating the loss due to fluid compressibility is: β = −V
Vair=.004% (default) Vair=.1% Vair=.5% Vair=1% Vair=2.5%
10
Energy Lost (J)
Ecomp =
Energy lost per cycle due to fluid compressibility (Yu Model) 12
(24)
The first term in Eq. (24) represents the power loss from the diverted flow, and the second term the power loss from the load flow. Rearranging Eq. (24) yields a constant loss component, 5
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5.5
6 x 10 System Response, 2 PWM cycles: PWM=10Hz, Duty Ratio=.5
Inlet Pressure Outlet Pressure
C
5 4.5
Pressure (Pa)
4
E
D
3.5 3 2.5 2 1.5 1
A
0.5
B 7.55
Figure 5. system
Figure 3.
7.6
7.65 Time (s)
Inlet and outlet presssures over
7.7
2
PWM cycles for a
10Hz
charge pressure of 4.1 × 106 Pa was used. In this setup, the pump was driven by an AC induction motor, which fixed the pump speed, eliminating the need for a large inertia. The hydraulic load used was an adjustable needle valve (H) and the inlet volume was about 40ml. The system also had pressure sensors (A, B, and E) on both sides of the check valve, and a flow meter (F) on the output line. The components chosen are not meant to be optimal but are used to determines the effects of different design parameters.
Dymola model
3.1
Figure 4.
System Operation A close-up view of the inlet and outlet pressures over 2 PWM cycles, for a switching frequency of 10Hz is shown in Fig. 5. The averaging effect of the accumulator can be clearly seen on this plot; between D and E, the accumulator is charging, and between C and D it is discharging. The difference between C and D (about 1.3 × 105 Pa) is the output ripple of the system. This is a characteristic of PWM controlled systems and cannot be completely eliminated. Between points A and B, the inlet pressure is low. The only load on the pump during this time is the loss across the fully open PWM valve, which was about 4.5 × 105 Pa. Between C and A, and again between B and D, the system is in transition. During these periods, the pressure is fairly high and the pump flow is going through the partially open on/off valve; this leads to a significant loss of energy.
Experimenal apparatus
3 Experimental Studies The experimental apparatus used for this paper is shown in Fig. 4. The PWM valve (C) was a linear proportional valve, which had a manufacturer specified transition time of 20ms. The fixed displacement gear pump (G) provided 5.68l/m, and components were sized for that flow. A nitrogen-filled diaphragm accumulator (D) with an initial gas volume of .16l and a pre-
3.2
Efficiency The efficiency of the hydraulic system was defined as the power out to the load divided by the power in from the pump. Therefore, any numbers presented here do not include any inefficiencies due to the pump or driving motor. The power into and out of the system was computed by multiplying the average 6
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Power Lost in a PWM and a Proportional Valve System 350 1 Hz 2.5 Hz 5 Hz 10 Hz Proportional
300
Power Lost (W)
250
200
150
100
50
0
0
20
40 60 % of Flow Sent to the Load
80
100
Figure 6. Power loss in a PWM system at various PWM frequencies and a valve controlled system (marked proportional)
(a) Power lost plane
ouput flow rate over 5 seconds and either the measured inlet or outlet pressure, and then taking the average of the product over 5 seconds (Eq. (27)). R
Power =
Qave P(t)dt ∆t
(27)
The power loss was measured at various duty ratios and at frequencies of 1, 2.5, 5, and 10 Hz. The efficiency was also measured for the system with the valve configured for valve control. Figure 6 shows the results, which indicate a significant improvement in efficiency over valve control. The proportional data was obtained by using the PWM valve as a proportional valve in a bleed-off configuration. These results demonstrate the potential of PWM controlled systems in creating efficient hydraulic systems. The power lost in the PWM system was fitted to a plane described by Eq. (13), which is shown in Fig. 7. The plane is also rotated to show the quality of fit. Table 1 lists the values of the coefficients a0 , a1 and a2 , used to define this plane, as well as the coefficients determined from the predictive models described in section 2.3 and Dymola with 0.5% air in the oil. In order to use the predictive models to estimate a0 , a1 and a2 , a model of the spool transition characteristics was needed. xe (t), the spool position during transition in Eq. (19), was modified until the transition pressure profiles during opening and closing of the valve were consitent between the model and experimental data. It was found that using a half sine wave velocity profile for the spool matched extremely well with the experimental results. Figure 8 shows the pressure profile of the experimental data, the Dymola model and Eq. (16) using the half sine velocity profile. The pressure profile was created using the experimen-
(b) Rotated power lost plane Figure 7.
Plane representing the power lost as a function of frequency
and duty ratio
tally determined values of 4.83 × 106 Pa for Phigh , 40ms for topen and 20ms for tclose . Using these values and Eqs. (15), (16) and (17), the predicted energy loss due to the valve transition was 5.95J. From Table 1, a1 was determined experimentally to be 8.3J. Given that the loss due to valve throttling was found to be 5.95J, the loss due to fluid compressibility is expected to be roughly 2.3J. From Fig. 2, this energy loss corresponds to the compression of hydraulic oil with .5%-1% air by volume for a load pressure of 4.83 × 106 Pa. This is a reasonable estimate for this system. These results suggest that a majority of the energy loss per cycle is in fact due to PWM valve throttling and fluid compressibility. 7
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Predicted
Dymola
Experiment
a0 (W )
41.1
42.1
39.7
a1 (J)
7.6
5.5
8.3
a2 (W )
-38.5
-37.6
-20.1
Valve Transition: Plow−>Phigh
6
5.5
x 10
Experiment, PWM=5Hz Dymola, PWM=5Hz, wn=150Hz, .5% Air Simulation, V=A*sine(wt), Tclose=40ms
5 4.5 4 Pressure (Pa)
Coefficient
Values of a0 , a1 and a2 from simulation and experiment. The analytical and Dynmola models assume 0.5% of air in the fluid. Table 1.
3.5 3 2.5 2 1.5
The energy loss due to duty ratio effects was calculated using Eq. (25) and the values of 4.34 × 105 Pa and 2.76 × 104 Pa for ∆Ppwm and ∆Pcheck respectively. a2 was calculated to be −38.5W for a nominal flow of 9.5 × 10−5 m3 /s (1.5gpm). Interestingly, the calculated value of a2 is more negative than the experimental value of −20.1W given in Table 1. This result suggests that there are additional unmodeled pressure drops enountered by the load flow. This is most likely due to unmodelled losses across the check valve or in the accumulator. Another cause for this discrepancy is that topen and tclose were assumed to be zero in Eq. (24) which is an idealization. Lastly, a0 , the constant loss coefficient, was found to be nearly identical in all three cases. Since fluid friction was only present in the experimental case, the simulated results indicate that the effect of fluid friction on the system was insignificant. Since large diameter hoses and relatively low flow rates were used, this conclusion seems reasonable. Using the models described in this paper, several design goals for creating an efficient PWM system can be extracted. a2 can be decreased by using a PWM valve which minimizes loss in the full open state. This may not be as simple as increasing the orifice size, since an increased orifice will lead to a longer transition time, thus limiting the maximum PWM frequency of the system and possibly increasing a1 . a1 can be decreased by using a faster valve or decreasing the inlet volume. While several simplifying assumptions are still included in the system model, at low frequencies, the model appears to match the experimental data, indicating that the significant sources of energy loss have been included.
1 0.5 0 0.02
(1+1/γ)
0.035
0.04 0.045 Time (s)
0.05
0.055
0.06
Valve Transition: Phigh−>Plow
6
5.5
x 10
Experiment, PWM=5Hz Dymola, PWM=5Hz, wn=150Hz, .5% Air Simulation, V=A*sine(wt), Topen=20ms
5 4.5
Pressure (Pa)
4 3.5 3 2.5 2 1.5 1 0.5 0 0.01
0.02
0.03
0.04 Time (s)
0.05
0.06
0.07
(b) PWM valve opening
Figure 8.
Inlet pressure profile during PWM valve transition
the volume and pre-charge pressure of the accumulator have a similar effect on the system dynamics. This term is the volumetric stiffness of the accumulator, and it was shown in [1] that a low pre-charge pressure, or equivalently a small volume have the effect of increasing the speed of the system while also increasing the size of the ripple on the output. Thus a tradeoff exists between the response time of the system and the magnitude of the output ripple. Note that the pressure in the accumulator must be higher than the pre-charge pressure in order for any smoothing to take place. The effect of the PWM frequency in this performance tradeoff is evident from Fig. 10, where the PWM frequency has been increased to 10Hz. This experiment showed a rise time of 2.47s and a ripple magnitude of 2.0% of the output pressure. While the
System Performance Output ripple size and response times are two important aspects of system performance. The open-loop step response of the system was observed by increasing the average output flow rate from 50% of full flow to 60% of full flow. The response of the system, using a PWM frequency of 5Hz, is shown in Fig. 9. The pressure settles to its final value in 3.01s and the ripple magnitude (peak to peak) is 4.3% of the output pressure. P0 V0
0.03
(a) PWM valve closing
3.3
It is evident from the coefficient γ P 1/γ
0.025
in Eq. (6) that both 8
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6
5.8
x 10
6
Open−Loop Step Response: 5Hz PWM Frequency 5.8
Closed−Loop Step Response: 5Hz PWM Frequency
5.6
5.6
5.4
Pressure (Pa)
5.4
Pressure (Pa)
x 10
5.2 5
5.2 5 4.8
4.8 4.6 4.6 4.4 4.4 4.2 4.2
0
5
10 Time (s)
Figure 9. Open-loop step response: Qout /Q f ull Frequency = 5Hz, trise = 3.01s, Ripple=4.3%
6
5.8
x 10
15
0
5
20
10 Time (s)
15
20
Figure 11. Closed-loop step response: Pdes = 4.52×106 Pa→5.65× 106 Pa, PWM Frequency = 5Hz, trise = .28s, Ripple=4.6%
= .511→.602, PWM
a pressure control algorithm with the duty ratio computed from the following equation:
Open−Loop Step Response: 10Hz PWM Frequency
5.6
s(t) = K f f
Pressure (Pa)
5.4 5.2
4.8 4.6 4.4
Figure 10.
0
5
10 Time (s)
15
20
Qout /Q f ull = .510→.608, = 2.47s, Ripple=2.0%
Open-loop step response:
PWM Frequency = 10Hz, trise
(28)
Where Pdes is the desired output pressure. The value of the feedback gain, K f b , was .025, which was the gain that provided the fastest response with no significant overshoot. The feedforward gain, K f f , was .0189 and was determined from a linearized model of the system. Values for Pdes were chosen to correspond to the initial and final pressures in the open-loop step response. The closed-loop step response for a system at 5Hz has a rise time of .28s and a ripple of 4.6%, demonstrating that closed-loop control can be used to improve the rise time without increasing the output ripple. This algorithm operates on controlling the pressure of the system, but feedback from a flowmeter on the load line can easily be used to create a flow-controlled system. An added benefit of closed-loop pressure or flow control is the ability of the control algorithm to compensate for the effect of load dynamics. Since the duty ratio under closed-loop control is constantly varying and determined by the control algorithm, the system can adapt continuously to changes in the system load. To demonstrate the ability of the system to track a pressure reference, a sinusoidal function was added to the nominal pressure, with the results shown in Fig. 12. The system tracked a .5Hz reference with little attenuation or phase shift. As the frequency of the reference was increased, the control signal began to saturate. Once the duty ratio is above 1 or below 0, the controller cannot improve the speed of the system, and bandwidth is limited by the dynamics of the accumulator. This lead to significant signal attenuation at frequencies of 1Hz and above.
5
4.2
p Pdes + K f b (Pdes − Pout (t))
rise time actually decresed slightly, the output ripple was reduced by more than half. This supports a conclusion of [1]: increasing the PWM frequency decreases the ripple size without increasing the rise time. This indicates that, while increasing the frequency might have a penalty on the system efficiency, it can also dramatically improve performance by reducing the output ripple. 3.4
Closed-Loop Control Li et al. [1] also proposed that adding closed-loop control to the system could improve this tradeoff by increasing the rise time without increasing the output ripple. This is shown experimentally in Fig. 11. In this experiment, the system was run with 9
c 2006 by ASME Copyright
6
5.4
x 10
Acknowledgment This material is based upon work supported by the National Science Foundation under grant number ENG/CMS-0409832.
Closed−Loop Sine Tracking: 5Hz PWM Frequency Reference Pressure Output Pressure
5.2
Pressure (Pa)
5
REFERENCES [1] P. Li, C. Li and T. Chase, “Software Enabled Variable Displacement Pumps” Proceedings of the 2005 ASME-IMECE, no. IMECE2005-81376, 2005. [2] B. Lehman and R. Bass, “Switching frequency dependent averaged models for pwm dc-dc converters,” IEEE Transactions on Power Electronics, vol. 11, no. 1, pp. 89–98, 1996. [3] L. Gu, M. Qiu, W. Feng, and J. Cao, “Switchmode Hydraulic Power Supply Theory,” Proceedings of the 2005 ASMEIMECE, no. IMECE2005-79019, 2005. [4] E. J. Barth, J. Zhang, and M. Goldfarb, “Control Design for Relative Stability in a PWM-Controlled Pneumatic System,” Journal of Dynamic Systems Measurement and ControlTransactions of the ASME, vol. 125, no. 3 pp. 504–508, 2003. [5] H. M. Paynter and E. P. Fahrenthold, “On the Nonexistence of Simple Polytropes and Other Thermodynamic Consequences of the Dispersion Relation,” ASME Bioengineering Division Publication, vol. 5, pp. 37–42, 1987. [6] N. Mohan, T. Undeland, and W. Robbins, Power electronics: converters, applications and design. John Wiley and Sons, 3rd ed., 2003. [7] Industrial Hydraulics Manual. Eaton Corporation, 4th ed., 2001. [8] J. Yu, Z. Chen, and Y. Lu, “The Variation of Oil Effective Bulk Modulus with Pressure in Hydraulic Systems,” Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, vol. 116, no. 1 pp. 146–150, 1994. [9] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part I - Nonlinear Mathematical Model,” Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, vol. 122, no. 3 pp. 416– 425, 2000. [10] N. Luo, F. Fronczak, and N. Beachley, “Comparison of Analytical and Experimental Investigations of a Hydraulic Multi-Circuit Sequential Apportioning System,” Society of Automotive Engineers Transactions, vol. 99, pp. 266–275, 1990.
4.8
4.6
4.4
4.2
0
2
4
6
8
10
Time (s)
Figure 12.
Closed-loop reference tracking: PWM Frequency = 5Hz
4 Conclusion Experimental validation of the feasibility of a softwareenabled variable displacement pump has been presented. It was shown that an on/off valve, check valve, and an accumulator can be used with a fixed-displacement pump to create a system with the functionality of a variable displacement pump. While a tradeoff between maximizing the system response and minimizing the output ripple will always exist, it was shown that increasing the PWM frequency and adding closed-loop control can dramatically improve the system performance. The efficiency of the proposed system was compared to that of a proportional valve, demonstrating the potential energy savings over valve control. While the PWM system dramatically improved the system efficiency, energy is still lost due to several effects, some of which are frequency dependent. Increasing the PWM frequency has the benefit of decreasing the output ripple and increasing the control bandwidth of the system. However, as higher frequency PWM systems are designed, attempts must be made to minimize the frequency dependent energy loss. The valve transition time is a significant source of energy loss which can be minimized through the design of faster on/off valves. In the system presented in this paper, the transition time of the valve was nominally 20ms, which lead to significant energy loss at 10Hz, and did not allow the system to function above 14Hz. The oil volume on the inlet side of the system also contributes to the inefficiency of the system. The volume in this paper was about 40ml, which lead to a significant loss of energy. This effect increases linearly with PWM frequency and inlet volume. This indicates the need to minimize the inlet volume as frequency is increased. Finally energy is lost in the dynamics of the accumulator and check valve. These effects were not examined in this paper and need to be fully investigated. 10
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IMECE2006-14973 SOFTWARE ENABLED VARIABLE DISPLACEMENT PUMPS - EXPERIMENTAL STUDIES ∗
Michael B. Rannow, Haink C. Tu, Perry Y. Li†and Thomas R. Chase Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis, MN 55455 Email: {rann0018,tuxxx021,pli,trchase}@me.umn.edu
ABSTRACT The majority of hydraulic systems are controlled using a metering valve or the use of variable displacement pumps. Metering valve control is compact and has a high control bandwidth, but it is energy inefficient due to throttling losses. Variable displacement pumps are far more efficient as the pump only produces the required flow, but comes with the cost of additional bulk, sluggish response, and added cost. In a previous paper [1], a hydromechanical analog of an electronic switch-mode power supply was proposed to create the functional equivalent of a variable displacement pump. This approach combines a fixed displacement pump with a pulse-width-modulated (PWM) on/off valve, a check valve, and an accumulator. The effective pump displacement can be varied by adjusting the PWM duty ratio. Since on/off valves exhibit low loss when fully open or fully closed, the proposed system is potentially more energy efficient than metering valve control, while achieving this efficiency without many of the shortcomings of traditional variable displacement pumps. The system also allows for a host of programmable features that can be implemented via control of the PWM duty ratio. This paper presents initial experimental validation of the concept as well as an investigation of the system efficiency. The experimental apparatus was built using available off-the-shelf components and uses a linear proportional spindle valve as the PWM valve. Experimental results confirm that the proposed approach can achieve variable control function more efficiently than a valve controlled system, and that by increasing the PWM frequency and adding closed-loop control can decrease system
response times and of the output ripple magnitude. Sources of inefficiency and their contributions are also investigated via modeling, simulation and are validated by experiments. These indicate design parameters for improving inefficiency.
1
Introduction All hydraulic systems require a power source, which usually consists of a prime-mover driving a pump. In most cases, the prime-mover runs at a constant speed, so variation of the flow sent to the load can only be achieved by varying the displacement of the pump or diverting a portion of the flow away from the load. Variable displacement is typically accomplished mechanically by varying the swash plate angle or eccentricity of the pump [7]. These techniques require moving a significant mass, which leads to a low control bandwidth, as well as bulky and expensive equipment [1]. Flow can also be varied using metering valve control, which usually consists of diverting excess flow across a relief valve or a restriction in a bleed-off configuration [7]. This involves little moving mass, so a high bandwidth can be achieved. However, the diverted flow is bled off at a high pressure, which results in a large amount of wasted energy. The goal of our research is to create a device that has the functionality and efficiency of a variable displacement pump but without the penalties in bandwidth, bulk, or cost. This will be achieved by applying the switch-mode power supply [6] from power electronics to a hydraulic system by mating a fixed displacement pump with a high-speed on/off valve, check valve, and 1
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an accumulator. Since on/off valves exhibit low loss when fully open or fully closed, the concept should be more energy efficient than metering valve-based control. Additionally, since the output flow is varied by changing the duty ratio of the PWM valve, it is simpler and more compact than traditional variable displacement pumps which rely on mechanical mechanisms to alter the flow rate. It is also expected to be more responsive. Proposals to use on/off valves to control fluid power systems have been around for awhile. For example, Luo et al. [10] proposed an apportioning system based on synchronized pilot operated poppet valves in the late 1980’s. In recent years, there has been a resurgence of activities in this area, partly due to the wide spread use of electrohydraulic concepts. For example, Gu et al. [3] uses the switch-mode converter concept to develop hydraulic transformers; and Barth et al. [4] substituted a PWM valve in place of a proportional valve to control a pneumatic load. Our current focus is on creating an integrated design that combines the various hydraulic elements (fixed displacement pump, flywheel, on/off valves, check valves and accumulator) as well as embedded closed-loop control to create a standalone system with multiple user-selectable modes of operation. Since these modes can be achieved via different control algorithms, they are implemented flexibly in software. Some examples of these software-enabled features include pressure control, flow/load control, the ability to run the prime-mover at its optimal speed, and the ability to compensate for non-ideal engine behavior. For further details please refer to [1]. In addition to the variable displacement pump, other applications of the hydraulic switch-mode concept are also currently being pursued. In this paper, the feasibility of a software enabled PWM variable displacement pump is experimentally validated. The paper explores the ripple size, response time, and the effect of closed loop feedback on their trade offs as discussed in [1], as well as the overall efficiency of the system when compared to metering valve control. A model is developed to describe several non-ideal effects of the system, and is verified through experiment and a multi-physic modeling software, Dymola. This study highlights the effects of various design parameters on efficiency and performance. The remainder of this paper includes the following: Section 2 gives an overview of the proposed concept, as well as a set of ideal and non-ideal models for the system, and for the prediction of energy loss. Section 3 presents experimental results concerning system efficiency and performance, and lastly, Section 4 contains some concluding remarks.
I−out I−in L Inductor
C PWM
Load Vout
Vin
Boost converter (a) DC-DC boost converter
(b) PWM variable displacement pump Figure 1.
Electrical DC-DC boost converter and its hydraulic analog
pump (and possibly an additional flywheel) is the functional equivalent of the inductor, and serves to smooth the load on the engine driving the pump. The check valve acts as a diode and ensures that there is no back flow from the load when the PWM valve is open. The accumulator, the analog of a capacitor, provides smoothing of the output flow. And finally, the high speed on/off valve has the role of a transistor. The PWM pump thus has two basic modes of operation. When the on/off valve is closed, the check valve opens, and the full flow of the pump is directed to the load and accumulator. Excess flow is stored in the accumulator. Once the on/off valve opens, the full flow of the pump is dumped to tank. This creates a drop in pressure on the inlet side which causes the check valve to close. The accumulator then provides the flow to the load until the valve closes again. 2.1
Idealized Model The system is modeled as a two state system, with the energy storage elements being the pump inertia and the accumulator. The idealized model contains several simplifying assumptions: the gas in the accumulator obeys the ideal gas law and operates adiabatically, the PWM and check valves are ideal (i.e. no time
2 Concept Overview and System Modeling The concept of a PWM based variable displacement pump comes directly from the DC-DC boost converter found in power electronics and is essentially its hydro-mechanical equivalent. Figure 1 shows a comparison between the two systems. In the hydraulic case, the inertia of the fixed displacement 2
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In steady-state, the equations become:
delay or pressure drop), and fluid friction and compressibility effects are negligible. Under these assumptions, the system can be modeled by the following equations:
Qout = D ˙ = −u(t) Pout + Γ(t) Jf ω 2π D Pout ˙ u(t) ω − Qout (t) Pout = γ V 2π | {z }
(1) (2)
(3)
if s(t) ≥ (t/T )mod1 if s(t) < (t/T )mod1
(4)
β(Pin ) Qadd (t) P˙in = Vin
where T is the period of the PWM cycle. In the analysis of switch-mode systems, the state-space averaging technique of using the duty ratio to average the system dynamics in the on and off states is often used. While there is evidence that this technique may introduce some inaccuracy and in some cases leading to instability [2], the design rule-of-thumb is that if the switching frequency is “fast enough”(10 times faster than the control bandwidth), the method is accurate. Using this method, the system dynamics can be written as: D Pout + Γ(t) 2π (1+1/γ) D Pout s(t) ω − Qout (t) = γ 1/γ 2π P0 V0 | {z }
(7)
Non-Ideal Model The model developed above demonstrates the feasibility of using PWM to create a software-enabled variable displacement pump. However, if an investigation of system performance and efficiency is to be undertaken, a more detailed model must be obtained that eliminates some of the simplifying assumptions made in the previous section. As described in [3], many effects must be considered in the analysis of switch-mode hydraulic systems. In a real system, efficiency and performance can be degraded by transition time in the PWM valve, fluid compressibility in the inlet oil volume, loss across a fully open PWM valve, closing time for the check valve, hysteresis in the accumulator bladder, fluid friction, leakage, and according to [5], variation in the adiabatic assumption of the gas in the accumulator during charging and discharging. In this paper, three of these effects are examined: the loss across the fully open PWM valve, the transition time of the PWM valve and the dynamics of the inlet oil volume. These three effects are dependent on the inlet pressure, which was not included in the idealized model, Eqs. (5) and (6). These effects can be included by adding the inlet pressure as a state with the following dynamics:
where Po and Vo are the initial pressure and volume of the gas in the accumulator. The input, u(t), is a pulse-width-modulation signal for the on/off valve, where u = 0 corresponds to the valve being fully open, and u = 1 corresponds to the valve being fully closed. u(t) is generated by comparing the desired duty ratio, s(t) ∈ [0, 1], to a periodic sawtooth wave: ( 1 u(t) = 0
s(t)D Pout 2π
2.2
where Pout is the pressure at the accumulator, ω is the rotational velocity of the pump, D is the pump displacement, J f is the rotary moment of inertia of the pump/flywheel, Γ(t) is the engine torque, Qout (t) is the output flow rate, γ is the ratio of specific heat at constant pressure and constant temperature of the gas in the accumulator, and V is the volume of compressed gas in the accumulator which satisfies: γ
Γ=
These equations show that the flow out of the system and torque on the engine are modified by the duty ratio s(t), representing a pump with an effective displacement of s(t)D.
−V˙
Pout V γ = P0V0
s(t)D ω; 2π
˙ = −s(t) Jf ω
(5)
P˙out
(6)
(8)
where β(Pin ) is the pressure dependent bulk modulus of the fluid, Vin is the volume of the oil which extends from the pump to the check valve and the PWM valve (see Fig. 1(b)), and Qadd (t) is the flow required to fill the void in the inlet volume once the oil is compressed. This is equal to the difference between the input flow from the pump and the output flows through the PWM and check valves. From Eq. (8), it is clear that the compressibility of the fluid causes the inlet volume to act like an accumulator with a volumetric stiffness, dP/dV , of β(Pin )/Vin . The inverse of the stiffness has a linear effect on the energy stored in the pressurized fluid, and thus, if Vin is increased, the energy stored in the fluid is also increased in a linear fashion. Once the PWM valve opens, this energy is lost and must be replaced in the next cycle. As the PWM frequency increases, this energy loss will occur more frequently, leading to a power loss that is increases linearly with frequency.
−V˙
3
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Energy is also lost due to throttling that occurs across the PWM valve as it transitions between fully open and fully closed. These losses are taken into account by modeling the valve as an adjustable orifice with area a(t). With this addition, the non-ideal model becomes:
to consist of two primary components: the energy lost due to throttling while the PWM valve is transitioning between fully open and fully closed, and the energy lost due to fluid compressibility. This gives the general form of a1 :
√ β(Pin ) D ω − a(t)K Pin − Qcheck (t) P˙in = Vin 2π D ˙ = − Pin + Γ(t) Jf ω 2π
a1 = Ethrottle + Ecomp
In calculating a1 it was assumed that the bulk modulus of the hydraulic oil in the system was pressure dependent, and the orifice of the PWM valve was circular. The energy lost due to throttling during valve transition was calculated using:
(10)
(1+1/γ)
P (Qcheck (t) − Qout (t)) P˙out = γ out 1/γ Po Vo
(11)
Z
where K is the discharge coefficient for the PWM valve, and the PWM valve orifice area, a(t), is the control input to the system. If the valve dynamics are known, a(t) can be determined from the valve model given some input signal. Qcheck (t) is the flow through the check valve, and, assuming that the check valve is ideal and that the inlet pressure is essentially static in the high pressure state, it is defined by:
Qcheck (t) =
( 0
√ D 2π ω − a(t)K Pin
if Pin < Pout if Pin ≥ Pout
Ethrottle =
Z
Pin Q pwm dt +
tclose
topen
Pin Q pwm dt
(15)
where tclose and topen are the times required for the system to transition from Plow → Phigh and Phigh → Plow respectively. Plow corresponds to the pressure drop across the fully open PWM valve. Phigh is simply the load pressure of the system. The inlet pressure of the system (Pin ) and the flow through the PWM valve (Q pwm ) are related to the open area of the PWM valve by the orifice equation:
(12)
( Pi Pin (t) = Phigh
2.3
Power Loss Model In order to investigate the sources of energy loss in a PWM system, the non-ideal model was extended to describe several of the predicted effects in detail: 1) the loss due to throttling during transition, 2) the loss due to compressibility in the fluid volume, and 3) the loss due to the pressure drop across the fully open PWM and check valves. The losses due to throttling during transition and compressibility in the inlet volume occur every PWM cycle and were predicted to be linear with frequency. The loss across the PWM and check valves is dependent on the amount of flow that passes over each valve and thus are dependent on the duty ratio. The overall power loss in the system modeled to be affine in both duty ratio and frequency: Powerlost = a0 + a1 f + a2 s
(14)
(9)
Q pwm (t) =
(
if Pi < Phigh if Pi ≥ Phigh
Dω 2π
if Pi < Phigh if Pi ≥ Phigh
p Ka(t) Phigh
(16)
(17)
Pi (t) is the predicted pressure drop across the on/off valve if with full flow across it given by
Pi (t) =
(13)
Dω 2πa(t)K
2
(18)
where K is the orifice discharge coefficient and a(t) is the open orifice area, determined using the geometric relationship derived by Richer and Hurmuzlu [9].
where a0 , a1 and a2 are coefficients for baseline power loss, frequency dependency, and duty ratio dependency respectively. f is the PWM frequency, and s is the duty ratio. The following sections describe a method for predicting the coefficients in Eq. (13).
r 2 a(xe ) = nh 2Rh arctan
xe 2Rh − xe
p − (Rh − xe ) xe (2Rh − xe )
(19) Rh is the radius of the circular orifice, and xe is the effective spool displacement. Eq. (19) was applied assuming no spool/orifice
2.3.1 Frequency Dependent Loss The frequency dependent coefficient a1 (the energy lost per cycle) was assumed 4
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overlap. xe as a function of time is determined from the dynamics of the PWM valve. Energy loss due to fluid compressibility, the other component of the frequency dependent energy loss, was determined by:
8
Z Vf
P dV
(20)
Vi
Ecomp = −V
Plow
2
0
1
2
3 4 Load Pressure (Pa)
5
6
7 6
x 10
Energy lost due to fluid compressibility
(21) P dP β(P)
(22) and the desired duty ratio component a2 .
where V is the total inlet volume of the system. β(P), the pressure dependent bulk modulus of the fluid, was determined using the model derived by Yu et al. [8]:
Dω Dω + (∆Pcheck − ∆Ppwm ) s 2π 2π Dω a2 = (∆Pcheck − ∆Ppwm ) 2π
Plost = ∆Ppwm
β(1 + 10−5 P)(1+1/γ) βe (P) = (1 + 10−5 P)(1+1/γ) + 10−5 R(1 − c1 P)(β/γ − 105 − P) (23) β is the bulk modulus of air free oil, P is the inlet pressure in Pa, γ is the ratio of specific heats for air, R is the ratio of entrained air volume to total volume, c1 is the variation of air bubble volume due to changing amounts of dissolved air, and βe is the effective bulk modulus of oil with air. Figure 2 depicts the results from numerically integrating Eq. (22) for various values of R and load pressure (Phigh ).
Dω Dω + ∆Pcheck s 2π 2π
(25) (26)
Eq. (26) reveals that pressure drops encountered by the load flow have a positive effect on a2 , while those encountered by the diverted flow have a negative effect. a0 , the contstant loss coefficient consists both of the constant term from Eq. (25) and any loss due to constant effects (i.e. fluid friction and leakage). These effects are difficult to predict and we did not attempt to model them in this paper. To verify the predictive models described in this section, the multi-physic modeling software package Dymola was used to create a complete system model. In addition to the three effects mentioned, Dymola was also used to model an idealized accumulator and load orifice, as well as the fluid volumes along the load branch of the circuit. The Yu bulk modulus model was also used, and air was included in the system. A schematic of the model used is shown in Fig. 3. The PWM valve in the Dymola was modeled with a circular orifice and second order spool dynamics, and it contains the open-valve throttling effect and the transition time effect. The Dymola model does not include any check valve dynamics, accumulator hysteresis, or fluid friction effects. The pressure drop across the accumulator orifice was also assumed to be negligible. The results of these simulations, as well as results obtained from an experimental setup, are presented in the following section.
2.3.2 Duty Ratio Dependent Loss The duty ratio dependent energy loss coefficient a2 was calculated by finding the power loss due to pressure drops encountered by the diverted flow, as well as non-load pressure drops encountered by the load flow. It was assumed that the dominant pressure drops would be due to the fully open PWM valve and the check valve. This leads to the power loss equation: Plost = ∆Ppwm (1 − s)
0
Figure 2.
dP dV
Z P high
6
4
Eq. (20) represents the energy required to compress the fluid in the inlet volume of the system from low pressure to high pressure. Using the definition of bulk modulus and performing a coordinate transformation on Eq. (20), the final equation for calculating the loss due to fluid compressibility is: β = −V
Vair=.004% (default) Vair=.1% Vair=.5% Vair=1% Vair=2.5%
10
Energy Lost (J)
Ecomp =
Energy lost per cycle due to fluid compressibility (Yu Model) 12
(24)
The first term in Eq. (24) represents the power loss from the diverted flow, and the second term the power loss from the load flow. Rearranging Eq. (24) yields a constant loss component, 5
c 2006 by ASME Copyright
5.5
6 x 10 System Response, 2 PWM cycles: PWM=10Hz, Duty Ratio=.5
Inlet Pressure Outlet Pressure
C
5 4.5
Pressure (Pa)
4
E
D
3.5 3 2.5 2 1.5 1
A
0.5
B 7.55
Figure 5. system
Figure 3.
7.6
7.65 Time (s)
Inlet and outlet presssures over
7.7
2
PWM cycles for a
10Hz
charge pressure of 4.1 × 106 Pa was used. In this setup, the pump was driven by an AC induction motor, which fixed the pump speed, eliminating the need for a large inertia. The hydraulic load used was an adjustable needle valve (H) and the inlet volume was about 40ml. The system also had pressure sensors (A, B, and E) on both sides of the check valve, and a flow meter (F) on the output line. The components chosen are not meant to be optimal but are used to determines the effects of different design parameters.
Dymola model
3.1
Figure 4.
System Operation A close-up view of the inlet and outlet pressures over 2 PWM cycles, for a switching frequency of 10Hz is shown in Fig. 5. The averaging effect of the accumulator can be clearly seen on this plot; between D and E, the accumulator is charging, and between C and D it is discharging. The difference between C and D (about 1.3 × 105 Pa) is the output ripple of the system. This is a characteristic of PWM controlled systems and cannot be completely eliminated. Between points A and B, the inlet pressure is low. The only load on the pump during this time is the loss across the fully open PWM valve, which was about 4.5 × 105 Pa. Between C and A, and again between B and D, the system is in transition. During these periods, the pressure is fairly high and the pump flow is going through the partially open on/off valve; this leads to a significant loss of energy.
Experimenal apparatus
3 Experimental Studies The experimental apparatus used for this paper is shown in Fig. 4. The PWM valve (C) was a linear proportional valve, which had a manufacturer specified transition time of 20ms. The fixed displacement gear pump (G) provided 5.68l/m, and components were sized for that flow. A nitrogen-filled diaphragm accumulator (D) with an initial gas volume of .16l and a pre-
3.2
Efficiency The efficiency of the hydraulic system was defined as the power out to the load divided by the power in from the pump. Therefore, any numbers presented here do not include any inefficiencies due to the pump or driving motor. The power into and out of the system was computed by multiplying the average 6
c 2006 by ASME Copyright
Power Lost in a PWM and a Proportional Valve System 350 1 Hz 2.5 Hz 5 Hz 10 Hz Proportional
300
Power Lost (W)
250
200
150
100
50
0
0
20
40 60 % of Flow Sent to the Load
80
100
Figure 6. Power loss in a PWM system at various PWM frequencies and a valve controlled system (marked proportional)
(a) Power lost plane
ouput flow rate over 5 seconds and either the measured inlet or outlet pressure, and then taking the average of the product over 5 seconds (Eq. (27)). R
Power =
Qave P(t)dt ∆t
(27)
The power loss was measured at various duty ratios and at frequencies of 1, 2.5, 5, and 10 Hz. The efficiency was also measured for the system with the valve configured for valve control. Figure 6 shows the results, which indicate a significant improvement in efficiency over valve control. The proportional data was obtained by using the PWM valve as a proportional valve in a bleed-off configuration. These results demonstrate the potential of PWM controlled systems in creating efficient hydraulic systems. The power lost in the PWM system was fitted to a plane described by Eq. (13), which is shown in Fig. 7. The plane is also rotated to show the quality of fit. Table 1 lists the values of the coefficients a0 , a1 and a2 , used to define this plane, as well as the coefficients determined from the predictive models described in section 2.3 and Dymola with 0.5% air in the oil. In order to use the predictive models to estimate a0 , a1 and a2 , a model of the spool transition characteristics was needed. xe (t), the spool position during transition in Eq. (19), was modified until the transition pressure profiles during opening and closing of the valve were consitent between the model and experimental data. It was found that using a half sine wave velocity profile for the spool matched extremely well with the experimental results. Figure 8 shows the pressure profile of the experimental data, the Dymola model and Eq. (16) using the half sine velocity profile. The pressure profile was created using the experimen-
(b) Rotated power lost plane Figure 7.
Plane representing the power lost as a function of frequency
and duty ratio
tally determined values of 4.83 × 106 Pa for Phigh , 40ms for topen and 20ms for tclose . Using these values and Eqs. (15), (16) and (17), the predicted energy loss due to the valve transition was 5.95J. From Table 1, a1 was determined experimentally to be 8.3J. Given that the loss due to valve throttling was found to be 5.95J, the loss due to fluid compressibility is expected to be roughly 2.3J. From Fig. 2, this energy loss corresponds to the compression of hydraulic oil with .5%-1% air by volume for a load pressure of 4.83 × 106 Pa. This is a reasonable estimate for this system. These results suggest that a majority of the energy loss per cycle is in fact due to PWM valve throttling and fluid compressibility. 7
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Predicted
Dymola
Experiment
a0 (W )
41.1
42.1
39.7
a1 (J)
7.6
5.5
8.3
a2 (W )
-38.5
-37.6
-20.1
Valve Transition: Plow−>Phigh
6
5.5
x 10
Experiment, PWM=5Hz Dymola, PWM=5Hz, wn=150Hz, .5% Air Simulation, V=A*sine(wt), Tclose=40ms
5 4.5 4 Pressure (Pa)
Coefficient
Values of a0 , a1 and a2 from simulation and experiment. The analytical and Dynmola models assume 0.5% of air in the fluid. Table 1.
3.5 3 2.5 2 1.5
The energy loss due to duty ratio effects was calculated using Eq. (25) and the values of 4.34 × 105 Pa and 2.76 × 104 Pa for ∆Ppwm and ∆Pcheck respectively. a2 was calculated to be −38.5W for a nominal flow of 9.5 × 10−5 m3 /s (1.5gpm). Interestingly, the calculated value of a2 is more negative than the experimental value of −20.1W given in Table 1. This result suggests that there are additional unmodeled pressure drops enountered by the load flow. This is most likely due to unmodelled losses across the check valve or in the accumulator. Another cause for this discrepancy is that topen and tclose were assumed to be zero in Eq. (24) which is an idealization. Lastly, a0 , the constant loss coefficient, was found to be nearly identical in all three cases. Since fluid friction was only present in the experimental case, the simulated results indicate that the effect of fluid friction on the system was insignificant. Since large diameter hoses and relatively low flow rates were used, this conclusion seems reasonable. Using the models described in this paper, several design goals for creating an efficient PWM system can be extracted. a2 can be decreased by using a PWM valve which minimizes loss in the full open state. This may not be as simple as increasing the orifice size, since an increased orifice will lead to a longer transition time, thus limiting the maximum PWM frequency of the system and possibly increasing a1 . a1 can be decreased by using a faster valve or decreasing the inlet volume. While several simplifying assumptions are still included in the system model, at low frequencies, the model appears to match the experimental data, indicating that the significant sources of energy loss have been included.
1 0.5 0 0.02
(1+1/γ)
0.035
0.04 0.045 Time (s)
0.05
0.055
0.06
Valve Transition: Phigh−>Plow
6
5.5
x 10
Experiment, PWM=5Hz Dymola, PWM=5Hz, wn=150Hz, .5% Air Simulation, V=A*sine(wt), Topen=20ms
5 4.5
Pressure (Pa)
4 3.5 3 2.5 2 1.5 1 0.5 0 0.01
0.02
0.03
0.04 Time (s)
0.05
0.06
0.07
(b) PWM valve opening
Figure 8.
Inlet pressure profile during PWM valve transition
the volume and pre-charge pressure of the accumulator have a similar effect on the system dynamics. This term is the volumetric stiffness of the accumulator, and it was shown in [1] that a low pre-charge pressure, or equivalently a small volume have the effect of increasing the speed of the system while also increasing the size of the ripple on the output. Thus a tradeoff exists between the response time of the system and the magnitude of the output ripple. Note that the pressure in the accumulator must be higher than the pre-charge pressure in order for any smoothing to take place. The effect of the PWM frequency in this performance tradeoff is evident from Fig. 10, where the PWM frequency has been increased to 10Hz. This experiment showed a rise time of 2.47s and a ripple magnitude of 2.0% of the output pressure. While the
System Performance Output ripple size and response times are two important aspects of system performance. The open-loop step response of the system was observed by increasing the average output flow rate from 50% of full flow to 60% of full flow. The response of the system, using a PWM frequency of 5Hz, is shown in Fig. 9. The pressure settles to its final value in 3.01s and the ripple magnitude (peak to peak) is 4.3% of the output pressure. P0 V0
0.03
(a) PWM valve closing
3.3
It is evident from the coefficient γ P 1/γ
0.025
in Eq. (6) that both 8
c 2006 by ASME Copyright
6
5.8
x 10
6
Open−Loop Step Response: 5Hz PWM Frequency 5.8
Closed−Loop Step Response: 5Hz PWM Frequency
5.6
5.6
5.4
Pressure (Pa)
5.4
Pressure (Pa)
x 10
5.2 5
5.2 5 4.8
4.8 4.6 4.6 4.4 4.4 4.2 4.2
0
5
10 Time (s)
Figure 9. Open-loop step response: Qout /Q f ull Frequency = 5Hz, trise = 3.01s, Ripple=4.3%
6
5.8
x 10
15
0
5
20
10 Time (s)
15
20
Figure 11. Closed-loop step response: Pdes = 4.52×106 Pa→5.65× 106 Pa, PWM Frequency = 5Hz, trise = .28s, Ripple=4.6%
= .511→.602, PWM
a pressure control algorithm with the duty ratio computed from the following equation:
Open−Loop Step Response: 10Hz PWM Frequency
5.6
s(t) = K f f
Pressure (Pa)
5.4 5.2
4.8 4.6 4.4
Figure 10.
0
5
10 Time (s)
15
20
Qout /Q f ull = .510→.608, = 2.47s, Ripple=2.0%
Open-loop step response:
PWM Frequency = 10Hz, trise
(28)
Where Pdes is the desired output pressure. The value of the feedback gain, K f b , was .025, which was the gain that provided the fastest response with no significant overshoot. The feedforward gain, K f f , was .0189 and was determined from a linearized model of the system. Values for Pdes were chosen to correspond to the initial and final pressures in the open-loop step response. The closed-loop step response for a system at 5Hz has a rise time of .28s and a ripple of 4.6%, demonstrating that closed-loop control can be used to improve the rise time without increasing the output ripple. This algorithm operates on controlling the pressure of the system, but feedback from a flowmeter on the load line can easily be used to create a flow-controlled system. An added benefit of closed-loop pressure or flow control is the ability of the control algorithm to compensate for the effect of load dynamics. Since the duty ratio under closed-loop control is constantly varying and determined by the control algorithm, the system can adapt continuously to changes in the system load. To demonstrate the ability of the system to track a pressure reference, a sinusoidal function was added to the nominal pressure, with the results shown in Fig. 12. The system tracked a .5Hz reference with little attenuation or phase shift. As the frequency of the reference was increased, the control signal began to saturate. Once the duty ratio is above 1 or below 0, the controller cannot improve the speed of the system, and bandwidth is limited by the dynamics of the accumulator. This lead to significant signal attenuation at frequencies of 1Hz and above.
5
4.2
p Pdes + K f b (Pdes − Pout (t))
rise time actually decresed slightly, the output ripple was reduced by more than half. This supports a conclusion of [1]: increasing the PWM frequency decreases the ripple size without increasing the rise time. This indicates that, while increasing the frequency might have a penalty on the system efficiency, it can also dramatically improve performance by reducing the output ripple. 3.4
Closed-Loop Control Li et al. [1] also proposed that adding closed-loop control to the system could improve this tradeoff by increasing the rise time without increasing the output ripple. This is shown experimentally in Fig. 11. In this experiment, the system was run with 9
c 2006 by ASME Copyright
6
5.4
x 10
Acknowledgment This material is based upon work supported by the National Science Foundation under grant number ENG/CMS-0409832.
Closed−Loop Sine Tracking: 5Hz PWM Frequency Reference Pressure Output Pressure
5.2
Pressure (Pa)
5
REFERENCES [1] P. Li, C. Li and T. Chase, “Software Enabled Variable Displacement Pumps” Proceedings of the 2005 ASME-IMECE, no. IMECE2005-81376, 2005. [2] B. Lehman and R. Bass, “Switching frequency dependent averaged models for pwm dc-dc converters,” IEEE Transactions on Power Electronics, vol. 11, no. 1, pp. 89–98, 1996. [3] L. Gu, M. Qiu, W. Feng, and J. Cao, “Switchmode Hydraulic Power Supply Theory,” Proceedings of the 2005 ASMEIMECE, no. IMECE2005-79019, 2005. [4] E. J. Barth, J. Zhang, and M. Goldfarb, “Control Design for Relative Stability in a PWM-Controlled Pneumatic System,” Journal of Dynamic Systems Measurement and ControlTransactions of the ASME, vol. 125, no. 3 pp. 504–508, 2003. [5] H. M. Paynter and E. P. Fahrenthold, “On the Nonexistence of Simple Polytropes and Other Thermodynamic Consequences of the Dispersion Relation,” ASME Bioengineering Division Publication, vol. 5, pp. 37–42, 1987. [6] N. Mohan, T. Undeland, and W. Robbins, Power electronics: converters, applications and design. John Wiley and Sons, 3rd ed., 2003. [7] Industrial Hydraulics Manual. Eaton Corporation, 4th ed., 2001. [8] J. Yu, Z. Chen, and Y. Lu, “The Variation of Oil Effective Bulk Modulus with Pressure in Hydraulic Systems,” Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, vol. 116, no. 1 pp. 146–150, 1994. [9] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part I - Nonlinear Mathematical Model,” Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, vol. 122, no. 3 pp. 416– 425, 2000. [10] N. Luo, F. Fronczak, and N. Beachley, “Comparison of Analytical and Experimental Investigations of a Hydraulic Multi-Circuit Sequential Apportioning System,” Society of Automotive Engineers Transactions, vol. 99, pp. 266–275, 1990.
4.8
4.6
4.4
4.2
0
2
4
6
8
10
Time (s)
Figure 12.
Closed-loop reference tracking: PWM Frequency = 5Hz
4 Conclusion Experimental validation of the feasibility of a softwareenabled variable displacement pump has been presented. It was shown that an on/off valve, check valve, and an accumulator can be used with a fixed-displacement pump to create a system with the functionality of a variable displacement pump. While a tradeoff between maximizing the system response and minimizing the output ripple will always exist, it was shown that increasing the PWM frequency and adding closed-loop control can dramatically improve the system performance. The efficiency of the proposed system was compared to that of a proportional valve, demonstrating the potential energy savings over valve control. While the PWM system dramatically improved the system efficiency, energy is still lost due to several effects, some of which are frequency dependent. Increasing the PWM frequency has the benefit of decreasing the output ripple and increasing the control bandwidth of the system. However, as higher frequency PWM systems are designed, attempts must be made to minimize the frequency dependent energy loss. The valve transition time is a significant source of energy loss which can be minimized through the design of faster on/off valves. In the system presented in this paper, the transition time of the valve was nominally 20ms, which lead to significant energy loss at 10Hz, and did not allow the system to function above 14Hz. The oil volume on the inlet side of the system also contributes to the inefficiency of the system. The volume in this paper was about 40ml, which lead to a significant loss of energy. This effect increases linearly with PWM frequency and inlet volume. This indicates the need to minimize the inlet volume as frequency is increased. Finally energy is lost in the dynamics of the accumulator and check valve. These effects were not examined in this paper and need to be fully investigated. 10
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