Optimal Design of a High-Speed On/Off Valve for a ...lixxx099/papers/Rannow_Li_optimization_Aachen 2010.pdfOptimal Design of a High-Speed On/Off Valve for a Hydraulic Hybrid Vehicle

Optimal Design of a High-Speed On/Off Valve for a Hydraulic Hybrid Vehicle Application

Michael Rannow

Perry Li

Thomas Chase

Haink Tu

Meng Wang

Center for Compact and Efficient Fluid Power, University of Minnesota, Minneapolis, MN, USA

ABSTRACT

Control of hydraulic systems using high-speed on/off valves has been proposed as a way to

avoid the inefficiency associated with throttling valves. However, on/off control has sources of

energy loss that must be considered when designing a system, such as transition throttling, full-

open throttling, compressibility, and leakage. A self-spinning rotary on/off valve achieving high

switching frequencies with low actuation power has been developed. However, the design of this

valve contains numerous tradeoffs that must be balanced to produce the most efficient system.

This paper outlines energy loss equations and design constraint equations that are needed to

apply optimization techniques to the valve design. The results of this optimization are presented

for a Virtually Variable Displacement Pump, and a Virtually Variable Displacement Pump/motor,

which is used as the wheel motor of a hydraulic hybrid vehicle.

7th International Fluid Power Conference Aachen 2010

1

NOMENCLATURE

Parameter Description 3-way Value 4-way Value Units

ρ Fluid Density 876 same kg/m3

Q Flow Rate 37.9 varies l/m

P Load Pressure 69 207 bar

Μ Fluid Viscosity 0.0387 same kg/ms

α Ratio of Shear Stress 0.2 same none

cd Orifice Coefficient 0.6 same none

βeq Fluid Bulk Modulus 3.7 x 108 1.2 x 109 Pa

Pc Check Valve Cracking Pressure 3.2 N/A bar

kspool Spool Coefficient 6.0 x 10-7 1.52 x 10-7 none

mspool Spool Flow Exponent 1.735 same none

nspool Spool Diameter Exponent 2.75 same none

Pr Relief Valve Pressure N/A 228 bar

ln Inlet Nozzle Length 4.2 9.5 mm

Dd Inlet/outlet Rail Diameter 6.8 same mm

Vin Inlet/line Volume 5 7 cm3

lmid Middle Land Thickness 1.5 same mm

ri,off Inlet Offset Fraction 0.79 same none

ro,off Outlet Offset Fraction 0.9 same none

fmin/fmean,min Minimum Frequency 75 35 (mean) Hz

de Pump/motor Displacement N/A 10 cc/rev

rm Engine Gear Reduction N/A 6.0 or 20.3 none

ωsw Transmission Shift Speed N/A 550 rpm

Vspool Fluid Volume inside Spool N/A 7 cm3


2

Figure 1: Diagram of 3-way rotary spool

1 INTRODUCTION

Hydraulic actuation systems are widely used due to their flexibility, durability, and high

power density. However, they also tend to be inefficient, primarily due to the use of

throttling valves for control. Valve control varies the output by restricting the fluid and

passing unwanted flow across valves at high pressure, generating heat and energy loss.

A different approach in hydraulic systems is to use a variable displacement pump for

each actuator, which has the potential to be more efficient. However, variable pumps

tend to be bulky and expensive which limits the spread of this control technique.

A different option for avoiding inefficient throttling valves is the use of high-speed on/off

valves and energy storage elements. Tomlinson and Burrows [1] and Li, Li, and Chase

[2] demonstrated the feasibility of using an on/off valve to vary the flow rate of a fixed-

displacement pump, and Rannow et al. [3] quantified the benefits of an on/off system in

comparison to a throttling system. Cao et al. [4] used a conventional on/off valve to

create hydraulic transformers, and Brown et al. [5] developed a high-speed, motor-

driven rotary on/off valve for use in a transformer.

The high-speed rotary on/off valve in Fig. 1 uses fluid forces to provide rotary actuation

[6]. This is a 3-way valve that directs flow from the center to either the top (load) or the

bottom (tank) section. Flow enters the center section through a rhombus inlet, and the

spool can move axially with respect to the inlet. The on/off valve pulse-width-modulates

(PWM) the fluid, with the duty ratio adjusted by the axial position.

This valve utilizes constant,

unidirectional motion to achieve the high-

speed on/off action. This reduces actuation

power by eliminating the acceleration and

deceleration of the spool in traditional valves.

Since fluid momentum drives the valve,

energy lost as a pressure drop across the

valve is scavenged.

Despite the reduction in acutation power

required, sources of energy loss still exist.


3

Figure 2: VVDP circuit

Figure 3: Pressure Profile for the VVDP

This paper quantifies these losses and

formulates an optimization problem to minimize

them, subject to constraints. This is done for a

3-way valve, and extended to a 4-way valve that

is currently being developed to be used with a

fixed-displacement pump/motor to create a

variable pump/motor for use in a hydraulic hybrid

vehichle. The optimization problem for the 4-way

valve will minimize the energy loss when the valve is used in a hydro-mechanical

transmission (HMT) driving the Urban Dynamometer Driving Schedule (UDDS).

2 VIRTUALLY VARIABLE DISPLACEMENT PUMP

The valve in Fig. 1 can be used in the circuit in Fig. 2. This configuration allows a

variable flow rate without a variable displacement pump. The on/off valve sends the full

flow to the load when it is on and to the tank when it is off. The accumulator stores

excess flow when the valve is on, and drives the load it is off.

On/off valves are efficient because the

pressure drop across the valve is low when

the valve is in either position (Fig. 3). In this

figure, the pressure drop is usually low;

however, during transition between on and

off, the pressure drop is high. Throttling due

to the full-open pressure drop and transition

throttling are two of the four sources of loss

in the valve. Energy is also lost from the

compressibility of the hydraulic fluid and

leakage across the barrier between the load and tank ends of the spool.

The valve design can be cast as an optimization problem, minimizing energy loss while

satisfying constraints:

transitionleakagecompressopenfulltotalx

LLLLL =min , 0)(s.t. 1 xc , 0)(2 xc , 0=)(3 xc , maxmin xxx (1)


4

Figure 4: Spool geometry with unwrapped center

where )(1 xc , )(2 xc , and )(3 xc are the constraint functions, and L is cycle-averaged

power loss. The parameter vector is given by endouthw lANclDRRx ,,,,,,,= , representing

the inlet orifice width ( wR ) and height ( hR ), the spool diameter (D ) and length ( l ), the

radial clearance ( c ), the number of PWM sections ( N ), the outlet area ( outA ), and the

length of the spool ends ( endl ).

These variables are incorporated

trade offs between the different

losses and constraints. Figure 4

shows relevant design parameters.

2.1 FULL-OPEN LOSS

The full open loss is caused by the

pressure drop across the inlet nozzle,

valve spool, and outlet turbine. The

inlet and outlet pressure drops are calculated using the orifice equation:

2

2

2

2

==outd

out

ind

inNAc

QP

NAc

QP

(2)

where hwin RRA 0.5= is the area of one inlet orifice.

The pressure drop across the spool results from the path through the turbine blades,

and the dependence of this path on the parameters is not obvious. Approximations are

to model the pressure drop as an orifice (2QP ), or pipe flow ( QP ). Since the true

pressure drop will be a result of both effects, a better model would combine the two;

spoolmQP , with [1,2]spoolm . If it is assumed that the equivalent area is dependent only

on the diameter, the pressure drop is given by:

spooln

spool

spoolm

spoolDk

QP = (3)

where [2,4]spooln defines the dependence of the pressure drop on the spool diameter.

The values of spoolm and spooln can be found through CFD analysis. The full-open loss is:


5

spooloutinopenfull QPQPQPL 1= (4)

where is the fraction of the rotation that the valve spends in transition, given by:

D

NRw

4= (5)

2.2 COMPRESSIBILITY LOSS

Energy is lost by compressing the fluid and then connecting the compressed fluid to

tank. The energy in the compressed fluid is given by Eq. (6) [3].

dPP

PVE highP

lowPcompcomp )(=

(6)

where )(P is the bulk modulus of the fluid, and Vcomp is the inlet volume:

innd

dncomp VNVD

DlDN

NV

4

21

=22

(7)

The inlet volume consists of the fluid volumes in the inlet pressure rail, the inlet nozzles

( ninn lAV = ), and volume between the inlet rail the pump ( inV ).

A common model for )(P is given by Yu, Chen, and Lu [7].

)10/(10)10(1

)10(1=)(

55)1/(15

)1/(15

PRP

PP

oil

oil

(8)

where oil is the bulk modulus of air free oil, is the ratio of specific heats for air, and

R is the fraction of air in the oil. Equations (6) and (8) can be integrated prior to the

optimization. Equation (6) can be replaced by a term using a constant bulk modulus:

eq

compcomp

PVE

2=

2

(9)

where eq is computed by equating Eqs. (6) and (9). With being the angular velocity

of the valve spool, the power loss due to the compressibility of the fluid is given by:

eq

compcompress

NPVL

4=

2

(10)


6

2.3 LEAKAGE LOSS

The 3-way spool has 2 leakage paths: the largest is across the barriers between the

high and low pressure sections; the other is across the land at the high pressure spool

end. Leakage across both paths is modeled as laminar flow between 2 plates [6].

The pressure across the center barrier is P , but the ends of the spool are connected to

a common chamber. Assuming no external leakage and identical ends, leakage from

the high pressure side equals leakage into the low side, and the chamber pressure is

P/2. The power loss across the two paths is given by:

endw

pleakage l

PcD

R

PclL

122

)(sin12=

23

23

(11)

where pl is the length of the path along the barriers, and is the angle of the barriers.

2.4 TRANSITION LOSS

The equation for the transition loss is derived by integrating the orifice equation

for each of the 4 transitions that occur each PWM period [6]. The loss is given by:

inc

c

cintransition PP

PP

PPQPL 2

1.52

2=

(12)

2.5 CONSTRAINTS

)(1 xc is a bound on the spool speed. The PWM frequency has a detrimental effect on

the efficiency, but a high frequency is desirable for other reasons. On/off control causes

a ripple, and a trade off exists between the system speed of response and ripple

magnitude. This can be improved by increasing the PWM frequency ([2],[3]). The bound

is selected using this consideration and is an input to the optimization:

2

=)(1

Nfxc min (13)

The velocity is derived from a torque balance between the viscous friction and the

angular momentum captured by the spool turbines [6]:


7

Figure 5: VVDPM circuit

outoffi

offo

inoffoeff

offi

Ar

r

AQr

cA

DN

DrQ

,

,

2,

2

,2

1

2

2=

(14)

To generate angular momentum, the inlet and outlet are offset from the spool center. In

Eq. (14), offir , and offor , correspond to these offset fractions. effA is the effective area:

endmidhmidendheff llRlDllRDA 22)2(= (15)

The first term is the bearing area at clearance, c . The next term is the rest of the spool,

modified by , which captures the effect of fluid swirl. In [6], CFD is used to estimate

)(2 xc states that κ must be less than one:

1=)(2 xc (16)

)(3 xc arises from the fact that the rhombus must have the same angle and width

as the helical land for a closed-center valve.

Nl

D

R

Rxc

th

w 2=)(3 (17)

where hendmidt Rllll /322= is the length of axial travel.

Equation (1) specifies variable bounds on the vector x . minx and maxx contain

boundary values for each parameter necessary to keep parameters in a feasible range.

1 VITUALLY VARIABLE DISPLACEMENT PUMP/MOTOR

The VVDP concept can be extended to create

a variable pump/motor (P/M) by adding a 4-way

on/off valve to a fixed-displacement P/M (Fig.

5). The VVDPM has a 4-way tandem center

rotary on/off valve and two 4-way valves. The

valve on the left switches between pump and

motor mode, and the valve on the right changes the direction of the shaft. With these

valves, the system is a 4-quadrant P/M. The direction switching valve keeps the flow

direction through the on/off valve constant to achieve self-spinning.

The 4-way valve is still being designed but works on similar principles to the VVDP. The

tandem center allows the P/M to freewheel and is the off position. A difference between


8

the designs is that the flow rate in the 4-way

valve varies, meaning that optimizing at one

PWM frequency is not meaningful.

For the 4-way valve, the problem is

formulated for an application: the P/M will be

designed to be the wheel motor of an HMT,

shown in Fig. 6. The P/M at the wheels

(Pump E) are realized by the VVDPM. The

flow rate through the valve is defined by the

drive cycle. For this paper, the VVDPM will be optimized for a vehicle driving over the

UDDS.

In this configuration, Pump E connects to the engine and wheel shafts through a

planetary gear set, specifying the torque and speed relationships between the 3 shafts:

w

m

enge r

r

2= rw

e

= (18)

where e is the pump speed, eng is the engine speed, w is the wheel speed, e and

w are the pump and wheel torques, and r is the reduction between the planetary and

pump/motor. mr has 2 gears, and switches when the wheel speed exceeds a threshold.

Ideally, the engine operates at its ideal speed, specifying eng . Since w and w are

specified, Eq. (18) defines the torque and speed of the P/M and the flow rate:

2= eedQ (19)

The absolute value results from the switching valve ensuring flow through the valve is

unidirectional. The P/M torque specifies the duty ratio for the valve:

rPdN

srPdN

sew

ep

ew

em

2=

2= (20)

where wN is the number of driven wheels on the vehicle. ms and ps are the motor and

pump duty ratios. Pump or motor mode is determined by the sign of the power demand

on the P/M, eeep = . Positive power implies motor mode, and negative implies pump.

The optimization problem minimizes the energy loss over the UDDS:

Figure 6: Architecture of an HMT


9

dtLtotalUDDSxmin , 0)(s.t. 1 xc , 0)(2 xc , 0)(3 xc , 0=)(4 xc , maxmin xxx (21)

where a new parameter, r, is optimized: rlANclDRRx endouthw ,,,,,,,,= .

2.6 FULL-OPEN LOSS

spooloutinopenfull QPQPsQPsL 111= (22)

where 1,0][s for pump mode and [0,1]s for motor mode. The s1 term results

from there being either 1 or 2 sets of turbines active, depending on the on/off state.

2.7 COMPRESSIBILITY LOSS

The 4-way valve does not connect pressurized fluid tank when switching, it connects the

sides of the P/M together. One side transitions between high pressure and some mid-

point, fwP , the other transitions between fwP and tank. The loss results from the

pressure falling from fwP to tank. High pressure fluid in one chamber raises the

pressure in the other when connected. fwP is computed from an energy balance:

PdPV

PdPV fwP

eq

P

fwPeq 0

21 =

(23)

where 1V and 2V are the volumes on either side. In motor mode (opposite for pump

mode), the volumes are given by:

inspooldd VVDDD

V

4=

22

1

,

inn

ddn VNVN

DDlDNV

4

21=

22

2

(25)

The compressibility loss is calculated from:

21

221

4=

VV

NPVVL

eqcompress

(26)

2.8 LEAKAGE LOSS

The 4-way valve has the same leakage paths, but, rather than being constant, the

pressure alternates between P and fwP . The duty ratio sets the cycle-average leakage.

Equation (27) gives the leakage in pump and motor modes.


10

end

fwpp

w

fwppppleakage l

PPs

PsDc

R

PPsPsclL

12

2)(1

2

)(sin12

))((1=

223

223

,

end

fwmm

w

fwmmpmleakage l

Ps

PsDc

R

PsPsclL

12

2)(1

2

)(sin12

)(1=

223

223

,

(27)

2.9 TRANSITION LOSS

The transition loss for the 4-way valve is derived in the same way as it is for the 3-way

valve and given for the pump and motor modes by:

)31.5(22

=, in

fwr

rfwrr

inptransition P

PP

PPPPP

PQL

)20.5

(2

=, in

r

r

fwr

rrfwr

inmtransition P

PP

PP

PP

PPPPP

PQL

(28)

2.10 CONSTRAINTS

)(1 xc is similar to the first constraint above. However, since the flow rate varies, the

PWM frequency varies. Thus, a bound is placed on the mean PWM frequency:

2

=)( ,,1

Nfxc meane

minmean (29)

where meane, is averaged over the drive cycle. Since there can be 1 or 2 sets of turbines

active, the cycle-averaged speed is calculated using the duty ratio:

outoffi

offo

inoffoeff

offi

e Ar

r

AQr

cA

DN

DrQs

,

,

2,

2

,2

1

2

21

=

(30)

The second inequality constraint is the same as )(2 xc for the 3-way valve. The third

inequality constraint results from having r as a variable; in order to meet the maximum

torque, rde must be large enough, given P. This results in the constraint:

2

=)(3e

w

max Prd

Nxc (31)


11

Figure 7: Power loss comparison in a 3-way valve

Table 1: Optimization results for the 3-way valve at low and high pressures

where max is the maximum torque required and wN is the number of driven wheels. The

final equality constraint, )(4 xc , is the same as the equality constraint in Eq. (17).

3 OPTIMIZATION RESULTS

The optimization software used is part of the Optimization Toolbox in MATLAB. The

function fmincon.m is a constrained nonlinear optimization algorithm using Sequential

Quadratic Programming with approximated gradients.

The optimization for the 3-way valve improved the design of an existing prototype. Table

1 shows the values from the current prototype and the optimal values.

x wR hR D l c outA endl

Units mm mm cm cm m 2mm mm

Initial 4.1 5.9 2.54 8.6 20 26.8 3.2

Optimal 6.3 10.8 2.39 10.39 29 22.5 1.9

Opt. - H. P. 5.2 11.7 1.99 11.17 16.8 15.9 1.8

Notice that N is not included in the list, as was originally formulated. It was decided that

having 3=N cuts provided significant benefit by balancing the spool. The initial guess

and the optimal values are for a system at bar69 . For the low pressure optimal values,

the power lost due to transition was the largest at W329 , followed by full-open at W152 ,

leakage at W78 , and

compressibility at W40 . Figure 7

compares these values with the

prototype values. The total loss is

decreased by ~ 23% , primarily

through a reduction in the

transition losses. The full-open

losses are reduced, while the

other losses are increased slightly. Notice the larger orifice area in the optimal case,

which reduces in the full-open and transition losses. This results in less torque

0

100

200

300

400

500

600

700

800

900

Full Open Compress. Leakage Transition Total

Pow

er Loss

(W)

Initial

Optimal


12

Table 2: Optimization results for the 4-way valve

Figure 8: Magnitude of the 4 valve losses on the

UDDS

generated, increasing c and decreasing Ao. Balancing these effects optimally is difficult

without optimization.

The optimization was also run at a higher pressure ( bar207 ), with the results shown in

table 1. High pressures increase leakage and compressibility, and the optimal values

indicate this; c is smaller, and the spool is longer and thinner to shrink Vin. Additional

friction generated by the tighter clearance is balanced with a smaller outlet area.

x wR hR D l c outA endl r

Units mm mm cm cm μm mm2 mm -

Optimal 3.3 7.7 1.27* 9.6 13 2.79 1.6* 4.6

The results for the 4-way valve are given in Table 2. A star indicates the value is hitting

the variable bounds; D and endl

are driven to the minimum

value. These bounds are

estimates of what is feasible.

The 4-way spool is designed to

run at high pressure, which the

optimal values reflect. The

clearance is small, and the

spool is long and thin as in the

3-way case. The outlet area is

smaller than in the 3-way valve

to make up for the small

clearance and the lower, variable flow rate.

CONCLUSION

Controlling hydraulic systems using high-speed on/off valves has shown promise for

reducing valve energy loss, but several mechanisms of energy loss must be considered.

A self-spinning rotary on/off valve has been proposed as a way of achieving high

switching frequencies without significant actuation power. However, the geometry and

self-spinning property create trade offs in minimizing the the full-open loss,


13

compressiblilty loss, leakage, and transition throttling. Power loss and constraint

equations for the valve have been presented for a 3-way valve acting as a VVDP, and a

4-way valve acting as a VVDPM in an HMT. The equations have been formulated to be

solved using optimization software. For the 3-way valve, the optimization procedure

achieved a 23% improvement in efficiency over a prototype designed using ad hoc

methods. The procedure applied to the 4-way valve resulted in a valve with a cycle

efficiency of 79% .

ACKNOWLEDGEMENT

This material is based upon work performed within the ERC for Compact and Efficient

Fluid Power, supported by the National Science Foundation under Grant No. EEC-

0540834.

REFERENCES

[1] S. P. Tomlinson and C. R. Burrows, 1992, ``Achieving a Variable Flow Supply by

Controlled Unloading of a Fixed-Displacement Pump,'' ASME Journal of Dynamic

Systems Measurement and Control, vol. 114, no. 1, pp. 166-171.

[2] P. Li, C. Li and T. Chase, 2005, ``Software Enabled Variable Displacement Pumps'',

Proceedings of the 2005 ASME-IMECE, Paper No. IMECE2005-81376.

[3] M. Rannow and H. Tu et. al.,``Software Enabled Variable Displacement Pumps -

Experimental Studies,'' Proceedings of the 2006 ASME-IMECE

[4] J. Cao, L. Gu, F. Wang, and M. Qiu, 2005, ``Switchmode Hydraulic Power Supply

Theory,'' Proceedings of the 2005 ASME-IMECE, Paper No. IMECE2005-79019.

[5] F. Brown, S. Tentarelli, and S. Ramachandran, ``A Hydraulic Rotary Switch-

Inertance Servo-Transformer,'' ASME Journal of Dynamic Systems, Measurement, and

Control, vol. 110, June 1988, pp. 144-150.

[6] H. Tu and M. Rannow et. al., ``High Speed Rotary Pulse Width Modulated On/Off

Valve'' Proceedings of the 2007 ASME-IMECE, Paper No. IMECE2007-42559.

[7] J. Yu, Z. Chen, and Y. Lu, 1994, ``The Variation of Oil Effective Bulk Modulus with

Pressure in Hydraulic Systems,'' ASME Journal of Dynamic Systems Measurement and

Control, vol. 116, no. 1, pp. 146-150.


14

Documents

Optimal Design of a High-Speed On/Off Valve for a ...lixxx099/papers/Rannow_Li_optimization_Aachen 2010.pdfOptimal Design of a High-Speed On/Off Valve for a Hydraulic Hybrid Vehicle