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(13)–(16) express the volumetric effi-ciency, mechanical efficiency, actual flow rate, and the actual torqueof the hydraulic motor, respectively:ηvM ¼ αDmaxωαDmaxω þ Qloss; ð13ÞηtM ¼ αDmaxΔp−TlossαDmaxΔp; ð14ÞQm ¼ QiηvM; ð15ÞTm ¼ ηtMαΔpDmax: ð16Þ 2.2.1.3. Electro-hydraulic displacement control mechanism. The mecha-nism in this study was fifth-order, although practical applicationsusually employ a reduced-order model [25]. We considered it to bea first-order system and expressed it asut ðÞ¼ τKsv_ x þ 1Ksvx; ð17Þwhere u(t) is the electric signal control, τ is the time constant, Ksv isthe DC gain, and x is the relative swivel angle of the pump/motor'sswash plate.2.2.2. Hydraulic accumulatorThe energy balance in the accumulator was based on the gas pre-sent in the accumulator, as in Eq. (18) [26]. For simplicity, T, p,…werepresented instead of T(t), p(t),… in this subsection.dUdt¼ −pdVdt−mfcfdTdt−hAw T−Tw ðÞ; ð18Þwhere Aw was the effective area of the accumulator for heat convec-tion. For a real gas, the time derivative of the internal energy Ucould be expressed using Eq. (19):dUdt¼ mcv p; T ðÞdTdtþ T∂p∂T v−p dVdt ; ð19Þwhere cv(p,T) is the volume-specific heat as a function of pressureand temperature [J/kg/K], m is the mass of the gas, V is the specificgas volume [m3/kg], and ∂p∂T vis the partial derivative with a constantgas volume [Pa/K]. For simplicity, the time constants of the accumula-tor were used, and cv was used instead of cv(p,T). The temperature ofthe accumulator was expressed as,dTdt¼ 1τwTamb−T ðÞ−TQacvm∂p∂T v; ð20Þwhere the accumulator time constant τw ¼ mcvαAcshould be measuredfor each type of accumulator, and the partial derivative with respectto time at a constant gas volume was expressed as,∂p∂T v¼ RV21 þ 2CVT3 V þ B0 1−bV ; ð21Þ where A0,B0, a, b, and R are constants. The Beattie–Bridgman Equa-tion expresses the gas pressure:p ¼ mV 2RT⋅ 1− mCVT3 Vmþ B0 1−mbV −A0 1−maV : ð22ÞThe gas volume could be estimated from the flow rate into the ac-cumulator as Eq. (23):V ¼ V0a þ ∫t0Qadt; ð23Þwhere p is the pressure of the gas in the accumulator, T is the temper-ature of the gas, and Qa is the flow rate into the accumulator. The pa-rameters of the accumulator are shown in Table 1.2.2.3. The connecting linesIn this study, losses due to pipe lengths were ignored. The dynam-ic of the directional control valve was ignored in the modeling butwas analyzed in the experiment. The pressure in the low-pressureline was similar to the pressure in large hydraulic accumulator HA2,which was considered to be constant pl. Hence, pl was similar to p2and p1 for driving and braking, respectively. If the pressures in thehigh-pressure lines, p1 for driving and p2 for braking, were lowerthan the gas pre-charge pressure of the high-pressure accumulatorHA1, they were modeled using the continuity equation. For simplicity,the pressure drop in the check valves was neglected, so the pressuresbefore and after the check valve were similar. The pressures p1 and p2were expressed as in Eqs. (24) and (26), respectively. Guided spoolTable 1Constants of the hydraulic accumulator.Constant Value UnitR 296.77 J/(kg K)a 9.33752.10−4m3/kgb −2.47359.10−4m3/kgA0 1.74116.102Jm3/kg2B0 1.8007.10−3m3/kgC 5.0948.10−4m3K3/kgcv 754.12 J/(kg K)τw 90 sm 1.25 kg type of the relief valve was used and the over flows via the valveswere modeled as in Eqs. (25) and (27).V0βdp1dt¼ Qpa þ Qb1 þ Qb2−Qr1−Qma; ð24ÞWhenV11 was activated; Qr1¼ kr1 p1−prs1 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffip1−plp if p1 > prs10 else:ð25ÞV0βdp2dt¼ Qmiþ Qb3 þ Qb4−Qr2−Qpi; ð26ÞWhenV12 wasactivated; Qr2¼ kr2 p2−prs2 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffip2−plp if p2 > prs20 else;ð27ÞQpa is the actual flow rate from the outlet of pump P1,Qma is theactual flow rate into the inlet of motor PM2,Qb1,2,3,4 are the bootsflow rates, V0 is the volume of the fluid in the hose between thepump and the motor, Qr1,2 are the flow rates via the relief valvesRV1,2,Qmi is the flow rate from the motor outlet, Qpi is the actualflow rate into the inlet of the pump, prs1,2 are the setting pressures,kr1,2 are the gains of the two relief valves, and β is the bulk moduleof the fluid.If the pressures in the high-pressure line were greater than the gaspre-charge pressure of HA1, they were considered to be similar to thegas pressure of HA1. The gas and fluid pressures were determinedusing the accumulator's parameters and its input flow rate. The flowrates into the high accumulator for the driving and braking wereexpressed as in Eqs. (28a) and (28b), respectively:Qa ¼ Qpa−Qr1−Qma; ð28aÞQa ¼ Qmi−Qr2: ð28bÞ2.2.4. FlywheelThe dynamic equation of the flywheel was obtained by applyingNewton's second law, as in Eq. (29):Tm ¼ J _ ω þ C þ Tex; ð29Þwhere J is the moment of inertia of the flywheel, C is the viscous fric-tion coefficient, Tm is as presented in Eq. (16), and Tex is the brakingtorque. Eq. (30) defines the pressure difference asΔp ¼ p1−p2: ð30ÞIn addition, the estimation of the recovery potential of the system-recovered efficiency was defined as in Eq. (31), the ratio of the max-imum kinetic difference of the flywheel FL after (i+1) and before (i)regenerative braking.ηr ¼ Eiþ1Ei 100 ¼12 Jω2max;iþ112 Jω20;i 100 ¼ω2max;iþ1ω20;i