# Modelling Internal Leakage in the Automatic Transmission Electro-Hydraulic Controller, Taking into Account Operating Conditions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{2}) and alumina (Al

_{2}O

_{3}), whose hardness on the Mohs scale is 7 and 9, respectively, and significantly exceeds the hardness of the materials used on the spool valve parts. The influence of mechanical impurities present in the hydraulic fluid is presented in the article [15]. Experimental studies of the effect of a particle size of 10 ± 2 µm and a mass concentration of impurity of 40 mg/L showed wear and jamming of the hydraulic distributor. Such a phenomenon was not observed for particle sizes of about 2 ± 1 and 25 ± 5 µm, suggesting that abrasive particles with a size commensurate with the radial clearance between the friction parts of the manifold are dangerous for the operation of the hydraulic manifold. The results of the above-mentioned studies are in line with those presented in [16].

## 2. Automatic Transmission (AT) Operational Properties

- no traditional clutch releases the driver from the need to operate it while driving,
- there is no need to engage gears, therefore, the driver keeps both hands on the steering wheel, thus increasing driving safety and comfort,
- possible travel at very low speeds (creep) in high urban traffic,
- changing gears with virtually no temporary loss of driving force,
- high durability and reliability, as well as low failure rate.

_{m}—radial mounting clearance [mm], Δc

_{e}—radial clearance increment resulting from abrasive wear during operation [mm].

_{re}is impacted by [43,44]:

- pressure acting upon a control component (mechanical strains) leading to reduced clearance values Δc
_{c}, - thermal expansion of a distribution valve spool—reduced clearance value Δc
_{Ts}, - thermal expansion of electro-hydraulic controller body (sleeve) leading to increased clearance value Δc
_{Tk}.

_{c}) caused by pressure can be neglected due to its minor values (≤1000 kPa). The impact of thermal expansion of hydraulic precision pair components on a change in the radial clearance is significant since the operating temperature T

_{e}of ATF and transmission components ranges from 70 to 90 °C. The temperature difference ΔT between the radial clearance measurement temperature c

_{r}(taken at an ambient temperature t

_{o}= 20 °C) and the temperature of hydraulic precision pair components in operation may be up to 70 °C. As a result, the relationship describing the effective radial clearance of a hydraulic precision pair (c

_{re}) has the following form:

_{Tk}, Δc

_{Ts}—clearance change resulting from thermal expansion of, respectively, controlled body and distribution spool [mm], β

_{k}, β

_{S}—linear thermal expansion coefficient for, respectively, body and spool material [1/°C], D

_{k}, d

_{s}—diameter of, respectively, opening in body and distribution spool [mm], ΔT

_{S}, ΔT

_{S}—temperature difference between the actual operating temperature of, respectively, body and distribution spool, and their temperature during the measurement.

_{p}= f(n

_{p}) of a pump can be determined based on geometrical dimensions and structural relationships. In the case of positive-displacement pumps, Q

_{p}can be calculated from the relationship:

_{p}—pump unit capacity (geometrical capacity) [mm

^{3}], n

_{p}—pump rotor speed (equal to engine speed) [rpm], η

_{vp}—pump volumetric capacity factor, η

_{vp}= 0.95.

_{p}is determined using the relationship:

_{p}= f(n

_{p}) of a passenger car transmission hydraulic pump for the following data—m = 4.5 mm, z = 12, b = 12 mm, h = 8.4 mm—is shown in Figure 2b.

## 3. Model of Fluid Flows through Internal Gaps of AT Hydraulic Precision Pairs

#### 3.1. Hydraulic Gap Models

_{l}—supplied side pressure [kPa], p

_{0}—leakage side pressure [kPa], (p

_{l}− p

_{0})—gap pressure drop [kPa], d

_{1}—hydraulic precision pair component (shaft) diameter [mm], c

_{r}—hydraulic precision pair radial clearance [mm], ρ(0)—hydraulic fluid density at atmospheric pressure [g/cm

^{3}], $\rho \left(\frac{{p}_{l}+{p}_{0}}{2}\right)$—hydraulic fluid density taking into account pressure action [g/cm

^{3}], µ—hydraulic fluid dynamic viscosity, µ = ν·ρ [Pa∙s], ν—kinematic viscosity [mm

^{2}/s], l

_{c}—length of contact between precision pair component and cylindrical surface [mm].

- temperature-related change in kinematic viscosity,
- hydraulic fluid thermal expansion—density change.

_{ve}decreases linearly along with increasing kinematic viscosity v, and increases to the third power along with growing radial clearance c

_{r}. The value of the fluid stream flowing through a hydraulic gap is impacted by the eccentric position of the distribution spool in the sleeve (Figure 5). In consequence, the equation for flow rate q

_{ve}through internal leakages in a hydraulic precision pair with an eccentric annular gap has the form:

_{1}(Figure 6). The interface surface depends on the degree of seal strain through hydraulic fluid pressure. As the component is used, the outer part of the seal is abraded (flattened) and permanently deformed. The consequence is an increased actual area of contact between the seal and the cylindrical surface, hence, higher unit pressure, which forces the ring into the gap, thus increasing its destruction rate. The greater the pressure and gap size, and the lower the sealing material hardness, the more intense the sealing ring is forced into the gap. As the operation time of such a connection elapses, the rubber sealing ring loses its pre-clamp and flexibility.

_{r}≥ 20 µm), we can observe active gap cross-section reduction, leading to a decreased flow rate of the hydraulic fluid flowing through the gap. The obliteration process is also dependent on the degree of hydraulic fluid contamination. The greater the concentration of hydraulic fluid impurities, the more intensive this process is. In the case of gaps larger than 20 µm, obliteration is virtually non-present, and it can be neglected in calculating the flow rate of a hydraulic fluid flowing through a precision pair gap [48].

#### 3.2. Assumptions to the Electro-Hydraulic Controller Internal Leakage Model

- the model takes into account the stationary states of distribution valve spool positions,
- pressure relationships (p
_{s}, p_{m}) in other sections, besides primary pressure p_{g}, were determined analytically (object measurement impossible), - pressure relationship for controller discharge channels was determined theoretically, as structurally assumed—a maximum 10% of the primary pressure p
_{g}, - the phenomenon of obliteration was neglected in the calculation due to high (c
_{r}> 20 µm) hydraulic precision pair clearances obtained during the measurements on actual objects and minor (≤1000 kPa) hydraulic gap pressure drops, - the impact of material thermal expansion on hydraulic precision pair radial clearance was taken into account (effective clearance c
_{e}).

- main supply pressure p
_{g}, - control pressure p
_{s}, - modulated pressure p
_{m}, - converter p
_{k}.

_{g}and converter p

_{k}pressures were determined experimentally during road tests, using a specially designed diagnostic tool, the main components of which were two piezoelectric ATF pressure sensors [10]: a temperature sensor, a rotational speed sensor employing the Hall phenomenon, and a signal processing module that enables recording, reading, and processing of measurement results on a PC computer.

_{s}and modulated p

_{m}pressures in a working transmission. The values of control p

_{s}and modulated p

_{m}pressure in an actual system should fall within a range of 0–500 kPa, since these parameters describe MV solenoid valves. The relationship for control pressure p

_{s}, which is directly proportional to primary pressure p

_{g}can be determined analytically using a model of a distribution valve spool in a force equilibrium state (Figure 8). Then, the force equilibrium equation takes the form:

_{s}

_{1}${p}_{s}$ resulting from the impact of control pressure on the spool is calculated from the relationship:

_{s}

_{2(3)}generated due to the impact of control pressure p

_{s}on surfaces perpendicular to the spool axis are calculated from the relationship:

_{2(3)W}—area of surface perpendicular to the spool axis [mm

^{2}].

_{17}—spring constant [N/mm], ${x}_{17}$—spring deflection for a spool in the equilibrium position.

_{s}of the controller relative to the primary pressure p

_{g}takes the form:

_{m}of the controller relative to primary pressure p

_{g}was determined to be linearly dependent on primary pressure p

_{g}. The maximum value of modulated pressure limited by the MV valve, p

_{m}= 500 kPa, corresponds to the maximum value of primary pressure value p

_{g}. Such an assumption enabled determining the relationship for modulated pressure p

_{m}in the form of:

_{g}for hydraulic dampers (i = 5, 7, 11) and distribution slide valves (i = 8, 9, 14, 15, 16, 17, 20, 23, 25, 26) in second gear.

_{s}for distribution slide valves (i = 13, 14, 15, 16, 17, 18, 19) in second gear.

_{m}for hydraulic dampers (i = 7, 12′) and distribution slide valves (i = 8, 12) in second gear.

_{k}of distribution slide valves (i = 20, 22, 23) in second gear.

#### 3.3. Determination of Hydraulic Precision Pair Geometric Dimensions

- distribution spool outer diameter—d
_{i}[mm], - distribution spool cylindrical opening (sleeve) outer diameter—D
_{i}[mm], - length of contact between cylindrical openings and the distribution spool working surface—l
_{c}[mm], - hydraulic damper piston rod outer diameter—d
_{i}[mm], - hydraulic damper cylindrical opening (sleeve) outer diameter—D
_{i}[mm], - length of contact between hydraulic damper precision pair components and a mating cylindrical opening surface—l
_{c}[mm].

_{r}of hydraulic precision pairs using the relationship:

_{1}-x

_{2}and y

_{1}-y

_{2}(Figure 10).

_{c}between cylindrical openings and the working surface of distribution spools and the length of contact between hydraulic damper piston rod precision pair components and the sleeve surface. The measurements were taken in a laboratory where the ambient temperature was 20 ± 0.5 °C. Spool and damper piston rod diameter was measured in a plane perpendicular to the spool axis, approximately at the half of the cylindrical length—minor cylindrical section lengths. Examples of measurement results are listed in Table 1.

_{c}between a precision pair component and the cylindrical surface is mostly determined by the width of the fin supporting the distribution valve spool (actual contact length—Figure 9a) and not the cylindrical surface length, as in the case of the hydraulic damper (Figure 9b).

## 4. Controller Internal Tightness Model Test Result Analysis

_{e}change entails changes to such input values for calculating internal leakage ATF flow rate ∑q

_{i}as kinematic viscosity v and hydraulic fluid density ρ, as well as effective radial clearance c

_{re}described by the relationship.

- Variant (W1)—assumed viscosity of fresh ATF, ATF operating temperature of 80 °C, effective radial clearance c
_{re}in hydraulic precision pairs at the values measured on an actual object. The impact of ATF temperature on component thermal expansion was taken into account in relationship (2). - Variant (W2)—assumed value of kinematic viscosity v as obtained in the measurements for the sample after an operational mileage S = 106,315 km [10].
- Variant (W3)—data as in (W1). Changed modelled wear by increasing radial clearances in all hydraulic precision pairs by 10%.
- Variant (W4)—data as in (W1). Changed modelled wear by increasing radial clearances in all hydraulic precision pairs by 50%.
- Variant (W5)—ATF operating temperature −40 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
- Variant (W6)—ATF operating temperature 20 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
- Variant (W7)—ATF operating temperature 40 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
- Variant (W8)—ATF operating temperature 100 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.

_{e}= −40 ÷ 100 °C were described by an approximated relationship [10]:

^{−0.043·Tp}[mm

^{2}/s],

^{2}= 0.9853.

_{e}= −40 ÷ 100 °C were described by a linear relationship [2].

_{P}+ 0.8409 [g/mL].

_{p}, the controller internal leakage q for individual variants (W1–W8) adopts a similar waveform but differs in terms of values. Internal leakage values grow with the increasing conventional wear of precision pairs and ATF temperature. The curves almost simultaneously shift towards higher recorded leak values. For example, in the case of W4 and the range t

_{p}= 0–0.8 s, internal leakages occur at a level q = 6.37–6.22 dm

^{3}/min, followed by a sudden drop to q = 5.09 dm

^{3}/min, and a rapid growth to q = 8.46 dm

^{3}/min. At t

_{p}= 2.1 s, the leakages reach their maximum value, q = 8.56 dm

^{3}/min.

_{p}depends on the engine speed, which rises rapidly during vehicle acceleration, hence the sudden increase in pump capacity Q

_{p}—Figure 13. After the engine obtains a specific speed n = 2500 rpm, pump capacity reaches its maximum value Q

_{p}= 33.6 dm

^{3}/min, and then stabilizes.

_{D/}

_{1→i+1}for electro-hydraulic controller internal leakages was marked with a green vertical dashed line, and the maximum values were marked with a red dashed line (Figure 13).

_{D/}

_{1→i+1}are obtained when modelling operational wear of hydraulic precision pair components—increase in radial clearance c

_{re}by 10% in the third variant (W3) and by 50% in variants four to eight (W4–W8). The fourth variant (W4) assumes very high abrasive wear of hydraulic controller precision pair components (50% increase in clearance) and operation at the hydraulic fluid operating temperature T

_{e}= 80 °C. Results of the model test in variant five (W5) exhibit the significant impact of hydraulic fluid temperature T

_{e}and temperature of controller components—spool T

_{s}and body T

_{k}(sleeve), regardless of, even considerable, values of radial clearances (increased by 50%)—on internal leakage values q

_{D/}

_{1→i+1}. Assuming a hydraulic fluid temperature T

_{e}of −40 °C and radial clearance c

_{re}increased by 50%, the obtained internal leakage values q

_{D/}

_{1→i+1}amount to just 0.021 dm

^{3}/min.

_{e}of −40 °C during cold start-up of the transmission (when ATF has not yet reached the operating temperature value), there are virtually no internal leakages q

_{D/}

_{1→i+1}through the radial clearances c

_{re}of the electro-hydraulic controller hydraulic precision pairs. However, as ATF temperature increases, entailing a reduction in its viscosity and density (assuming a constant value of conventional degree of wear Z

_{sb}= 50%), the maximum internal leakage values increase (Figure 14). Temperature growth within the range of T

_{e}= −40 ÷ 20 °C does not cause a significant increase in internal leakages. A sudden increase in the maximum leakages takes place upon exceeding temperature T

_{e}= 40 °C (q

_{D}= 1.41 dm

^{3}/min), and when the fluid temperature is T

_{e}= 100 °C, leakages reach values severalfold higher than at the level of q

_{D}= 11.23 dm

^{3}/min.

_{D/}

_{1→i+1}for this model variant (W6) were virtually non-existent—maximum q

_{D/}

_{1→i+1}= 0.38 dm

^{3}/min. The objective of the model studies for the assumptions set out in variant seven (W7) was to demonstrate that an increase in ATF temperature and the heating of transmission components initiates the increase in internal leakage values q

_{D/}

_{1→i+1}. Assuming a 50% wear of the hydraulic precision pair coupling (increased radial clearance c

_{re}) and hydraulic fluid temperature T

_{e}= 40 °C, internal leakages q

_{D/}

_{1→i+1}take insignificant values—a maximum of 1.41 dm

^{3}/min. The objective of the W8 model tests was to demonstrate an extreme case—wherein the hydraulic fluid overheats at T

_{e}= 100°C and hydraulic precision pair coupling wear is 50% (increased radial clearance c

_{re}). Internal leakages q

_{D/}

_{1→i+1}in this model variant take high values ranging from 6.67 to 11.23 dm

^{3}/min, which may constitute more than 50% of the instantaneous capacity Q

_{p}of the hydraulic pump.

## 5. Conclusions

_{e}on the electro-hydraulic controller hydraulic precision pair internal leakage flow rate q

_{D/}

_{1→i+1}. The following assumptions were drawn based on the obtained model study results:

- (1)
- The value of internal leakages q
_{D/}_{1→i+1}is significantly impacted by hydraulic fluid temperature T_{e}and the temperature of hydraulic precision pair components above 40 °C. A temperature increase in the range of 40–100 °C leads to a considerable reduction in the kinematic viscosity v and density ρ of the fluid, and hence, an eightfold increase in internal leakages. - (2)
- AT operation at low ambient temperatures makes electro-hydraulic controller hydraulic precision pair internal leakages q
_{D/}_{1→i+1}adopt low values, which do not exceed a 3% share of the internal leakage relative to the stream generated by the hydraulic pump. - (3)
- Excessive internal leakages q
_{D/}_{1→i+1}caused by high wear of electro-hydraulic controller hydraulic precision pairs in the event of AT overheating (above 80 °C) may cause malfunctioning of the automatic transmission. In such a case, internal leakages account for a significant part (above 50%) of the instantaneous hydraulic pump capacity Q_{p}, which leads to primary pressure p_{g}decreasing below the value required for the correct operation of the automatic transmission hydraulic controller system (variant W8). - (4)
- ATF viscosity v reduced in the course of operation (approx. 20%) causes significantly greater electro-hydraulic controller precision pair internal leakages q
_{D/}_{1→i+1}(approximately 25% of the instantaneous hydraulic pump capacity Q_{p}). Controller precision pair leakages, combined with internal leaks in other system components and reduced pump volumetric capacity factor η_{vp}, may lead to a decrease in the primary pressure p_{g}below the value required for the correct operation of an automatic transmission hydraulic control system. - (5)
- The wear of controller hydraulic precision pair components (increased radial clearance c
_{re}) significantly affects the internal leakage flow rate q_{D/}_{1→i+1}, especially when the temperature T_{e}of ATF in the transmission reaches or exceeds the operating value (T_{e}≥ 80 °C), which has been evidenced by a model test for an ATF with a temperature value of T_{e}= 100 °C. - (6)
- Minor wear (radial clearance c
_{re}up by 10%) of controller hydraulic precision pair components (between the distribution spool and sleeve) and the operation of the AT hydraulic system under operating temperature conditions (T_{e}= 80 °C) means that the hydraulic precision pair internal leakage value flow rate q_{D/}_{1→i+1}may adversely impact AT operation. In such a case, internal leakages may reach a value of up to 20% of the hydraulic pump instantaneous capacity Q_{p}—variant W3.

_{r}between the spool and sleeve of a distribution valve, and its value should be compared with the assembly clearance c

_{m}value in the hydraulic precision pair couplings of an electro-hydraulic controller. Measurements of controller precision pair components should be taken after analysing the results of road tests and assessing the degree of hydraulic fluid (ATF) contamination.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**AT hydraulic control system diagram in a passenger car with ATF flow paths, for the selector lever in the “D” position and gear ration “1”: 1—hydraulic pump, 2—hydraulic filter, 3—converter, 4—ATF cooler: 5—clutch “A” hydraulic damper, (distribution slide valves: 6—“B” clutch with a hydraulic damper, 7—“C” brake hydraulic damper, 8—brake “C

_{1}”, 9—“C” brake with a hydraulic damper, 10—“D” brake with a hydraulic damper, 11—“E” clutch hydraulic damper, 12—“F” brake with a hydraulic damper, 13—reverse gear, 14, 15, 16—gear change 1 ↔ 2 (2 ↔ 3) (3 ↔ 4), 17—“Kick—down” function, 18, 19—pressure, 20—converter, 21—lubrication system overflow valve, 22—lubrication pressure control, 23—“Lock—up” clutch control valve, 24—pressure modulation valve, 25—main pressure distribution valve, 26—mechanically controlled driving mode selection, 27—MV1/MV2/MV3 dual-position solenoid valve block, 28—MV4 proportional solenoid valve, 29—EAT controller [22].

**Figure 2.**Internal toothed wheel hydraulic pump with a sickle insert: (

**a**) general view (1—inner wheel, 2—sickle insert, 3—outer wheel), (

**b**) speed characteristics Q

_{p}= f(n

_{p}) [10].

**Figure 3.**Simplified block diagram of an automatic transmission electro-hydraulic controller internal leakage model.

**Figure 4.**Diagram and distribution of pressure in a hydraulic precision pair with a concentric annular gap: (

**a**) transverse direction of the gap; (

**b**) longitudinal direction of the gap [45].

**Figure 5.**Diagram and distribution of pressure in a hydraulic precision pair with an eccentric annular gap: (

**a**) transverse direction of the gap; (

**b**) longitudinal direction of the gap [45].

**Figure 7.**Structural model of AT electro-hydraulic controller internal leakages in second gear. Hydraulic control system components are designated as per Figure 1.

**Figure 8.**Model of distribution valve “17” with a spool in the state of force equilibrium for a fully open control pressure section channel.

**Figure 9.**Electro-hydraulic controller hydraulic precision pair diagrams: (

**a**) distribution slide valve; (

**b**) hydraulic damper.

**Figure 11.**Electro-hydraulic controller blocks: (

**a**) block No. 1, (

**b**) block No. 2, (

**c**) block No. 3, (

**d**) block No. 4. Designation corresponds to the diagrams in Figure 1.

**Figure 12.**Results of diameter and contact length components, and results of radial clearance calculations.

**Figure 13.**Electro-hydraulic controller internal ATF leakage flow rates q

_{D/}

_{1→i+1}under the minimum acceleration test conditions in gear “1” and “2” of the transmission in variants W1–W8.

**Figure 14.**Maximum flow rates q

_{D/}

_{1→i+1}of electro-hydraulic controller internal leakages depending on hydraulic fluid temperature T

_{e}and for minimum acceleration test in gear “1” and “2”, assuming the wear of a hydraulic precision pair coupling (a 50% increase in radial clearance c

_{re}).

No. Acc. to Figure 8 | Component | Measurement Plane | Measurement Direction: | Coordinate x1 (y1) [mm] | Coordinate x2 (y2) [mm] | Diameter: x1-x2 (y1-y2) [mm] | Diameter Mean Value d _{i} (D_{i}) [mm] |
---|---|---|---|---|---|---|---|

6 | spool | 1 | 0° | 89.605 | 79.666 | 9.939 | 9.939 |

79.666 | 89.605 | 9.939 | |||||

90° | 86.422 | 76.483 | 9.939 | ||||

76.483 | 86.422 | 9.939 | |||||

2 | 0° | 84.371 | 74.430 | 9.941 | 9.942 | ||

74.430 | 84.371 | 9.941 | |||||

90° | 85.121 | 75.178 | 9.943 | ||||

75.178 | 85.121 | 9.943 |

Variant No. | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 |
---|---|---|---|---|---|---|---|---|

ATF fluid temperature: T _{e} [°C] | 80 | 80 | 80 | 80 | −40 | 20 | 40 | 100 |

Nominal degree of wear Z _{sb} [%] | 0 | 0 | 10 | 50 | 50 | 50 | 50 | 50 |

Kinematic viscosity: ν [mm ^{2}/s] | 10.93 | 8.40 | 10.93 | 10.93 | 1923.0 | 145.0 | 42.0 | 7.0 |

Density: ρ_{c} [g/cm^{3}] | 0.783 | 0.783 | 0.783 | 0.783 | 0.870 | 0.826 | 0.812 | 0.768 |

**Table 3.**Values of internal leakage q

_{D}through hydraulic precision pair couplings—test results for model variants W1–W8.

Modelling Variant | ${\mathit{q}}_{\mathit{D}/1\to \mathit{i}+1}$ [dm^{3}/min] | Maximum Share of Internal Leakage Relative to the Stream Generated by the Hydraulic Pump U_{p} [%] |
---|---|---|

W1 | 2.16 | 8.89 |

W2 | 2.81 | 11.6 |

W3 | 3.64 | 14.9 |

W4 | 8.56 | 35.2 |

W5 | 0.021 | 0.086 |

W6 | 0.38 | 1.56 |

W7 | 1.43 | 5.89 |

W8 | 11.23 | 56.2 |

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## Share and Cite

**MDPI and ACS Style**

Dziubak, T.; Szczepaniak, P.
Modelling Internal Leakage in the Automatic Transmission Electro-Hydraulic Controller, Taking into Account Operating Conditions. *Energies* **2023**, *16*, 7667.
https://doi.org/10.3390/en16227667

**AMA Style**

Dziubak T, Szczepaniak P.
Modelling Internal Leakage in the Automatic Transmission Electro-Hydraulic Controller, Taking into Account Operating Conditions. *Energies*. 2023; 16(22):7667.
https://doi.org/10.3390/en16227667

**Chicago/Turabian Style**

Dziubak, Tadeusz, and Paweł Szczepaniak.
2023. "Modelling Internal Leakage in the Automatic Transmission Electro-Hydraulic Controller, Taking into Account Operating Conditions" *Energies* 16, no. 22: 7667.
https://doi.org/10.3390/en16227667