Dynamic Analysis of Fluid Power Drive-trains for Variable
Speed Wind Turbines: a Parameter Study
A. Jarquin Laguna, N.F.B. Diepeveen
TU Delft, The Netherlands
E-mail:a.jarquinlaguna@tudelft.nl,n.f.b.diepeveen@tudelft.nl
Abstract
In the pursuit of making wind energy technol-ogy more economically attractive, the application of fluid power technology for the transmission of wind energy is being developed by several par-ties all over the world. This paper presents a dynamic model of a fluid power transmission for variable speed wind turbines and shows a para-metric study on the dynamic behaviour below rated wind speed. A pressure control strategy is proposed to achieve a variable speed operation. The rotor of the NREL 5 MW reference turbine is used to perform time domain simulations. Dif-ferent values of the hydraulic line length, trans-mission efficiency and rotor mass moment of in-ertia are considered for the same wind condi-tions. The results show that the amount of oil in the system has a relatively large influence in pressure transients and controllability. Lower-ing the volumetric efficiency of the hydraulic mo-tor leads to more damping of this pressure fluc-tuation, however its influence is minor and un-likely to be advantageous when compared to the power loss. A higher rotor mass moment of in-ertia implies a slower but smoother response of the system.
Keywords: hydraulic transmission, wind
en-ergy, fluid power.
1
Introduction
1.1
Background
As an alternative to geared or direct drive power transmissions, hydraulic power transmission is particularly attractive for large offshore wind tur-bines due to its compactness and the option of a continuous variable transmission ratio.
Having a more compact and light transmission means less mass at the top of the tower. This
mass reduction leads to a significant reduction in the required amount of support structural steel. Furthermore, a continuous variable transmission ratio eliminates the necessity of a voltage fre-quency converter.
So far most of the (pre)commercial concepts go for a solution where all the drive train is placed in the nacelle, see figure 1. However, the mod-ularity and flexibility of hydraulic systems gives rise to the option of further weight reduction by relocating the hydraulic motor and generator from the nacelle to the bottom of the tower.
Figure 1: Different concepts of fluid power trans-missions according to the location of compo-nents - Nacelle solution (A) and Tower base so-lution (B)
A comparative study has shown that, for a 5MW turbine with only a pump in the nacelle, the mass reduction of the nacelle with respect to a geared transmission is up to 66%. Using the same rotor, this leads to a reduction in top mass of up to 50%. This resulted in a reduction of more than 24% of the amount of steel required for the support structure [1]. The aforementioned research shows how this result stretches the
ap-plicability of the monopile for deeper waters and larger turbines. But how feasible is this concept and what are its limitations from an operational point of view?
The further the hydraulic motor is placed from the pump, the greater the amount of oil in the system and the greater the friction losses in the pipelines.
Internal leakages in the hydraulic pump & mo-tor are also present and imply power losses. Note that lower efficiencies mean increased damping and are therefore not necessarily an unfavorable aspect for the dynamic performance of the turbine.
1.2
Objective
The objective of the research presented in this paper is to determine to which extent specific parameters have an effect on the behavior of a variable speed wind turbine with a fluid power transmission system. The results demonstrate the influence of the parameters (1) the length of hydraulic line, (2) the volumetric efficiency of hy-draulic motor and (3) the rotor inertia on the dy-namic performance of a horizontal axis wind tur-bine.
1.3
Approach
The rotor of the NREL 5MW baseline turbine [2] is used as starting point to build a numerical model of a wind turbine with hydraulic transmis-sion. This fictitious turbine has become the stan-dard reference turbine for numerous academic studies mainly because, in contrast to existing turbines, all its properties are freely available. The relevant properties of the rotor are listed in Appendix C.
The drive train consists of a low speed, fixed displacement radial pump (directly coupled to the rotor), a fluid transmission line and a vari-able displacement hydraulic motor coupled to a fixed-speed synchronous generator.
The fluid power transmission system is de-signed based on the same operational param-eters as the reference rotor.
Based on validated theory on the behavior of hydraulic systems, a state-space model of the fluid power transmission system and its con-troller is presented.
In order to obtain a realistic response of the hydrostatic transmission, accurate simulations of the aerodynamic rotor and wind conditions
are required. The numerical model created in MATLAB-Simulink is converted to a Dynamic Link Library (DLL) which is implemented as an external controller for the model of the NREL 5MW reference turbine in the commercial soft-ware package GH Bladed. The multi-body dy-namics software package Bladed from energy consultancy GL Garrad Hassan is the industry standard integrated software package for the de-sign and certification of onshore and offshore turbines [3].
An overview of how the simulation of the rotor and drive train is configured is given in figure 2.
Figure 2: GH Bladed interface with the external controller.
1.4
Assumptions and Conditions
• The effect of the different configurations of
the hydraulic transmission system on the total mass of the nacelle is not taken into account (same structure and top mass is used)
• The diameter and number of hydraulic lines
were selected in order to have laminar flow in the hydraulic transmission
• The pipe walls are assumed to be rigid (for
elastic pipes, a modified sound speed is used to include the elasticity effect)
• Thermodynamic effects are negligible • Properties of the hydraulic fluid are constant
2
Rotor & Drive Train Models
2.1
Aerodynamics
Although the aerodynamic and elastic character-istics of the rotor blades are obtained through GH Bladed based on the Blade Element Momen-tum method (BEM) [3], the general static rela-tionships of the aerodynamic torque, thrust and power of the rotor are described in Appendix A.
2.2
Fluid power transmission
2.2.1 Hydraulic pump & motor
The primary function of hydraulic machines is to convert mechanical energy into hydraulic energy and vice versa. The modeling of these hydraulic system components is done as described in [4]. Pumps and motors are characterized by a vol-umetric displacement Vp, which describes the
amount of volumetric fluid obtained per rotational displacement of the driving shaft. The net gen-erated flow from a pumpQpis expressed as:
Qp= Vpω |{z} Qideal − Cv,p∆pp | {z } Qs (1)
∆pp [Pa] pressure difference over pump
ω [rad/s] rotation speed of pump
Cv [-] laminar leakage coefficient
Qideal [m3
/s] ideal flow from pump
Qs [m3/s] slip flow due to internal
leakages
Vp [m3/rad] volumetric displacement
For the purpose of this study and for simplicity of analysis, a simple model is assumed where the total leakage flowQsis directly proportional
to the pressure difference across the hydraulic drive∆p[5]. The effective torque of the pumpτp
is modeled as: τp= Vp∆pp | {z } τideal + Bpω | {z } τd + Cf,pVp∆pp | {z } τf (2) τd [Nm] damping torque τf [Nm] friction torque
τideal [Nm] ideal torque
Bp [Nms/rad] viscous damping coefficient
Cf,p [-] Coulomb friction coefficient
The damping torqueτd is a torque loss required
to shear the fluid in the small clearances be-tween mechanical elements in motion. It is in-dependent of the load and is assumed to be pro-portional to the pump speed and viscous damp-ing coefficientBp[6].
Friction torque τf simulates the effect of dry
friction forces on the pump pistons that oppose their motion. The resulting friction torque is pro-portional to the volumetric displacement and the pressure difference across the hydraulic drive. For a quasi-static analysis a constant Coulomb friction coefficientCf is defined to describe the
friction torque independently of the speed [7].
2.2.2 Hydraulic lines
The dynamics of the pressure and flow through a hydraulic line are described through a modal ap-proximation of the 1D dissipative model of lam-inar viscous flow. The model is derived from a variational method of the distributed parameter pipeline model as described by [8]. The
pro-Figure 3: Distributed parameters pipeline repre-sentation
posed distributed parameter model is a two-port model based on volumetric flow rates and pres-sures at the upstream and downstream side of the pipeline.
The great advantage of this model is that it allows to describe viscous effects in the friction term in comparison with other models. Linear friction models and lumped parameter models use a steady-state friction term which under-estimates dissipation losses in the transient re-sponse. The difference is especially important for pipelines with dissipation numbersDngreater
than 0.001 [9].
The acoustic velocity in the fluid c0 and the
dissipation numberDnare obtained through the
pipeline parameters: c0= s E ρ, Dn= νL r2c 0 (3) Some additional pipeline parameters are defined as: Z0= ρ c0 πr2, T = L c0 , ǫ = 8νL r2c 0 (4)
L [m] transmission line length
r [m] transmission line internal
radius
ρ [kg/m3] fluid density
ν [m2/s] fluid kinematic viscosity
E [Pa] fluid bulk modulus
c0 [m/s] acoustic velocity in the fluid
Dn [-] dissipation number Z0 [Pas/m3 ] characteristic impedance of transmission line T [s] wave period ǫ [-] dimensionless friction coefficient
The impedance model of a pipeline is defined when the inputs of the model are specified as the flow rates at the two ends of the pipeline. The outputs from the model are the pressures at the upstream and downstream side of the pipeline. The linear dynamic equations of the proposed model are rewritten in state-space form accord-ing to [8]:
˙x = Ax + Bu
y= Cx (5)
Here, x is the state vector, u is the input vec-tor andy is the output vector. For the hydraulic line with flow rates as inputs these vectors are defined as: x= p0 r1 p1 .. . rn pn , u= Qin Qout y= ∆pp ∆pm (6)
The matricesA,BandCare defined as:
A= diag 0, A1, A2, . . . An , Ai= 0 −ωi T2 1 −ǫi T (7) B= B0 B1 .. . Bn (2n+1)×2 B0=ZT0 1 −1 B2i−1= 2Z0Tw22i−1 ǫ bn ǫ bn T T B2i=2Z0T2w2i ǫ −ǫ T −T (8) C= 1 0 1 0 1 0 . . . 1 0 −1 0 −1 0 . . . 2×(2n+1) (9) The modal natural frequencyωi and the modal
damping coefficientǫiare given by:
ωi= αi− 1 4√αiǫ + 1 16ǫ (10) ǫi= 1 2√αiǫ + 1 8ǫ (11) αi= iπ, i = 1, 2, 3, . . . , n (12)
The Gibbs effect introduces spurious oscilla-tions in the transient response. The following attenuation factors with Riemann windowing are used to filter this effect [8].
wi= sin βi βi , βi = iπ n + 1 (13) bn= 8 n−1 X i=1,3,... wi ω2 i −1 (14) Furthermore for the impedance model, the flow rates are obtained from the hydraulic drives char-acteristics:
nlinesQin= Vpωr− Cv,p∆pp (15)
nlinesQout= Vmωmem+ Cv,m∆pm (16)
2.2.3 Variable displacement actuator
The actuator, which gives the relative volumetric displacement of the motor, is described through a first order linear system. The model is char-acterised only by the time constant Tctrl. The
input is the demand for the relative volumetric displacement of the motoredemand, and the
out-put is the actual relative volumetric displacement of the motorem.
Tctrl ˙em= edemand− em (17)
2.3
State-space system model
2.3.1 Hydraulic transmission model
The overall dynamics of the assembled hydraulic transmission model without control (pump + hy-draulic line + hyhy-draulic motor + actuator) are represented through a linear state-space model, see Appendix B.
The dynamic interaction between the different subsytems of the complete model is observed in
Figure 4: Subsystem block diagram of the wind turbine with hydraulic transmission
the block diagram of figure 4. Note that the hy-draulic transmission model described in this sec-tion, together with the load and pitch controllers, are implemented in MATLAB-Simulink and then compiled into a discrete time Dynamic Link Li-brary file. This DLL file is used by the external controller interface of GH Bladed and called with a timestep of 0.01 ms. The different output pa-rameters of the hydraulic transmission are also available through the external controller interface of GH Bladed.
3
Control System
In accordance with modern wind turbines, the control system for power production is based on a variable-speed configuration with full-span rotor-collective pitch-to-feather blades. For be-low rated wind speeds, optimisation of power capture is possible by allowing the rotor speed to vary linearly with the wind speeds up to rated rotor speed. The rotor speed is modified in the proposed fluid power concept by adjusting the volumetric displacement of the hydraulic motor. Above rated wind speeds, a collective pitch con-trol is used to limit the aerodynamic power [10]. In this situation, the fluid power transmission lim-its the torque by maintaining a maximum pres-sure in the system, and the pitch control is limited to regulate the rotor speed to rated conditions.
3.1
Pressure control for variable
speed
A variable displacement hydraulic motor coupled to a fixed speed synchronous generator is able
to modify the pressure in the hydraulic line and therefore the transmitted torque from the pump to the rotor. By controlling this pressure, the ro-tor speed can be regulated to the desired condi-tions. In order to obtain maximum aerodynamic efficiency, a constant tip speed ratio should be maintained. While the rotor speed must change linearly with the wind speed, the torque will change with the square of the wind speed and thus the square of the rotor speed. Hence the required torque-speed operation to obtain maxi-mum aerodynamic efficiency ofCP = 0.485at a
tip speed ratio of around7.55[2], should have a quadratic relationship according to
τaero= CP λ3 1 2ρairπR 5ω2 = Kλω2 (18)
The rotor speed is limited by the maximum speed of the blades tip, which for this rotor is rated at 80 m/s [2], resulting in a rated rotor speed of 12.1rpm. Once the rated rotor speed is achieved, the rotor speed should be maintained while the torque keeps increasing up to rated wind speed. Because the last statement implies a vertical line in the resulting torque-speed curve which cannot be implemented by means of a function or lookup table with the torque as a func-tion of rotor speed, a transifunc-tion region just below rated rotor speed is implemented. This region is implemented as a linear relationship such that the rated rotor speed is obtained at rated wind speed of 11.4m/s[2]. The resulting operational torque-speed curve is shown in the following fig-ure:
0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Power [kW]
Mech power rotor shaft Hydraulic power pump side Mech power generator shaft
0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wind speed [m/s] Efficiencies [−] Pump Pipeline Motor Total 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 Torque [kNm] Ideal Real 0 5 10 15 20 25 0 2000 4000 6000 8000 10000
Volumetric flow rate [lpm] IdealReal
0 5 10 15 20 25 0 100 200 300 400 Pressure [bar] Work pressure Charge pressure 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Wind speed [m/s]
Motor relative displacement [−]
Figure 5: Steady-state values for operational wind speeds
2 4 6 8 10 12 14 16 0 1000 2000 3000 4000 5000 11.4 m/s 11 m/s 10 m/s 9 m/s 8 m/s 4,000 kW 5,000 kW 6,000 kW Rotor speed [rpm] Torque [kNm]
Figure 6: Torque-speed curve for variable speed operation
transmissions where a minimum generator speed is imposed for low wind speeds, lower rotational speeds are possible when using hy-draulic pumps & motors (low-speed, high-torque radial units). When operating at low rotational speeds is mostly the partial load efficiency of the hydraulic drives which may determine the cut-in wind speed.
The variable speed controller uses a proportional-integral (PI) pressure control of the high-pressure line by adjusting the motor displacement and thus the flow and pressure at the hydraulic motor side. Reference values for the pressure are taken from the correspond-ing steady-state values for different rotational speeds of the rotor [11](the pressure-speed curve is obtained from the torque-speed curve
shown in figure 6). A schematic of the proposed controller is shown in figure 7.
Figure 7: Pressure control loop with outer speed feedback for variable speed operation
4
Parameter Study for a 5MW
Turbine
4.1
Reference
properties
and
Steady-state operation
To demonstrate the implementation of the pro-posed model and controller, the rotor of the NREL offshore 5MW reference turbine is cou-pled to a fluid power transmission with a fixed-speed generator. The same rotor specifications are used as in [2]. The required dimensions of hydraulic pumps & motors in the 5MW class are not yet commercially available. Therefore
the performance properties (such as efficiencies, nominal and maximum operating pressures) are derived from technology which is currently the state of the art [12] and applicable for smaller class turbines (1-2MW). The specific values ap-plied in the model are listed in Appendix D.
4.2
Parameter study results
The rotor speed and the pressure in the hy-draulic line at the pump side are shown for dif-ferent stepwise wind speed inputs ranging from cut-in to rated wind speed. The control parame-ters are kept constant for all the cases as well as the pressure of the return line.
4.2.1 Length of the hydraulic line
This parameter is directly associated to the amount of oil present in the system. When more oil is used, the effective stiffness is reduced due to compressibility of hydraulic fluids, and the fluid inertia is increased. The 10m and 100m line correspond to the nacelle and tower based solu-tion respectively. Depending on the type of con-figuration this effect might be of importance as shown in figure 8. Friction losses in the lines are minor when having a laminar flow.
Pressure fluctuations are relatively high for the 100 m pipeline, with overshoots from 10 to 40% with respect to the reference; the 50 m shows 4 to 20% difference while the 10 and 20 m only shows fluctuations from 1 to 2%.
4.2.2 Volumetric efficiency of the hydraulic motor
One of the main concerns of hydraulic transmis-sions for wind turbine is the efficiency of the vari-able displacement hydraulic motor(s), specially at partial load operation. However the oil leak-ages, which depend on the pressure difference across the unit, provide damping in the pressure fluctuations as shown in figure 9.
As observed in the results, the pressure fluctu-ations are reduced from varifluctu-ations of 11 to 50%, to 7 to 30% when the volumetric efficiency of the motor is reduced from 95 to 60% . The amount of increased damping is not considered significant especially when compared to the total effciency of the transmission.
4.2.3 Rotor mass moment of inertia
A dynamic wind speed input is used to observe the response of a lighter or heavier rotor to dy-namic loads below rated speed.
0 100 200 300 400 500 600 4 6 8 10 12 Time [s] Wind speed [m/s]
Figure 11: Hub height wind speed of 8m/s with 17.67% TI
The different values of the mass moment of inertia account for a rotor which is effectively ten times lighter or heavier with respect to the refer-ence turbine using the same rotor geometry. If the comparison is made in terms or rotor diame-ter, the inertias correspond to those of a 80m-2MW, 63m-5MW and 200m-12.5MW turbines. However this approach can not be compared di-rectly in the results, since each rotor would have different operating rotational speeds.
The rotor mass moment of inertia has a direct influence in the time response of the system, it is observed that larger inertias lead to lower fluctu-ations of pressure and rotational speed.
5
Conclusion
The amount of oil in the system has a signifi-cant influence in the stiffness of the transmis-sion and therefore in the pressure transients; the main reason is that the high fluid inertia of the system is able to produce high pressure tran-sients whenever there are sudden variations of volumetric flow. It is seen that small damping in the system is introduced due to the oil leak-ages in the hydraulic transmission, mainly repre-sented by the motor volumetric efficiency.
On the other hand, the rotor mass moment of inertia has an important influence in the time re-sponse of pressure and rotational speed, leading to a slower but smoother response of the system. In general a solution with long pipelines is prone to higher pressure fluctuations and minor friction losses for laminar flow. A low volumetric efficiency of the hydraulic transmission provides a pressure fluctuation damping however its influ-ence is minor and unlikely to be advantageous when compared to the loss in power production.
0 50 100 150 200 250 300 0 100 200 300 400 Time [s]
Pump pressure [bar]
L= 10 m L= 20 m L= 50 m L= 100 m 0 50 100 150 200 250 300 4 6 8 10 12 14 Time [s] Rotor speed [rpm]
Figure 8: Transient response to a series of wind steps for different pipeline lenghts;ηvol,m= 90%
0 50 100 150 200 250 300 0 100 200 300 400 Time [s]
Pump pressure [bar]
ηvol,m= 60% ηvol,m= 80% ηvol,m= 90% ηvol,m= 95% 0 50 100 150 200 250 300 4 6 8 10 12 14 Time [s] Rotor speed [rpm]
Figure 9: Transient response to a series of wind steps for different motor volumetric efficiencies lenghts;L = 100m 0 100 200 300 400 500 600 0 100 200 300 400 Time [s]
Pump pressure [bar]
J r= 3.88e6 kgm 2 J r= 3.88e7 kgm 2 Jr= 3.88e8 kgm2 0 100 200 300 400 500 600 6 8 10 12 14 Time [s] Rotor speed [rpm]
Figure 10: Transient response to dynamic wind speed for different rotor mass moments of inertia;
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Estimates of the Impact of
Fluid Power Drive Trains on the
Levelized Cost of Energy
This appendix was added at the request of the EWEA 2013 conference organizing committee. Its content is not in line with the rest of the paper. A fluid power drive train such as the tower base solution displayed in figure 1 has an eco-nomic impact in several ways. The numbers pre-sented here are estimations based on the cost model for offshore wind described by [13] using the electricity cost calculator from EWEA.
Capital Expenditure As mentioned in the in-troduction, the study in [1] showed a reduction of 24% in the amount of support structure steel (from tower and monopile). This results in a 7.7% reduction of CAPEX (e/kW).
Turbine installation cost reduction is assumed to be 10%. This cost represents 9% of the CAPEX, resulting in 0.9% reduction. The elim-ination of the need for power electronics leads to a reduction of 5.6% of turbine cost (this cost rep-resents 51% of the CAPEX), resulting in 2.9% reduction. Note that the cost of the gearbox is assumed to be of the same cost as the hydro-static transmission. Overall CAPEX reduction: 11.5%
Operational Expenditure Maintenance (ser-vice and spare parts) cost reduction is assumed to be 30% (this cost represents 39% of the OPEX according to [13]), resulting in 11.7% re-duction of OPEX (e/kWh). Note that the re-liability is assumed to be the same as refer-ence.Overall OPEX reduction: 11.7%
Energy production In comparison with the reference annual energy production, the produc-tion of a turbine with a fluid power drive train will be: 8.6% less energy is produced when using a motor with 90% efficiency (reference), result-ing in capacity factor of 0.32. 4.7% less energy when using a motor with 95% efficiency (likely), resulting in capacity factor of 0.33.
Note that this is for a 10.1 m/s average wind speed and reference of 0.35 capacity fac-tor.Overall Annual energy production reduction: 4.7 to 8.6%
Levelized Cost of Energy: The reference value for offshore wind energy is 89.62e/MWh. Three cases were examined for the fluid power drive train, yielding three different results.
1. Containing a motor with 90% efficiency: 83.15e/MWh→7.2% cost reduction
2. Containing a motor with 95% efficiency: 80.46e/MWh→10.2% cost reduction
3. Assuming same energy production as ref-erence: 77.5e/MWh→13.5% cost reduction
Appendix A
The torque and power characteristics of a tur-bine rotor are described through their non di-mensional coefficients. These coefficients are expressed as a function of the tip speed ratioλ
and the blades pitch angleβ. The tip speed ra-tio is a non-dimensional parameter which relates the tangential velocity of the blade tip and the upstream undisturbed wind speed,
λ = ωrR
U (19)
Hence, the aerodynamic loading of the rotor is described through quasi-static relations:
τaero= Cτ(λ, β) 1 2 ρairπ R 3U2 (20) Paero= CP(λ, β) 1 2 ρairπR 2U3 (21)
Since the last equations are valid only for steady characteristics, further extensions to the BEM method are implemented to account for un-steady aerodynamic behaviour. Although not de-scribed in this paper, these unsteady character-istics are included in GH Bladed.
Appendix B
The state-space representation of the hydraulic transmission is given by:
x˙ ˙em = " A − Cv,p nlinesb1cT1+ Cv,m nlinesb2cT2 Vmωmnlinesb2 0 − 1 Tctrl # x em + " Vp nlinesb1 0 0 1 Tctrl # ωr edemand (22)
Where the following outputs are defined:
τ pump ∆pp = " Vp(1 + Cf,p) cT1 0 cT 1 0 # x em + B p 0 0 0 ω r edemand (23)
Appendix C
Properties of the NREL 5MW reference turbine, acquired from [2]:
Parameter Symbol Unit Value
Rotor mass moment of inertia Jr [kg m2] 3.9e7
Rotor diameter Drotor [m] 126
Tower height ztower [m] 90
Cut-in wind speed vci [m/s] 3
Rated wind speed vrated [m/s] 11.4
Cut-out wind speed vco [m/s] 25
Optimal tip speed ratio λopt [-] 7.55
Maximum power coefficient CP,max [-] 0.485
Appendix D
Reference values for the proposed 5MW hydro-static transmission wind turbine.
Parameter Symbol Unit Value
Pump mass moment of inertia Jp [kgm 2
] 3680 Pump volumetric displacement V˜p [L/rev] 800
Motor volumetric displacement V˜m [L/rev] 5.82
Nom. press. of pump & motor pnom [Pa] 350e5
Nom. volumetric eff of pump ηvol,p [-] 0.98
Nom. volumetric eff of motor ηvol,m [-] 0.90
Pump nom. speed np [rpm] 12
Motor nom. speed nm [rpm] 1500
Pump laminar leakage coeff. Cv,p [m3/s/Pa] 9.1e-11
Motor laminar leakage coeff. Cv,m [m 3
/s/Pa] 4.16e-10 Pump dry friction coeff. Cf,p [-] 0.02
Motor dry friction coeff. Cf,m [-] 0.02
Pump viscous damping coeff. Bp [Nms] 50e3
Motor viscous damping coeff. Bm [Nms] 3.5
Actuator time constant Tctrl [s] 0.1
Number of pipelines nlines [-] 10.0
Pipe line internal diameter D [m] 0.25 High pressure line length L [m] 10 Density of fluid ρ [kg/m3
] 917 Kinematic viscosity of fluid ν [m2
/s] 40e-6 Effective bulk modulus of fluid Ef luid [Pa] 1.0e9
