Urban rail transit systems can effectively alleviate urban traffic congestion due to their ability to carry large loads at high speeds for long distances. More than 300 cities around the world have developed urban rail transit systems, and their passenger volumes have reached 50%–80% in many developed cities. As of 31 December 2018, there was a total of 5,139 km of urban rail transit in 34 cities of mainland China. It is clear that city rail transit has entered a period of vigorous development.

Due to the short distances between stations on urban lines and the high density of vehicles, considerable braking energy is generated during frequent starting/braking processes. Vehicle speeds have increased in recent years, such that greater braking energy is involved [1–4]. Generally, about 30% of the electricity consumed by urban rail transit is used for auxiliary equipment such as lighting and air conditioning, 10% is used by other components, and the remaining 60% is used for traction power. Among them, the energy generated by regenerative braking accounts for at least 40% of the traction energy. To make better use of regenerative braking energy and suppress fluctuations in DC traction grid voltages, onboard supercapacitor control systems have been introduced [5].

Automotive supercapacitor control systems are nonlinear and uncertain control systems that are susceptible to end-to-end disturbance [6]. By switching the control quantity in sliding mode control, the system can slide along the sliding surface and gain good robustness and interference resistance. resistance. The algorithm used is independent of the system parameters and disturbances, so sliding mode control can be considered for the control of vehicle supercapacitor control systems. A hybrid vehicle supercapacitor system based on DC-DC electricity and batteries was proposed in Gao et al. [7] for energy recovery and utilization. A regenerative braking energy storage system for subway networks based on supercapacitors (SC) was proposed in Shetty et al. [8]. A method of optimal braking-force distribution based on an artificial neural network and sliding mode control was proposed in Ma et al. [9]. A controller composed of a radial basis function (RBF) neural network and sliding mode controller (SMC) was designed in Mao et al. [10] to recover energy from the braking process of mobile vehicles to extend driving distance and save energy. According to the requirements for fast and robust braking in automobile anti-lock braking systems, a sliding mode controller based on an RBF neural network was designed by Hung et al. [11]. An adaptive neural network sliding mode controller design method based on a decoupling method was proposed, and a trolley rod system and ball beam system were simulated.

However, in the design of practical control systems, chattering is prone to occur on both sides of the sliding mode surface when under sliding mode control. To alleviate the chattering problem effectively in traditional sliding mode control, the robustness of the neural network adaptive control system was improved to a certain extent. This study uses the advantages of RBF neural networks (excellent nonlinear function approximation, adaptiveness and self-learning abilities) to propose an onboard supercapacitor system with adaptive and robust sliding mode control. This can be used to recover regenerative braking energy and reduce the line losses of traction grids.

A structural diagram of an onboard ultracapacitor control system is shown in Fig. 1.

Due to the state discontinuity caused by frequent on-off events, the converter has nonlinear characteristics and is easily affected by disturbances, so the control effect is not ideal. A sliding mode variable structure control algorithm with independent parameter inputs and disturbances is used to design the controller [21–24], which is then switched by the control quantities in order to make the system slide along the sliding mode surface. To overcome the chattering phenomenon on both sides of the sliding mode surface, a neural network is added to approximate the nonlinear relationship between the sliding mode surface and the control quantity, so the system has good robustness and anti-interference characteristics.

According to the characteristics of the on-board supercapacitor system, a single, hidden-layer, three-layer, feed-forward, RBF neural network is adopted in this paper, which can approximate arbitrary nonlinear and linear functions. Its structure is shown in Fig. 2.

In this paper, an RBF neural network is used to approximate the nonlinear mapping relationship between the sliding mode surface and the control quantity, with the switching function s and the derivative s˙ used as inputs to the RBF neural network [25–29], while the sliding mode controller is the output. The structure of its adaptive sliding mode control system based on RBF is shown in Fig. 3.

The input and output algorithms of the RBF network are supposed as:

(12) hj=exp⁡(‖x−cj‖22bj2)

(13) f=W∗Th(x)+ε

Among them, x is the input of the neural network, i is the network input of the first input layer, j is the network input of the hidden layer, h=[hj]T is the output of the Gaussian basis functions, W∗ is the ideal network weight, ε is the network approximation error ( ε≤εN ) and f is the output of the neural network.

The input of the neural network is x=[e e˙]T , then its output is:

(14) f^(x)=W^Th(x)

The control law is:

(15) u=1g(−f^+θ¨d+ce˙+ηsgn(s))

Substituting control law Eq. (15) into Eq. (5), we can obtain:

(16) s˙=θ¨d−f−gu−d(t)+ce˙=θ¨d−f−[−f^+θ¨d+ce˙+ηsgn(s)]−d(t)+ce˙=−f+f^−ηsgn(s)−d(t)=−f~−ηsgn(s)−d(t)

Among them, f~=f−f^=W∗Th(x)+ε−W^Th(x)=W~Th(x)+ε , W~=W∗−W^ .

The Lyapunov function is designed as:

(17) L=12s2+12γW~TW~

Among them, γ>0 .

From Eqs. (17) and (16), it can be obtained that:

(18) L˙=ss˙+γW˜TW˜˙=s(−f˜−d(t)−ηsgn(s))−γW˜TW^˙=s(−W˜Th(x)−ε−d(t)−ηsgn(s))−γW˜TW^˙=−W˜T(sh(x)+γW^˙)−s(ε+d(t)+ηsgn(s))

The adaptation law can be derived as follows:

(19) W^˙=−1γsh(x)

Then,

(20) L˙=−s(ε+d(t)+ηsgn(s))=−s(ε+d(t))−η|s|

Because the RBF network’s approximation error ε is very small, we set η≥εN+D , then L˙≤0 . When L˙=0 , s≡0 , so according to the Lasalle principle of the invariant, the closed-loop system is asymptotically stable. When t→∞ , s→0 . It can be seen that the role of the robust term ηsgn(s) in the control law is to overcome interference and neural network approximation errors to ensure the stability of the system.

A simulation test was carried out on a certain track line with an operating interval of 1.98 km. The parameters of the supercapacitor are: capacitance = 30 F, rated voltage = 500 V and working current = −400–400 A. The parameters of the bidirectional DC/DC converter are: energy storage inductance = 7.5 mH and filter capacitance = 30,000 uF. The IGBT switching frequency = 10 kHz and the standard supply voltage of the DC traction network = 1500 V [30,31].

Simulink was used to establish a simulation model of a single train with an on-board ultracapacitor (Fig. 4), which can be used to robustly verify the absorption and utilization of regenerative braking energy by the designed control system.

It can be seen from the simulation results (Fig. 5) that when the system is not connected to the on-board supercapacitor energy storage system and the train accelerates and tows, the network voltage fluctuation is about 400 V. The voltage difference between the train deceleration stage and the traction network is about 200 V. It can be seen that the voltage fluctuation of the traction network is high and braking energy cannot be recovered during this process. When the system is connected to the onboard supercapacitor energy storage system, the voltage difference between the train acceleration stage and the traction network is about 150 V, while that between the train deceleration and braking stages is about 90 V. Hence, the designed neural network adaptive sliding mode control system obviously reduces the dynamic amplitude of the traction grid voltage and, at the same time, effectively absorbs and utilizes regenerative braking energy.

It can be seen from Fig. 6 that, the designed control system can follow the ideal train operating speed curve well. This significantly improves the response speed and steady-state error of the control system, enhances the external anti-interference ability and improves its robustness, making driving more comfortable, safe and stable.

Figs. 5–7 show that the designed on-board ultracapacitor control system can alleviate drops in traction grid voltage during train starts and absorb energy during braking, and can also prevent increases in grid pressure. When the braking resistor is used before optimization, the line losses of the traction grid are about 1.395 kWh. When the onboard ultracapacitor control system is used and the neural network adaptive sliding mode control method is adopted, the line loss of the traction grid is about 0.38 kWh. Hence, losses are reduced by 37.4%, which avoids energy wastage.

From the above analysis, it can be seen that the vehicle-mounted ultracapacitor energy storage system can effectively utilize regenerative braking energy, suppress fluctuations in grid voltage and reduce losses in the traction grid.

(1) With their high power density and long service life, ultracapacitors have unique advantages and development prospects in utilising regenerative energy in urban rail trains. They can improve the braking power of high-speed trains and allow operation without feeders, making trains more comfortable, safe and stable.

(2) An adaptive, neural network-based, sliding mode control system for supercapacitors was proposed in this paper. Through simulation, it was verified that the system can suppress fluctuations in network pressure, reduce losses in traction grids, and improve the utilization rate of regenerative braking energy.