Energy — Eﬃcient Operation in Subway Systems: Tracking Optimal Speed Proﬁle with on Board Supercapacitor Energy Storage System

Objectives : To verify the energy eﬃciency operation of electriﬁed trains on the certain metro line, in Vietnam by combining two solutions to recover regenerative braking energy with on-board supercapacitors and tracking the optimal speed proﬁle. Methods : This study proposes an integrated optimization method: applying Pontryagin’s maximum principle (PMP) ﬁnds the optimal speed proﬁle with ﬁxed running time and recuperating regenerative braking energy by designing the control method — Current Mode Control (CMC) to manage charge/discharge process of the on-board supercapacitor energy storage system (SCESS) tracking the optimal speed proﬁle. Findings : With this approach, a considerable reduction in consuming energy obtained for Cat Linh-Ha Dong metro line, Vietnam has been veriﬁed by simulation results on MATLAB and MAPLE software indicating that applying PMP, the highest operation energy saving is 10.15%, but if both solutions PMP and SCESS are applied, the energy saving level increases up to 14.7% in comparison with simulation results of the case of original speed proﬁle. Novelty : Combining two energy saving solutions simultaneously: applying PMP to determine the optimal speed proﬁle and using super-capacitors with CMC algorithm have recuperated the regenerative braking energy. The level of energy saving is higher than other saving solutions.


Introduction
While subway systems, as an important part of urban transport, carry passengers to their destination safely, quickly, and conveniently, they also consume a lot of energy; namely, 40% to 50% of the total energy consumed is for traction power (1) . Therefore, the studies of energy-saving in electric railways have an important economic value, https://www.indjst.org/ many energy saving methods have been published by researchers, manufacturers worldwide, typically in recuperating regenerative braking being able to reach up more than 30% of the traction energy (2) . Surplus energy in braking phases recovered by the installation of onboard or wayside energy storage systems with designing controllers to control charge/discharge process of SCESS suitable for operation characteristic of train. Studies, applications of super capacitor energy storage system with tested to practical utilization (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) . Iannuzzi et al (5) using onboard SCESS integrated with traction drive system saves energy to 38%, reducing peak power up to 50% in accelerating regime, stabilizing voltage on DC bus to 1%, increasing power supply distance among traction substations ; Dominguez et al released a study of energy consumption reduction to 24% in the Metro de Madrid (7) ; Michael Steiner et al (8) shows Bombardier installed Mitrac energy saver being able to reduce the consumption of the traction energy to 30% and line current peak and voltage drop by 50% ; Diego Iannuzzi, Flavio Ciccarelli, Davide Lauria (12) use stationary ultra-capacitor storage device for improving energy saving and voltage profile of light transportation networks; reversible substations (17,18) ; maximizing the regenerative energy exchange between trains by synchronizing their accelerating and braking phases as much as possible (19)(20)(21) , Fathy Ahmed et al. (22) applied parasitism-predation algorithm (PPA) in the energy management strategy for hybrid photovoltaic/fuel cell/battery/supercapacitor to minimize the hydrogen consumption of fuel cell; Jamadar Najimudin et al (23) developed regenerative braking system (RBS) and braking energy management techniques, considering different driving situations and road conditions which employed in addition to mechanical braking for increasing the braking efficiency of the electric vehicle system. Additionally, another energy saving approach which has also attracted more attention from experts is to find optimal speed profile by two methods: the Mathematical and Optimal theory (Maximum principle, Dynamic programming, Linear programming); the other one is the Computational Intelligence (Fuzzy neural networks, Genetic Algorithm, Predictive control, Colony Optimization Algorithm) (24)(25)(26) . A group of scientists at the University of South Australia Howlett et al. (27,28) , Vu (29) , and Albrecht et al (30,31) systematically have researched optimal strategies using mathematical approaches and optimal theory to propose control laws to detect optimal switching points, then finding the optimal speed profile, and others, Khmelnitsky (32) attempted to solve the problem of optimal control by applying the Pontryagin's maximum principle and adjoint variables, which consider regenerative braking, Liu et al. (33) uses the maximum principle for finding a set of optimal controls, the control switching graphs, and complementary conditions of optimality. Bao Huy et al. (34) applied Pontryagin's minimum principle to develop an optimal energy management strategy (EMS) for battery/supercapacitor hybrid energy storage systems replaced Dynamic Programming (DP), and the proposed optimal EMS is hundreds of times faster than DP with better results. The novel strategy is based on formulating the problem in terms of power and energy which forms a state-constrained optimal control problem. PMP is then applied with a penalty function, in which the inequality state constraints are reformulated to deduce a new state-unconstrained problem. The proposed optimal EMS is hundreds of times faster than DP with better results. Moreover, the optimal solution is piecewise constant that could give significant insights to develop real-time strategies in future studies. Shaofeng Lu (35) applied three optimization algorithms (Colony Optimization Algorithm, Dynamic Programming, GA) to search for the optimal speed trajectory, Dominguez et al. (36) designed the speed profile giving guarantees of running time and consumption energy saving, then Dominguez et al. (7) designed train operation speed profiles for an Automatic Train Operation System (ATO) to select the optimal speed profiles with reducing energy consumption, Wong and Ho (37) used genetic algorithm, Açikba, Söylemez (38) utilized artificial neural networks and genetic algorithms to determine the optimal coasting point, yet all studies have not mentioned fixed running time, as well as combined solutions to enhance energy saving percentage in electric train operation. Therefore, in this paper, a combined approach between the optimal control theory-Pontryagin's maximum principle determining the optimal speed profile, fixed running time and the onboard supercapacitor energy storage system recuperating regenerative braking energy with charging or discharging of SCESS tracking the optimal speed profile is proposed.

Urban Railway System Model
Urban electric railway system in Figure 1 consists of parts: traction substations, overhead lines or the third rail, feeders, trains as electric loads, traction drive motors, and onboard supercapacitor energy storage system including a bank of supercapacitors with bidirectional DC-DC converter. Modelling some main parts has been performed briefly.

Train model
The train using Onboard supercapacitor energy storage system is represented by the following continuous -space model (39)(40)(41) https://www.indjst.org/ Where v,t,x,m represent respectively train speed(m/s), operation time(s), train position (m), full load translating mass of train (tone) and F tr , F b ,W 0 , F grad are traction, electrical braking, resistance, gradient resistance forces applied on the train. Furthermore, differing with the conventional continuous -space model of train is supercapacitor power P sc (t) in ences in body same as Equation (1) P sc (t) accumulating/releasing energy by charging/discharging of supercapacitors tracks operating characteristics of trains.Regenerative braking energy recuperation is possible when the electric train speed is greater than 10km/h in braking phase, while mechanical braking force only works at a speed less than 10km/h in braking phase energy recover the mechanical braking force is omitted in equation (1). Figure 2 describes discharge/charge process of SCESS accordance with operation of trains in three phases. The SCESS's discharging energy supports for operating trains in accelerating regime with P sc (t)being negative; the SESSs do not charge/ discharge in coasting regime with P sc (t) equal to 0kW, and charging regenerative braking energy in braking regime with P sc (t) being positive. https://www.indjst.org/

Traction, braking forces
With characteristic curves of traction force, braking force performed by manufacturers, using the identification method: Least Square Method to find equivalent polynomials.
The maximum traction force corresponding to the speed is The maximum braking force corresponding to the speed is

Basic resistance, and gradient forces
Forces acting on the train in which the basic resistance force w 0 (v) consisting of the air resistance, the friction resistance is represented in David's formula (42) .
Where a,b,c are coefficients of train's resistance force. The gradient force F grad caused by slope of road Where g, a are the gravity acceleration and the rail track slope respective.

Supercapacitor energy storage system mode
The energy storage system consists of an interleaved bidirectional DC-DC converter with an on-board supercapacitor bank, as shown in Figure 5. The DC-DC converter being able to exchange energy bi-directionally placed between high voltage DC bus and low voltage SCESS operates in buck or boost mode: In boost mode, SBS and DBS are the operating switches, and the low-voltage side delivers energy to the high-voltage side (DC bus); super-capacitor modules get discharged by low voltage. In buck mode, SBK and DBK are the operating switches, and the high voltage side transfers energy to the low voltage side; the super-capacitor modules get charged from the DC bus.
Averaged model of bidirectional DC-DC converter is shown in Figure 6, switches are replaced by an ideal transformer with d(t) being transformer factor, and u

Control Design for Urban Railway System
The target is to minimize train operation energy by two solutions: using PMP finds optimal speed profile, and using SCESS recovers regenerated braking energy; so does control design. https://www.indjst.org/

Optimal Speed Profile determination based on PMP
Many solutions for the energy effective control are outlined to detect the optimal speed profile with minimizing train's operation energy consumption, ensuring fixed trip time as well. In the part 3, utilizing PMP computing optimal switching points of operation modes of speed trajectory applies for the train model recuperating regenerative braking energy by the onboard supercapacitor energy storage system.

Problem formulation
In proposed method, regenerative braking energy has recovered by SCESS. Train motion equation is rewritten: Where V(x) is the maximum allowable speed, X is the terminal of the train operation; v(0), v(X) are the speed at the beginning, at the end of the route; T is duration of the trip is also given by the timetable. Assume that accelerating phase happens between {t 1 , t 2 } ; coasting phase happens between {t 2 , t 3 } ; regenerative braking phase happens between {t 3 , t 4 } , thus supercapacitor power per unit mass p sc (t) is given To ensure absolute schedule time, objective function is defined as: Where T = T actual (X) − T demand (X), λ is an unknown Lagrange multiplier. Given Finding the value of Lagrange multiplier λ is to deliver the required running time T dem and (X), T actual is actual running time of train. Therefore, object function is

Solution
Pontryagin's Maximum Principle is applied to solve the train energy-efficient operation problem by seeking optimal switching points of the train's operation modes. Combining (7)-(13), the Hamiltonian function can be expressed in the form as follow: Where p 1 , p 2 are adjoint variables. Adjoint variable differential equations are reformed: https://www.indjst.org/ Hamiltonian function is reformulated as: Therefore, Hamiltonian function is maximized by the following values of u tr and u br From the above analysis, five optimal control laws are designed: Full power mode: p > 1, u br = 0, u tr = 1 finding accelerating time t a , accelerating distance x a . Using equation (22) From (7) finding the differential equation to determine x a , t a : With initial conditions: x(0) = 0,t(0) = 0.

Control Design For DC-DC Interleaved Converter
The charge/discharge of SCESS tracking the optimal speed profile is performed by designing controllers based on the principle CMC for DC-DC interleaved converter. CMC is shown in Figure 7 with the dual-loop control structure. https://www.indjst.org/  From the first equation of (6), setting

Design of the PI current-loop control algorithm
In steady state, u sc may be considered as constant and therefore acts as constant disturbance for the control loop. It can be compensated by feed-forward control. Therefore, the transfer function relating the inductor current with the duty cycle d(t) is computed: Where k = The corresponding PI current controller transfer function is given by: The closed-loop transfer function is shown in (38) G Si (s) = 1 + T I s (T II T /k pi k) s 2 + T Ii [1 + (1/k p k)] s + 1 https://www.indjst.org/ A first-order transfer function with the gain equal to 1 is stable, having static error equal to 0 and short transient process if inertia time constant is small, so authors determined in order to closed-transfer function (38) can be defined as (39): Where T' -the smaller is the better The best PI controller performance was gained when the plant's dominant pole was cancelled by the controller (39). Thus, the zero at -1 T Ii was assigned to the time constant of the plant, which was T Ii = T = L r L , and T k pi k = 2 × 10 −4 (s), so

Design of the PI voltage-loop control algorithm
For the small perturbations, the current loop acts extremely fast, and it can be assumed ideally with a gain of unity. Form the second equation of (6) the transfer function relating the voltage u DClink with the inductor current is computed: With The transfer function of outer loop is type of the integral form. However, the system still exists disturbance, so digital PI controller may be effectively used to ensure both zero steady-state error and controlled bandwidth.
The transfer function of PI G cv (s) = k pv The closed-loop transfer function of Figure 9 is shown: Using Symmetry Optimal Method with norm function finds values of k PV , T IV Where w n -Oscillation cycle, x-Damping ratio (select x=0.71) https://www.indjst.org/

Simulation Result and Analysis
To verify the effectiveness of the control strategy, a simulation is formulated. The train parameters and line in the simulation are based on the data of Cat Linh-Ha Dong metro line, Vietnam. There are 12 stations, 01 depot, 06 traction substations, and two-side power supply mode (45) .  Operation modes of electrified train includes: Accelerating (R) Coasting (R) Braking. Depending on track conditions, constraints, the speed from a station to another station is different, but it is always smaller than limit speed 80km/h. While the train runs in Optimal speeds slower than in original ones from 1 to 3 km/h, optimal switching points change, so do optimal accelerating, coasting, braking distances significantly, but keeping running time unchanged as in Figures 11 and 12. Tables 4 and 5 show that saving energy for train operation in optimal speed trajectory applied PMP up to 10.15% in comparison with original speed trajectory, level of energy saving if combining two solutions PMP and SCESS increases up to 14.7%, higher than that of energy saving of original speed only using PMP.
When the train operates in braking phase, surplus energy is absorbed by SCESS, and this kind of energy which is released to support the train in accelerating phase is represented by supercapacitor power shown Figure 10. In Figure 10 described discharge/charge process of SCESS from Cat Linh to La Thanh station (with other stations, discharge/charge process of SCESS is similar). The SCESS can discharge energy in 32 s (from 0 to 32th second) with equal to -225kW, the SESS does not charge/ discharge within 31s of coasting phase (from 33th second to 63th second), and charge in regenerative braking phase in 16s (from 63th second to 79th second) with equal to 447kW. From 80 th second to 88 th second, electric train speed is slower than 10km/h, so the regenerative braking energy is not recuperated. https://www.indjst.org/

Conclusion
The paper proposed two integrated solutions to minimize total input energy of metro network: using the optimal speed profiles and onboard-SCESS with charging/discharging processes tracking the optimal speed profile. The simulation results with studied cases for Cat Linh -Ha Dong metro line, Vietnam show that the biggest saving energy of trains' operation tracking the optimal speed profile is the highest about 10%, while if applying both solutions (tracking optimal speed profiles, and recovering regenerated braking energy by SCESS), energy saving increases significantly to 14.7%. Furthermore, this paper also has provided the foundation research for enhancing levels of energy-efficient operation by applying integrated energy saving methods which are feasible and efficient in Urban railways of Vietnam in the near future.