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
Urban electric railway system in
The train using Onboard supercapacitor energy storage system is represented by the following continuous  space model
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
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
Forces acting on the train in which the basic resistance force
Where a,b,c are coefficients of train’s resistance force.
The gradient force
Where g, a are the gravity acceleration and the rail track slope respective.
The energy storage system consists of an interleaved bidirectional DCDC converter with an onboard supercapacitor bank, as shown in
The DCDC converter being able to exchange energy bidirectionally 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 lowvoltage side delivers energy to the highvoltage side (DC bus); supercapacitor 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 supercapacitor modules get charged from the DC bus.
Averaged model of bidirectional DCDC converter is shown in
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.
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.
In proposed method, regenerative braking energy has recovered by SCESS. Train motion equation is rewritten:
Where
Boundary conditions are given by
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
To ensure absolute schedule time, objective function is defined as:
Where
Given
Finding the value of Lagrange multiplier
Therefore, object function is
Pontryagin’s Maximum Principle is applied to solve the train energyefficient 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
Substitute
Hamiltonian function is reformulated as:
Therefore, Hamiltonian function is maximized by the following values of
From the above analysis, five optimal control laws are designed:
Full power (FP):
Partial power (PP):
Coasting (C):
Full braking (FB):
Partial braking (PB):
Substitute (16), (19) in (18), finding the differential
equation for p(x).
Full power mode:
finding accelerating time
Using equation (22)
From (7) finding the differential equation to determine
With initial conditions:
Partial Power mode:
Using equation (23):
Where
Easily, from (15),
If l is chosen previously, solve to find the holdspeed
Coasting mode:
Where
From (7) finding the differential equation to determine
with
Partial braking mode:
Using equation (23):
Full braking mode:
Using equation (23)
From (7) finding the differential equation:
with
The charge/discharge of SCESS tracking the optimal speed profile is performed by designing controllers based on the principle CMC for DCDC interleaved converter. CMC is shown in
The inner loopthe current loop captures the inductor current dynamics; namely, managing charge or discharge of supercapacitor system, while the outer loop  the voltage loop is designed to keep DClink voltage at a certain constant value regardless of the variations of load and input voltage.
From the first equation of (6), setting
In steady state,
Therefore, the transfer function relating the inductor current with the duty cycle d(t) is computed:
Where
The corresponding PI current controller transfer function is given by:
The closedloop transfer function is shown in (38)
A firstorder 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 closedtransfer 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 
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 steadystate error and controlled bandwidth.
The transfer function of PI
The closed loop transfer function of
Using Symmetry Optimal Method with norm function finds values of
Where
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 LinhHa Dong metro line, Vietnam. There are 12 stations, 01 depot, 06 traction substations, and twoside power supply mode
Parameters of metro train 
Unit 
Value 
Train formation 
2M2T 

Number of electrical traction units 

08 
Max acceleration/braking rates 

0.94/1 
Maximum speed 
km/h 
80 
Base speed 
km/h 
32 
Parameters of route 
Unit 
Value 
Length of simulation route 
m 
12661 
Running time 

992 
David’s coefficients of train’s resistance 







Parameters of metro train 
unit 
value 
Train gandup 
2M2T 

Full load translating mass 
[kg] 
246700 
Number of electrical traction unit (N) 

08 
Max speed (V_{max}) 
km/h 
80 
Base speed (V_{b}) 
km/h 
40 
Dwell time 

30 
Max acceleration/braking rates 
m/s^{2} 
0.94/1 
Wheel diameter (D_{wh}) 
[m] 
0.84 
Parameters of supercapacitor BMOD0063 P125 B08 63F/125V 

Operation modes of electrified train includes: Accelerating
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
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
Interstation length Distance (m) Trip time (s) Practical energy consumption (kWh) Energy consumption optimization applied PMP (kWh) Energy saving (%) Cat LinhLa Thanh 931 88 8.31 7.50 9.75 La ThanhThai Ha 902 78 10.20 9.40 7.84 Thai HaLang 1076 91 10.20 9.86 3.33 LangThuong Dinh 1248 103 11.73 10.60 9.63 Thuong Dinh Ring Road 3 1010 79 13.41 12.23 8.80 Ring Road 3Phung Khoang 1480 104 16.75 15.82 5.55 Phung KhoangVan Quan 1121 86 13.85 12.66 8.59 Van Quan Ha Dong 1324 97 15.74 14.17 9.97 Ha DongLa Khe 1110 84 14.30 13.18 7.83 La KheVan Khe 1428 101 16.75 15.53 7.28 Van KheYen Nghia 1032 81 13.40 12.04 10.15
Interstation length Distance (m) Trip time (s) Practical energy consumption (kWh) Energy consumption optimization applied PMP and onboardSCESS (kWh) Energy saving (%) Cat LinhLa Thanh 931 88 8.31 7.16 13.84 La ThanhThai Ha 902 78 10.20 8.91 12.65 Thai HaLang 1076 91 10.20 9.41 7.75 LangThuong Dinh 1248 103 11.73 10.18 13.21 Thuong Dinh Ring Road 3 1010 79 13.41 11.59 13.57 Ring Road 3Phung Khoang 1480 104 16.75 15.19 9.31 Phung KhoangVan Quan 1121 86 13.85 12.05 13.00 Van Quan Ha Dong 1324 97 15.74 13.57 13.79 Ha DongLa Khe 1110 84 14.30 12.54 12.31 La KheVan Khe 1428 101 16.75 14.90 11.04 Van KheYen Nghia 1032 81 13.40 11.43 14.70
The paper proposed two integrated solutions to minimize total input energy of metro network: using the optimal speed profiles and onboardSCESS 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 energyefficient operation