In typical HHV applications, the main power is produced by internal combustion engine in the form of mechanical torque and rotational speed [28]. This input mechanical power (P_i) to the hydraulic unit is characterized by its mechanical torque (τ) and angular velocity (ω), as follows:

The corresponding hydraulic unit output power (P_o) is given by fluid pressure (p) and volumetric flow (Q), as:

The ideal relation between fluid pressure and mechanical torque is described as [29]:

where D is pump displacement and Δp is pressure difference between unit input pressure p₀ and output pressure p. Note that for open loop hydraulics systems, p₀ is equal to the environment pressure (if the resistance in the return line can be neglected).

In the latter case, the pressure difference is equal to the unit output pressure Δp ≈ p. Note also that the unit displacement (D) is related to the unit volume (V) according to D = V/2π. The theoretical relationship between angular speed (ω) and volumetric flow (Q_t) is given by [28]:

Figure 1 shows the schematic representation of a hydraulic pump unit model which includes two types of power losses. The first source of losses is related to the so-called hydro-mechanical effects, which include viscous friction losses, mechanical friction losses, and hydrodynamic losses. These losses may be represented by the hydro-mechanical efficiency coefficient (η_mt). The second source of losses is associated with volumetric losses which include internal and external leakage and compressibility losses [28]. These losses can be lumped together and represented as additional leakage flow (Q_l), or, alternatively, as equivalent volumetric efficiency coefficient η_vol = (Q_t – Q_l)/Q_t = Q/Q_t [10]. Overall unit efficiency can than be represented as the product of hydro-mechanical and volumetric efficiency η_tot = η_mtη_vol.

Figure 1.

A schematic representation of hydraulic pump unit

According to Figure 1, the effect of hydro-mechanical efficiency on the developed torque can be directly included into the torque vs. pressure relationship in the following manner:

using the relationship between angular speed (ω) and volumetric flow (Q) which takes into account the volumetric efficiency, which is given by:

Inserting eq. (5) and eq. (6) into eq. (1), yields the following power equation:

Typically, the HHV transmission consists of hydraulic axial piston units (pump and motors) in a so-called swash-plate or bent axis design [21]. After an exhaustive catalogue search, a Parker-Hannifin’s hydraulic variable displacement pump from P1 series [30] has been selected for this study as a representative example of the axial piston pump in the category of swash-plate design. The aforementioned P1 pump series can be ordered in 7 different sizes (corresponding 7 different displacement values), as shown in Table 1, and with power ratings starting with 30 kW for P1-018 model, up to 140 kW for P1-140 model.

Figure 2 shows the catalogue data and derived maps for the P1-160 pump unit and the case of maximum displacement. These data are used herein to illustrate the pump mechanical parameters, i.e., its total efficiency (η_tot) (Figure 2a), input power (P_i) (Figure 2b) and output flow (Q) (Figure 2c) with respect to pump pressure drop and driving axle rotational speed (n) [rpm]. Magenta-coloured ‘+’ markers in Figure 2 denote the catalogue data points, whereas the remaining map data are obtained by using cubic interpolation between data points. Catalogue [30] also provides the same data for each unit model.

Figure 2.

P1-060 unit maximum displacement catalogue data for: overall efficiency (a); input power (b) and output flow (c)

Based on eq. (1) and eq. (5), the pump hydro-mechanical efficiency map may be obtained from the input power map P_i(ω, Δp) as follows:

where the angular speed (ω) [rad/s] is obtained from rotational speed (n) [rpm] as follows:

Similarly, from the Figure 2c, and flow (Q) vs. speed (ω) relationship in eq. (6), the volumetric efficiency may be calculated as:

Willans Line rule has been applied in [25] to describe the relationship between the load and steam consumption for a turbine governed by means of throttling. Subsequently, the method has been extended to describe the energy conversion devices in hybrid electric and conventional vehicles, which included electric machines [26], and internal combustion engines [27]. In particular, [26] has proposed that this rule might be applicable to any power converting unit under stationary (steady-state) conditions in order to estimate its efficiency.

By definition, the Willans Line approximation for a power converter unit is represented as a straight line with line slope (k) which represents the internal energy conversion efficiency of the converter, and line offset (P_wl) which represents external (parasitic) losses [26]:

The total efficiency power converter is calculated from the relationship between the input and output power given by eq. (7), which yields after rearranging:

If all power losses are lumped together as a single loss source (P_l), then the output power is given by P_o = P_i – P_l, and the total efficiency can be then expressed as:

Combining eq. (12) and eq. (13), yields the following relationship between the overall power losses (P_l), slope (k) and offset parameter (P_wl) as follows:

According to [31], the above overall power losses of the prime mover (i.e., electric or internal combustion motor) are described by a power losses map P_l(τ, ω), dependent on motor torque and speed. These losses are then approximated by a second-order torque vs. speed polynomial relationship as follows:

where c₂(ω), c₁(ω) and c₀(ω) are angular speed-dependent coefficients of the above polynomial approximation. These coefficients are determined in [32] by means of convex optimization. This approach can be iteratively combined with the dynamic programming optimization algorithm, as shown in [33], in order to provide more precise estimates of approximation parameters values.

Even though the hydraulic unit power conversion process is typically defined by its efficiency [eq. (7)], an alternative approach based on overall power losses P_l(τ, ω) may also be used:

The power losses data points P_l(τ,ω) for a P1-060 hydraulic pump unit are represented by the map shown in Figure 3a, which has been derived from the overall unit efficiency map previously introduced in Figure 2a. Note that the unit efficiency map in Figure 2a has been presented in terms of rotational speed (n) and pressure drop (Δp), whereas the power losses map in Figure 3a is given in terms of angular speed (ω) and driving torque (τ). Hence, in order to equate the aforementioned maps, conversion from n to ω and Δp to τ needed to be carried out, taking into account the hydro-mechanical efficiency defined by eq. (8).

Figure 3.

P1-060 hydraulic unit original: power losses map (a); approximation coefficients (b); approximation map (c) and approximation error (d)

For each angular velocity (ω) and torque (τ) value pair, proper values of coefficients c₀, c₁ and c₂ of the second-order polynomial approximation in eq. (15) need to be found. This can be done by numerically solving a system of equations for each torque vs. speed data point, and the resulting maps of speed-dependent approximation coefficients are shown in Figure 3b. By using eq. (15) and coefficient values from Figure 3b, the power losses approximation map in Figure 3c is obtained. The resulting approximation percentile error (i.e., relative error between the original power losses map in Figure 3a and approximated map in Figure 3c) is presented in Figure 3d. It shows that the proposed approximation based on Willans Line is accurate within 5% over a wide range of hydraulic unit operating regimes.

In order to derive the rules for model up/down-scaling, and, thus, to achieve straightforward model scalability reported in the aforementioned references (see e.g. [32]), the scaling factor (s_p) is introduced. This factor is defined as the ratio between original unit size which is used to derive coefficient values c₀(ω)…c₂(ω), and approximated unit size, wherein for hydraulic units, size directly corresponds to device displacement (D). Hence, the scaling factor is defined as:

where D_app is approximated unit displacement and D_orig is original unit displacement.

Based on the above relationship, the scalable power losses (P_ls) can be re-written as:

Since the input argument of the approximation polynomial in eq. (15) is the pump unit torque, while its coefficients are speed-dependent and do not depend on unit torque, the aforementioned scaling factor only refers to torque values as follows:

By combining eq. (19) and eq. (15), and inserting them into eq. (18), the following form of scalable polynomial approximation map is obtained [32]:

which, after multiplication with the scaling factor (s_p) yields the following form of overall losses map relationship [32]:

For each hydraulic unit from the Parker-Hannifin’s P1 series, the power losses map P_l(τ,ω) is calculated first from the total unit efficiency map (Figure 2a) by using P_l= P_i(1 − η_tot) from eq. (13), while also taking into account the aforementioned relationships between rotational speed (n) and angular speed (ω) and between pressure drop (Δp) and output torque (τ). The resulting (i.e., converted) total efficiency map for P1-160 in τ vs. ω coordinate frame is presented in Figure 4a. This result has been accompanied by the speed-dependent approximation coefficient values c₀(ω)…c₂(ω), which have been determined from the power losses map (see discussion on Willans Line approximation above, and Figures 3a and 3b for P1-060 unit). To verify the accuracy of the approximation method, the scaled polynomial approximation in eq. (21) has been utilized with different values of polynomial approximation coefficients c₀(ω)…c₂(ω), and sizing coefficient (s_p). The approximated power losses map, obtained by using eq. (13), is then converted to approximated total efficiency map, and the resulting difference between the original map and approximated is calculated.

Figure 4.

P1-060 hydraulics unit: original efficiency map (a); approximated and scaled efficiency map (b) and approximation error (c)

Figure 4 shows the comparison between the original P1-060 pump efficiencies obtained from the catalogue data (Figure 4a), and the approximate efficiency map shown in Figure 4b. The latter approximation is obtained by using approximation coefficients c₀(ω)…c₂(ω) obtained from the P1-075 pump unit data [30], and scaled down by the scaling factor s_p= D_app/D_orig = 60/75 = 0.8 [eq. (17)]. Figure 4c shows the relative difference (percentile error) between the original and approximated efficiency maps, as well as the mean error value, maximum error value and error standard deviation. The latter results point out to rather small approximation error bounds, which indicates that the approximation methodology based on Willans Line can predict the efficiency data values obtained from the manufacturer’s catalogue with very good accuracy.

In order to determine the overall model approximation accuracy, efficiency maps of all types of P1 units within the series have been compared, wherein the comparison has been performed between the catalogue efficiency data and scaled-model approximated efficiency data. The comparison results are shown in Figure 5 where D_orig represents displacement of hydraulic unit which has been used to define coefficients c₀(ω)…c₂(ω), while D_app represents the unit displacement which has been used for the scaled approximation efficiency calculation for the purpose of comparison. Notably, the mean error plot in Figure 5a shows that the mean value of approximation error for all combinations is less than 5.5%. The maximum error plot in Figure 5b shows that for combination D_orig = 100 cm³/rev and D_app = 75 cm³/rev the maximum error may reach up to 20%. However, this maximum error value refers to the boundary of the efficiency difference map (similar to the characteristic shown in Figure 4c).

Figure 5.

Overall approximation errors: mean value (a); max value (b) and standard deviation (c)