Energy Optimal Control of Electromechanical Systems: Trade-off Demands

The driving strategy of electromechanical systems has a significant influence on their energy consumption. The original control strategies, which relied on maximum acceleration and speed, were highly energy-demanding. This prompted the development of energy-efficient control methods. These methods can significantly reduce the drive’s energy consumption, improving environmental sustainability and lowering operational costs.

To verify the effect of practicable manoeuvre time selection, this study presents two different control strategies that are capable of minimising the drive’s electrical and mechanical losses. The first strategy, energy optimal control (EOC), is based on the Euler-Lagrange principle approach (Pontryagin, 2018), while the second strategy, energy near-optimal control (ENOC), is based on the prediction of drive losses (Tolle, 1975). ENOC exploits a symmetrical trapezoidal speed profile with truly finite settling time. Both position control strategies are designed to complete a position manoeuvre in the prescribed manoeuvre time (T_m). The description of the route between two stops, including passive resistance to the movement of a given system, is required for the design of optimal energy-saving output state variables. The value of the average gradient of the route is added to the constant train resistance to avoid the computation of necessary switching points for changing the slope on the route. Regenerative braking is assumed for both control strategies. The energy-saving reference position generator produces optimal time functions of position, speed and acceleration between two stops. These outputs are then faithfully followed using a feedback control based on field orientation (Novotny and Lipo, 1996) and accomplished with a matched zero dynamic lag pre-compensator, yielding the required closed-loop dynamics.

Possible energy savings through minimising the drive’s speed depend on load torque, which can consist of constant (Coulomb), linear (viscous) and quadratic (drag coefficient) components. Propelling mechanisms with typical linear friction have a load torque directly proportional to the angular speed (ω). The power as a product of torque and speed is proportional to the square of speed (ω²). By halving the drive’s speed to ω/2, energy demands are reduced by 50% compared with the original speed requirements, despite the manoeuvre time being twice as long [Eq. (1a)].

Higher energy savings can be achieved by halving the speed of mechanisms with quadratic torque (such as pumps and fans), where the load torque is proportional to the square of the speed (ω²). In this case, power is proportional to ω³, and by halving the drive’s speed to ω/2, energy demands are reduced by 75% compared with the original speed requirements, despite the manoeuvre time being twice as long [Eq. (1b)]. (1a) $\frac{ω^{2} T - {(ω / 2)}^{2} 2 T}{ω^{2} T} 100 = 50 %$ {{{{\rm{\omega }}^2}T - {{\left( {{\rm{\omega }}/2} \right)}^2}2T} \over {{{\rm{\omega }}^2}T}}100 = 50\% (1b) $\frac{ω^{3} T - {(ω / 2)}^{3} 2 T}{ω^{3} T} 100 = 75 %$ {{{{\rm{\omega }}^3}T - {{\left( {{\rm{\omega }}/2} \right)}^3}2T} \over {{{\rm{\omega }}^3}T}}100 = 75\%

These two examples of load torque influence on energy consumption offer a general conclusion: the maximum speed of electromechanical systems should be kept to the minimum practicable value. Since the load torque of most drives includes all three aforementioned components, choosing a practical manoeuvre time significantly impacts both energy consumption and the balance between the drive’s cost and users’ benefits.

The increasing demand for energy-saving technologies is reflected in numerous researches dedicated to this subject. The first application of Pontryagin’s maximum principle (PMP) to train operations dates back to 1968 (Ichikawa, 1968). The developed train and route model was restricted to a flat track and a load torque containing only a linear component.

To solve the problem of finding the optimal control strategy for trains equipped with diesel-electric locomotives, a combination of two different approaches was proposed (Howlett, 2000). For continuous control problems, PMP is used, while for discrete control issues, which are common in diesel-electric locomotive fuel consumption, optimal switching times are determined by utilising the Kuhn-Tucker conditions to minimise fuel usage.

The PMP approach is also used to derive energy-optimal trajectories by Heineken et al. (2023). A special code, Leda, was developed to construct the unique minimum energy trajectories as the solution. As an example of energy savings, the sensitivity to deviations of the train speed from the calculated optimal speed profile is studied in detail.

Several models and numerical algorithms were developed for traction applications to minimise losses of traction drives by adjusting switching points for the complete optimal trajectory design.

An algorithm optimising the driving style of a train, enabling energy optimisation, is described by Franke et al. (2000). A non-linear point-mass model of the train, incorporating the motion equation, is arranged to facilitate a piecewise analytical solution to the non-linear control problem. This designed algorithm creates a suitable base for a non-linear model predictive controller to operate in real-time.

An optimal control algorithm for the reduction of energy consumption of subway trains based on proximal policy optimisation (PPO) is proposed by Chen et al. (2023). The reinforcement learning architecture model for optimal control of the train is constructed as the first step. This model is then trained by exploiting PPO and reward scaling. As a result, the effective reduction of train energy consumption was achieved.

A new cellular automation model of train movement under mixed-traffic conditions and loss minimisation is proposed by Ping et al. (2023). The space-time diagram of the existing traffic flow is investigated, as well as the train movement, to obtain updated rules for the driver of the train.

A complex analysis of the operation strategies and their impact on the overall system design of the metro train is described by Su et al. (2016). The optimal train control model considers parameters, such as train mass reduction, minimising traction resistances, designing energy-saving gradients and implementing regenerative braking, along with corresponding timetable optimisation. An optimal train control simulator, based on an energy-efficient system, enables the analysis of practical operational data from the Beijing metro line. Quantitative analysis of possible energy reduction for different vehicle parameters and control strategies helps the decision-makers to adjust the balance between system costs and passenger benefits.

To investigate the influence of manoeuvre time selection on energy demands, the simulation of the suburban train unit propelled by induction motors was chosen as an example. The results gained are valid for any electric drive load torque that is speed-dependent, as described further. The control system of this unit, shown in Figure 1, exploits principles of field-oriented control (FOC) due to its fast dynamic response and the possibility of controlling torque and field components of stator current separately (Orlowska-Kowalska and Dybkowski, 2016; Perdukova et al., 2004). The FOC technique also offers significant potential for a considerable reduction of drive losses.

The mathematical description of the control system for optimising the drive’s energy consumption consists of four differential equations that account for vehicle dynamics as well as the traction motor (TM) position, speed and current dynamics (Brandstetter and Dobrovsky, 2013). The resulting energy-saving speed profiles consist of the following modes: acceleration, cruising and braking. The designed control structure for verification is shown in Figure 1 and consists of an EOC or ENOC energy saving profile generator (position, speed, acceleration and jerk) accomplished with zero-lag pre-compensator and a position-controlled induction TM capable of precise tracking of generated state variables (Vittek et al., 2017a,b).

2.

Theoretical Background

In previous work (Ftorek et al., 2021; available as Open Access Journal), the background of EOC and ENOC for rotational drives was described in detail. Therefore, only the parts necessary for the implementation of these control strategies are listed further.

The movement of the suburban train unit is described by the following differential Eq. (2): (2) $m_{t} \frac{d^{2} S_{T D}}{d t^{2}} = F_{t} (t, v) - F_{p} (v) - F_{s} (S_{T D}) - F_{b} (v)$ {m_t}{{{{\rm{d}}^2}{S_{TD}}} \over {{\rm{d}}{t^2}}} = {F_t}\left( {t,v} \right) - {F_p}\left( v \right) - {F_s}\left( {{S_{TD}}} \right) - {F_b}\left( v \right) where m_t is the total mass of the train, S_TD is the travel distance between two stops, F_t is the traction force, F_p is the force covering vehicle resistance, F_s is the force covering slope resistance and F_b is the braking force, respectively.

The relationship between travel distance (S_TD) and TM rotor position θ_r is given in Eq. (3): (3) $θ_{r} = \frac{2 u}{D_{w a}} S_{T D}$ {{\rm{\theta }}_r} = {{2u} \over {{D_{wa}}}}{S_{TD}} where D_wa is the average diameter of the driving wheel, and u is the gear ratio between the rotor of TM and the driving wheel. The following Eq. (4) determines TM rotational speed (ω_r) as a function of vehicle velocity (v_V) (4) $ω_{r} = \frac{u}{1.8 D_{w a}} v_{V}$ {{\rm{\omega }}_r} = {u \over {1.8{D_{wa}}}}{v_V}

Constant, linear and quadratic traction resistances to unit movement as a function of vehicle speed are defined in Eq. (5): (5) $p_{0} = A + B v_{V} + C v_{V}^{2}$ {p_0} = A + B{v_V} + Cv_V^2 and for suburban unit has form $p_{0} = 3.1 + 0.025 v_{V} + 0.00038 v_{V}^{2}$ {p_0} = 3.1 + 0.025{v_V} + 0.00038v_V^2 . The relationship between TM torque (Γ_TM) and train unit traction force (F_t) is given in Eq. (6). (6) $Γ_{1 M} = \frac{D_{w a}}{2 m u} F_{t}$ {\Gamma _{1M}} = {{{D_{wa}}} \over {2mu}}{F_t}

The parameters of the train unit are listed in Table A1 in the Appendix. By analysing these strategies, this study aims to evaluate their effectiveness in minimising energy consumption while maintaining optimal operational performance.

2.1.

EOC

EOC is based on Euler–Lagrange optimisation. The dynamical system for optimisation of AC drives [Eqs (7) and (8)] consists of the differential equations for rotor position (θ_r), rotor speed (ω_r) and electric torque (Γ_e). In Eq. (7), J_r stands for the moment of inertia reduced to the shaft. The load torque (Γ_L) is treated as the state variable; therefore, the system is completed with its derivative, which is the function of rotor speed. The load torque can be written as the sum of the constant term A, linear term Bω_r and quadratic component $C ω_{r}^{2}$ C{\rm{\omega }}_r^2 . The time derivative of the load torque has the form: ${\dot{Γ}}_{L} = (B + 2 C ω_{r}) {\dot{ω}}_{r}$ {\dot \Gamma _L} = \left( {B + 2C{{\rm{\omega }}_r}} \right){{\rm{\dot \omega }}_r} . (7) $[\begin{matrix} {\dot{θ}}_{r} \\ {\dot{ω}}_{r} \\ {\dot{Γ}}_{e} \\ {\dot{Γ}}_{L} \end{matrix}] = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 / J_{r} & - 1 / J_{r} \\ 0 & - k_{ω} & - k_{Γ} & 0 \\ 0 & 0 & B + 2 C ω_{r} / J_{r} & - B + 2 C ω_{r} / J_{r} \end{matrix}] [\begin{matrix} θ_{r} \\ ω_{r} \\ Γ_{e} \\ Γ_{L} \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ k_{U} \\ 0 \end{matrix}] u_{q}$ \left[ {\matrix{ {{{{\rm{\dot \theta }}}_r}} \cr {{{{\rm{\dot \omega }}}_r}} \cr {{{\dot \Gamma }_e}} \cr {{{\dot \Gamma }_L}} \cr } } \right] = \left[ {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & {1/{J_r}} & { - 1/{J_r}} \cr 0 & { - {k_\omega }} & { - {k_\Gamma }} & 0 \cr 0 & 0 & {B + 2C{{\rm{\omega }}_r}/{J_r}} & { - B + 2C{{\rm{\omega }}_r}/{J_r}} \cr } } \right]\left[ {\matrix{ {{{\rm{\theta }}_r}} \cr {{{\rm{\omega }}_r}} \cr {{\Gamma _e}} \cr {{\Gamma _L}} \cr } } \right] + \left[ {\matrix{ 0 \cr 0 \cr {{k_U}} \cr 0 \cr } } \right]{u_q} (8) $y = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}] {[\begin{matrix} θ_{r} & ω_{r} & Γ_{e} & Γ_{L} \end{matrix}]}^{T}$ y = \left[ {\matrix{ 1 & 0 & 0 & 0 \cr } } \right]{[\matrix{ {{{\rm{\theta }}_r}} & {{{\rm{\omega }}_r}} & {{\Gamma _e}} & {{\Gamma _L}} \cr } ]^T}

Constants of Eq. (7) for the IM are defined as: R_c = R_s + R_r (L_m / L_r)², k_Γ = R_c/σL_s, k_ω = pk_tΨ_D (c₂/σL_s + 1/L_m) and k_U = k_t / σL_s. The IM torque constant, k_t = 3pL_mΨ_D / L_r and u_q, is the q-axis excitation voltage for both motors. For simplification of calculations, a new constant (k_m) was defined as $k_{m} = k_{t}^{2} / 2 R_{c}$ {k_m} = k_t^2/2{R_c} .

If Euler–Lagrange calculus is applied to the system (7), a highly non-linear system (9) is obtained. This system consists of two derivatives of Lagrange coefficients λ_1,2 and the derivative of rotor speed and rotor position, which can only be solved numerically (9) $[\begin{matrix} {\dot{λ}}_{1} \\ {\dot{λ}}_{2} \\ {\dot{ω}}_{r} \\ {\dot{θ}}_{r} \end{matrix}] = [\begin{matrix} B / J_{r} & - 1 / J_{r} & 2 B / J_{r} & 0 \\ 0 & 0 & 0 & 0 \\ K_{m} / J_{r} & 0 & - B / J_{r} & 0 \\ 0 & 0 & 1 & 0 \end{matrix}] [\begin{matrix} λ_{1} \\ λ_{2} \\ ω_{r} \\ θ_{r} \end{matrix}] + [\begin{matrix} 1 / J_{r} \\ 0 \\ - 1 / J_{r} \\ 0 \end{matrix}] A + [\begin{matrix} 2 C λ_{1} \\ 0 \\ 0 \\ 0 \end{matrix}] ω_{r} + [\begin{matrix} 3 C / J_{r} \\ 0 \\ - C / J_{r} \\ 0 \end{matrix}] ω_{r}^{2}$ \left[ {\matrix{ {{{\dot \lambda }_1}} \cr {{{\dot \lambda }_2}} \cr {{{{\rm{\dot \omega }}}_r}} \cr {{{{\rm{\dot \theta }}}_r}} \cr } } \right] = \left[ {\matrix{ {B/{J_r}} & { - 1/{J_r}} & {2B/{J_r}} & 0 \cr 0 & 0 & 0 & 0 \cr {{K_m}/{J_r}} & 0 & { - B/{J_r}} & 0 \cr 0 & 0 & 1 & 0 \cr } } \right]\left[ {\matrix{ {{\lambda _1}} \cr {{\lambda _2}} \cr {{{\rm{\omega }}_r}} \cr {{{\rm{\theta }}_r}} \cr } } \right] + \left[ {\matrix{ {1/{J_r}} \cr 0 \cr { - 1/{J_r}} \cr 0 \cr } } \right]A + \left[ {\matrix{ {2C{\lambda _1}} \cr 0 \cr 0 \cr 0 \cr } } \right]{{\rm{\omega }}_r} + \left[ {\matrix{ {3C/{J_r}} \cr 0 \cr { - C/{J_r}} \cr 0 \cr } } \right]{\rm{\omega }}_r^2

The derived system (9) must satisfy boundary conditions (10), which are as follows: (10) $\begin{array}{l} \begin{matrix} for t = t_{0} & \to & ω_{r} (t_{0}) = ω_{0} \\ θ_{r} (t_{0}) = θ_{r 0}, \end{matrix} \\ \begin{matrix} for t = T_{m} & \to & ω_{r} (T_{m}) = ω_{m} \\ θ_{r} (T_{m}) = θ_{r} \end{matrix} \end{array}$ \eqalign{ & \matrix{ {{\rm{for }}t = {t_0}} & \to & {{{\rm{\omega }}_r}\left( {{t_0}} \right) = {{\rm{\omega }}_0}} \cr {} & {} & {{{\rm{\theta }}_r}\left( {{t_0}} \right) = {{\rm{\theta }}_{r0}},} \cr } \cr & \matrix{ {{\rm{for }}t = {T_m}} & \to & {{{\rm{\omega }}_r}\left( {{T_m}} \right) = {{\rm{\omega }}_m}} \cr {} & {} & {{{\rm{\theta }}_r}\left( {{T_m}} \right) = {{\rm{\theta }}_r}} \cr } \cr}

The solution to system (9) with boundary conditions (10) is obtained using the MATLAB function, bvp4c.m (boundary value problem). By providing approximation points for the expected solution, this solver creates a mesh and determines a numerical solution by solving the differential equations resulting from the boundary conditions and the collocation conditions imposed on all subintervals. Subsequently, the solver estimates the error of the computed solution on each subinterval. If the solution does not meet the tolerance criteria, the solver adapts the mesh and repeats the process. The resulting functions for acceleration, speed and position meet the requirements for minimum energy consumption.

2.2.

ENOC

The ENOC approach is based on the detailed prediction of copper and friction losses while using the symmetrical trapezoidal profile of rotor speed. The total drive energy [Eq. (11)], including predictable drive losses, is then defined as: (11) $W_{T} = \frac{3}{2} \int_{0}^{T_{m}} R_{c} i_{q}^{2} d t + \int_{0}^{T_{m}} (A + B ω_{r} + C ω_{r}^{2}) ω_{r} d t$ {W_T} = {3 \over 2}\int\limits_0^{{T_m}} {{R_c}i_q^2{\rm{d}}t + } \int\limits_0^{{T_m}} {\left( {A + B{{\rm{\omega }}_r} + C{\rm{\omega }}_r^2} \right){{\rm{\omega }}_r}{\rm{d}}t}

After integration over the manoeuvre time T_m, as a function of acceleration interval T_ε for the demanded position setpoint θ_rd, the total consumed energy W_T results in Eq. (12): (12) $W_{T} = \frac{θ_{r d}^{2}}{{(T_{m} - T_{ε})}^{2}} [\frac{k_{1}}{T_{ε}} + k_{s} (T_{m} - \frac{4}{3} T_{ε}) + k_{3} \frac{θ_{r d}^{2}}{{(T_{m} - T_{ε})}^{2}} (T_{m} - \frac{8}{5} T_{ε}) + k_{5} \frac{θ_{r d}}{(T_{m} - T_{ε})} (T_{m} - \frac{3}{2} T_{ε})] + k_{A} T_{m} + k_{6} θ_{r d} + A θ_{r d}$ {W_T} = {{{\rm{\theta }}_{rd}^2} \over {{{({T_m} - {T_{\rm{\varepsilon }}})}^2}}}\left[ {{{{k_1}} \over {{T_{\rm{\varepsilon }}}}} + {k_s}\left( {{T_m} - {4 \over 3}{T_{\rm{\varepsilon }}}} \right) + {k_3}{{{\rm{\theta }}_{rd}^2} \over {{{({T_m} - {T_{\rm{\varepsilon }}})}^2}}}\left( {{T_m} - {8 \over 5}{T_{\rm{\varepsilon }}}} \right) + {k_5}{{{{\rm{\theta }}_{rd}}} \over {\left( {{T_m} - {T_{\rm{\varepsilon }}}} \right)}}\left( {{T_m} - {3 \over 2}{T_{\rm{\varepsilon }}}} \right)} \right] + {k_A}{T_m} + {k_6}{{\rm{\theta }}_{rd}} + A{{\rm{\theta }}_{rd}} where $k_{1} = 3 R_{c} \frac{J_{r}^{2}}{{(c Ψ)}^{2}}$ {k_1} = 3{R_c}{{J_r^2} \over {{{(c\Psi )}^2}}} , $k_{2} = \frac{3}{2} R_{c} \frac{A^{2}}{{(c Ψ)}^{2}} + B$ {k_2} = {3 \over 2}{R_c}{{{A^2}} \over {{{(c\Psi )}^2}}} + B , $k_{3} = \frac{3}{2} R_{c} \frac{C^{2}}{{(c Ψ)}^{2}}$ {k_3} = {3 \over 2}{R_c}{{{C^2}} \over {{{(c\Psi )}^2}}} , $k_{4} = 3 R_{c} \frac{A C}{{(c Ψ)}^{2}}$ {k_4} = 3{R_c}{{AC} \over {{{(c\Psi )}^2}}} , $k_{5} = 3 R_{c} \frac{B C}{{(c Ψ)}^{2}} + C$ {k_5} = 3{R_c}{{BC} \over {{{(c\Psi )}^2}}} + C , $k_{6} = 3 R_{c} \frac{A B}{{(c Ψ)}^{2}}$ {k_6} = 3{R_c}{{AB} \over {{{(c\Psi )}^2}}} , $k_{A} = \frac{3}{2} R_{c} \frac{A^{2}}{{(c Ψ)}^{2}}$ {k_A} = {3 \over 2}{R_c}{{{A^2}} \over {{{(c\Psi )}^2}}} and k_s = k₂ + k₄.

The optimised acceleration time (T_ε _opt) yielding energy demand minimisation [Eq. (13)] is determined via the derivative: dW_T/dT_ε = 0 as: (13) $\begin{array}{l} - \frac{4}{3} k_{s} T_{ε}^{5} + [\frac{10}{3} k_{s} T_{m} + 3 k_{5} θ_{r d}] T_{ε}^{4} + [3 k_{1} - \frac{2}{3} k_{s} T_{m}^{2} - \frac{9}{2} k_{5} θ_{r d} T_{m} - \frac{24}{5} k_{3} θ_{r d}^{2}] T_{ε}^{3} + \\ + [- 7 k_{1} + \frac{2}{3} k_{s} T_{m}^{3} + \frac{3}{2} k_{5} θ_{r d} T_{m}^{2} + \frac{12}{5} k_{3} θ_{r d}^{2} T_{m}] T_{ε}^{2} + 5 k_{1} T_{m}^{2} T_{ε} - k_{1} T_{m}^{3} = 0 \end{array}$ \eqalign{ & - {4 \over 3}{k_s}T_{\rm{\varepsilon }}^5 + \left[ {{{10} \over 3}{k_s}{T_m} + 3{k_5}{{\rm{\theta }}_{rd}}} \right]T_{\rm{\varepsilon }}^4 + \left[ {3{k_1} - {2 \over 3}{k_s}T_m^2 - {9 \over 2}{k_5}{{\rm{\theta }}_{rd}}{T_m} - {{24} \over 5}{k_3}{\rm{\theta }}_{rd}^2} \right]T_{\rm{\varepsilon }}^3 + \cr & + \left[ { - 7{k_1} + {2 \over 3}{k_s}T_m^3 + {3 \over 2}{k_5}{{\rm{\theta }}_{rd}}T_m^2 + {{12} \over 5}{k_3}{\rm{\theta }}_{rd}^2{T_m}} \right]T_{\rm{\varepsilon }}^2 + 5{k_1}T_m^2{T_{\rm{\varepsilon }}} - {k_1}T_m^3 = 0 \cr}

Newton’s formula is used to solve this fifth-order algebraic equation and find the optimised acceleration time (T_ε _opt). The near-optimal acceleration time (T_ε _opt) also determines the optimal acceleration (ε_{p opt}) as given by Eq. (14): (14) $ε_{p o p t} = \frac{θ_{r d}}{T_{ε o p t} (T_{m} - T_{ε o p t})}$ {{\rm{\varepsilon }}_{p\,\,\,opt}} = {{{{\rm{\theta }}_{rd}}} \over {{T_{{\rm{\varepsilon }}\,\,opt}}\left( {{T_m} - {T_{{\rm{\varepsilon }}\,opt}}} \right)}}

During the analysis of energy consumption, it was observed that the drive’s cruising speed can be maintained at the minimum practicable values in the presence of any speed-dependent torque components (linear or quadratic), resulting in reduced energy consumption. In contrast, for any combination of these two components with constant friction, there exists only one unique solution for the manoeuvre time (T_m).

3.

Influence of Travel Time on Energy Consumption of Near-Optimal Control

To investigate the influence of travel time on the minimisation of energy consumption, the proposed control strategies were implemented into a traction unit with maximum speed v_max = 100 km/h and two induction TMs with a total nominal power of P_T = 1 440 kW, with parameters listed in Table A1 in the Appendix. The overall control system, shown in Figure 1, has a cascade structure and consists of energy-saving profile generators (EOC and ENOC) and a position control system based on FOC (Orlowska-Kowalska and Dybkowski, 2016), completed with a dynamic-lag pre-compensator (Vittek et al., 2017a,b) that ensures precise tracking of prescribed state variables, using near-optimal energy behaviour.

The maximum speed for the ENOC simulations was adjusted to match the maximum speed of the EOC, as shown in Figure 2. The geometry of the symmetrical trapezoidal speed profile was used to adjust the corresponding acceleration time T_ε_i. (15) $T_{ε i} = T_{m i} - \frac{S_{T D}}{v_{t i}}$ {T_{{\rm{\varepsilon }}i}} = {T_{mi}} - {{{S_{TD}}} \over {{v_{ti}}}}

The computed acceleration times were T_ε1 = 60.932 s, T_ε2 = 62.713 s and T_ε1 = 65.082 s. The simulation results gained for a travel time T_m₂ = 300 s, corresponding to an average speed of 60 km/h, are compared with the results from the other two travel times. Time functions of the prescribed state variables (acceleration, speed and position) for both EOC and ENOC travel times (T_m₂) are shown in Figure 3.

The functions of real control variables, including speed vs. position, speed vs. time, torque current component vs. time and energy consumption vs. time, for both EOC and ENOC are presented in Figures 4 and 5. The energy consumption profiles highlight both the peak value and total energy consumption associated with the entire vehicle, while the torque current component profile pertains to a single TM.

Tables 1 and 2 summarise the chosen maximum and minimum values of current and consumed energy for a given run. Short current switching transients were filtered, while their influence on total energy consumption was retained. The peak energy during the corresponding run is denoted as W_max. The useful energy delivered to the shaft of the TM to overcome traction resistances is referred to W_sh. The total energy consumed during a given run is indicated as W_T. The difference between W_max and W_T represents the recuperated energy during electrodynamical braking.

Table 1.

Peak and minimum values of speed, current and energy for different manoeuvre times under the EOC strategy.

V_max (km/h)	I_max (A)	I_min (A)	W_sh (Wh)	W_max (Wh)	W_T (Wh)
T_m₁ = 270 s
86.094	371.8	–352.3	4468.2	8052.2	4714.9
T_m₂ = 300 s
75.858	325.1	–305.3	4062.8	6767.9	4251.8
T_m₃ = 330 s
67.946	289.0	–268.9	3758.0	5855.7	3907.6

EOC, energy optimal control.

Table 2.

Peak and minimum values of speed, current and energy for different manoeuvre times under the ENOC strategy.

V_max (km/h)	I_max (A)	I_min (A)	W_sh (Wh)	W_max (Wh)	W_T (Wh)
T_m₁ = 270 s
86.094	228.1	–187.8	4659.4	9293.0	4921.8
T_m₂ = 300 s
75.858	196.2	–159.1	4207.3	7773.7	4406.7
T_m₃ = 330 s
67.946	170.7	–135.8	3876.7	6704.5	4032.3

ENOC, energy near-optimal control.

Subplot (a) of Figure 4 confirms that the required travel distance (S_TD = 5000 m) was reached for all three prescribed travel times. Subplot (b) presents the time functions of the three speed profiles corresponding to the prescribed travel times. Subplot (c) shows the time functions of the current torque components for the TM. Finally, subplot (d) shows the total energy consumption W_T as a function of time, along with the peak energy (W_max). To facilitate the identification of the maximum and minimum values of the selected variables, these values are summarised in Table 1.

If T_m₂ = 300 s is taken as the base for investigating the influence of travel time, a 10% decrease in travel time (T_m₁ = 270 s) results in an increase of 14.36% in the motoring torque current component and 15.39% in the generating torque current component. The magnitude of consumed energy increases by 18.97%, while the total consumed energy increases by 10.89%.

On the contrary, a 10% increase in manoeuvre time (T_m₃ = 330 s) results in a decrease of 11.1% in the motoring torque current component and 11.92% in the generating torque current component. The magnitude of consumed energy decreases by 13.48%, and the total consumed energy decreases by 8.81%.

Additionally, ENOC subplot (a) of Figure 5 confirms that the required travel distance (S_TD = 5000 m) was reached for all three prescribed travel times. Subplot (b) shows the time functions of the speed profiles for different travel times. Subplot (c) shows the time functions of the current torque components for a single TM, while subplot (d) displays energy consumption (both peak and total) as a function of time. The maximum and minimum values of the chosen variables for ENOC are summarised in Table 2.

If T_m₂ = 300 s is taken as a baseline, a 10% decrease in travel time (T_m₁ = 270 s) results in a 16.26% increase and a 15.03% increase in the magnitudes of motoring and generating torque current components, respectively. The magnitude of consumed energy increases by 19.54%, and the total consumed energy increases by 11.69%. On the contrary, a 10% increase in manoeuvre time (T_m₃ = 330 s) results in a decrease of 12.99% in the motoring torque current component and 14.64% in the generating torque current component. The magnitude of consumed energy decreases by 13.75%, and the total consumed energy decreases by 8.5%.

The influence of wide-spectrum changes in travel time, due to the necessary numerical solution of the EOC strategy, was limited to ENOC, resulting in 3D graphs that show the effects of such changes.

Consumed energy as a function of prescribed travel time (T_m) and corresponding acceleration time (T_ε) is shown in Figure 6. To show the computed real minimum energy consumption graph (6a) extends the coordinates T_m and T_ε to non-realistic values of 5000 s and 250 s, respectively. The coordinates of the computed minimum energy consumption (marked by the red point) are [T_{m min} T_{ε min} W_{T min}] = [4000.025 s 96.788 s 2104 Wh]. To obtain more realistic travel time values, T_m was halved twice and the corresponding results are summarised in Table 3.

Table 3.

Total consumed energy.

Point colour	T_m (s)	T_εε (s)	W_T (Wh)	Energy increase (%)
Red (min. point)	4100.025	96.788	2104	0.0
Blue	2050.012	48.394	2157	+2.52
Green	1025.006	24.197	2409	+14.5

Halving the coordinates T_m and T_ε twice of the computed energy minimum in Figure 6(a) results in their substantial reduction, bringing them closer to realistic values for manoeuvre time, T_m. The corresponding increase in consumed energy is only +2.52% and +14.5%, respectively. To approach real values for travel time T_m, the new coordinates in Figure 6(b) are significantly limited to [T_m _max T_ε _max W_{T max}] = [500 s 100 s 9000 Wh]. Energy consumption, computed for minor changes in travel time T_m, is now visible as dots (green, blue and red). It is evident that for shorter times, such as T_m₃ = 270 s, the energy consumption follows a steep slope.

As observed from both the evaluations of minor changes and substantial changes in travel time, a common feature for energy savings is to maintain the maximum speed of electromechanical systems at the minimum practicable values.

4.

Conclusion

This study presents significant research on electromechanical systems and the potential for energy savings through the choice of prescribed manoeuvre time. As a case study, the movement of a suburban train unit and its energy consumption are evaluated using two different energy-saving control strategies (EOC and ENOC). The train unit control system consists of an energy-saving profile generator, followed by a dynamic-lag pre-compensator, and a position-controlled induction motor drive operating under the principles of FOC. This control structure effectively eliminates dynamic lag, ensuring that pre-computed energy-optimal and near-optimal reference input functions are precisely followed, resulting in ENOC performance by the control system. The potential for experimental implementation of the proposed control strategies under laboratory conditions was presented in previous work.

The comparisons of energy consumption for travel time T_m₂ = 300 s revealed that a 10% increase in travel time (T_m₃ = 330 s) resulted in energy savings of 8.81% for EOC and 8.5% for ENOC.

Conversely, a 10% reduction in travel time T_m₁ = 270 s led to an increase in energy consumption, 10.89% with EOC and 11.69% with ENOC. In conclusion, the evaluation of energy savings confirms that the maximum speed of position-controlled electromechanical systems should be maintained at the minimum practicable level.

The findings of this study are applicable to any electromechanical system that defines the three components of load torque and uses automatic control of torque and speed. The total energy consumption for both control strategies was verified in two ways: first, by integrating the motor input power over time, and second, by summing the individual energy components, including motor losses and the energy delivered to the motor’s shaft. Simulation results confirm that the EOC strategy leads to the lowest energy consumption compared with ENOC.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Computer Sciences, Artificial Intelligence, Engineering, Electrical Engineering, Electronics

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Energy Optimal Control of Electromechanical Systems: Trade-off Demands

Branislav Ftorek

Vladimír Vavrúš

Ján Šimon

Ján Vittek

Article Category: Research Paper

Published Online: Jun 21, 2025

Page range: 177 - 188

Received: Mar 03, 2025

Accepted: May 27, 2025

DOI: https://doi.org/10.2478/pead-2025-0012

KeywordsEnergy near-optimal position control, field-oriented control, dynamic lag pre-compensator, train position control

© 2025 Branislav Ftorek et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Energy near-optimal position control, field-oriented control, dynamic lag pre-compensator, train position control