# Regenerative energy management of electric drive based on Lyapunov stability theorem

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## Abstract

In recent years, urban rail systems have developed drastically. In these systems, when induction electrical machine suddenly brakes, a great package of energy is produced. This package of energy can be stored in energy storage devices such as battery, ultra-capacitor and flywheel. In this paper, an electrical topology is proposed to absorb regenerative braking energy and to store it in ultra-capacitor and battery. Ultra-capacitor can to deliver the stored energy to DC grid and to charge the battery for auxiliary applications such as lighting and cooling systems. The proposed system is modeled based on large signal averaged modeling, which leads to the simplicity of calculations. The control system is based on Lyapunov stability theorem which guarantees system stability. Also, an energy management algorithm is proposed to control energy under braking and steady-state conditions. Finally, the simulation results validate the effectiveness of the proposed control and energy management system.

## Keywords

DC/DC converter Lyapunov stability Bidirectional converter Energy management system (EMS) Ultra-capacitor Battery Switching function## 1 Introduction

Capacity, reliability and safety of urban rail systems make these devices suitable for public transportation in developed countries [1, 2]. Considering energy price and climate change, energy saving has become an important subject for research studies. Consumed energy in urban rail systems is divided into two parts, traction usage and non-traction usage. In such systems, about 50% of total consumed energy is related to the traction requirements and the rest is related to non-traction usage or auxiliary systems, such as cooling systems and lighting systems [3, 4], and therefore designing a power electronic topology capable of providing energy for these usages, apart from many benefits, can be useful to the economy.

The topic of energy saving in urban rail systems has been investigated in different aspects. In [5], an energy management strategy for capacitor is proposed to adjust charging and discharging threshold voltage based on analysis of train operation states. The main parameter for energy calculations is state of charge (SOC) of energy storage device. In [6], capacitor is used for energy saving in train systems and a hierarchical control strategy is proposed based on energy management section and converter control section. The energy management system works based on an introduced machine and converter control mainly consist of a proportional-integral (PI) closed-loop strategy. Also an optimization algorithm is proposed to estimate the control parameter values at different operations. In [7], a train system considering renewable energy sources (photovoltage and wind power) and the capabilities of using regenerative braking energy is investigated. Apart from these aspects, uncertainties of renewable energies are considered through different scenarios and the whole problem is considered and solved as a large-scale nonlinear optimization problem. Energy and economic energy saving of the proposed system under different strategies is also studied.

In this paper, a topology for saving regenerative braking energy in storage devices is proposed and control system is designed. A bidirectional DC/DC converter and a unidirectional DC/DC converter are connected in series. Also, ultra-capacitor and battery are used as main energy storage devices. Regenerative energy generated by induction electrical machine (IEM) is a high power density package of energy which occurs during a very short period of time, so must be stored in a device with high power density such as ultra-capacitor [8, 9, 10]. To increase the reliability and system efficiency, ultra-capacitor is connected to DC link via a bidirectional DC/DC converter [11, 12, 13, 14].

To control the proposed system, switching functions are extracted based on state-space equations [15]. Extraction of switching functions is a well-known method to control switching process of power electronic devices, in which, switching functions are obtained based on system’s requirements [16]. In this paper, switching functions are extracted using fundamentals of Lyapunov stability theorem. Fast and accurate tracking of reference values and maintaining system’s stability are main advantages of this method.

## 2 Modeling and control of proposed system

*I*

_{m}is the current from IEM to DC link capacitor. \( I_{{L_{ 1} }} \) is the current of bidirectional converter and is positive if the converter works in buck mode, or negative if the converter works in boost mode. \( I_{{L_{ 2} }} \) that is either positive or zero, is the current of buck converter.

*V*

_{dc}and

*C*

_{dc}are the voltage and capacitor of DC link, respectively. Also,

*C*

_{uc},

*R*

_{uc}and

*V*

_{uc}are capacity, resistance and voltage of ultra-capacitor, respectively.

*V*

_{b}is voltage of the battery.

*d*

_{1},

*d*

_{2},

*d*

_{3}are the duty cycles of switches S1, S2 and S3, respectively.

*L*

_{1}and

*L*

_{2}are the inductors of bidirectional and unidirectional converters, respectively. Moreover, there is a dynamic resistor

*R*

_{dynamic}that must dissipate surplus energy when DC link capacitor and ultra-capacitor are fully charged. Therefore,

*S*

_{d}and

*u*

_{d}are the switch and its duty cycle of the circuit that connect the dynamic resistor to the DC link.

*k*is described as:

*d*

_{12}combined of

*d*

_{1}and

*d*

_{2}is generated as [18]:

*d*

_{12}is the switching function of bidirectional converter.

### 2.1 Switching functions extraction using Lyapunov stability theorem

*V*: ℝ

^{n}→ ℝ ≥ 0 for \( \dot{\varvec{x}} = \varvec{f}(\varvec{x}) \) is a continuously differentiable function such that there exist \( \alpha \), \( \beta \) belong to class

*Κ*

_{∞}, a continuous positive definite function\( \gamma \): ℝ

^{n}→ ℝ ≥ 0 for

*∈ ℝ*

**x**^{n}[19]. State variables of the system must be defined as a form of their errors:

*x*

_{1}to

*x*

_{4}are the errors of state variables; superscript * represents the reference values of corresponding variables. Matrix \( \dot{\varvec{X}} \) is introduced as:

*is the input matrix and includes system inputs and constant values of state matrix, calculated as:*

**B***V*must be between the smallest and the largest eigenvalues of

*[20]. Therefore \( \alpha \) and \( \beta \) in (4) are equal to the smallest and the largest eigenvalues of*

**P***, respectively, namely*

**P***λ*

_{min}and

*λ*

_{max}:

*λ*

_{max}:

*V*satisfies the following inequality:

*d*

_{3}and

*d*

_{12}must be calculated in a way that (18) stays negative and system remains globally stable, therefore:

### 2.2 Energy management algorithm

*SOC*

_{b},

*SOC*

_{uc}and

*V*

_{dc}. The maximum and minimum values of

*SOC*

_{uc},

*SOC*

_{b}and

*V*

_{dc}are chosen according to the systems’ requirements. In this case,

*SOC*

_{uc},

*SOC*

_{b}are chosen as a value between 0 and 100%, and

*V*

_{dc,max}and

*V*

_{dc,min}are voltage parameters based on operator’s choice. The system has three operational modes that are not enabled together, and priority of these modes is based on the followings: ① ultra-capacitor charging by DC link voltage when

*V*

_{dc}>

*V*

_{dc,max},

*SOC*

_{uc}<

*SOC*

_{uc,max}; ② DC link capacitor charging by ultra-capacitor when

*V*

_{dc}<

*V*

_{dc,min},

*SOC*

_{uc}>

*SOC*

_{uc,min}; ③ battery charging by ultra-capacitor when

*V*

_{dc,min}<

*V*

_{dc}<

*V*

_{dc,max},

*SOC*

_{uc}>

*SOC*

_{uc,min},

*SOC*

_{b}<

*SOC*

_{b,max}. And an auxiliary mode with following conditions:

*V*

_{dc}

*>*

*V*

_{dc,max},

*SOC*

_{uc}

*>*

*SOC*

_{uc,max}.

*u*

_{1},

*u*

_{2}and

*u*

_{3}are gating signal of S1, S2 and S3, respectively.

## 3 Simulation results

^{2}). As observed in the figure, the speed reaches 100 rad/s in 2 s and when braking, it decreases from 150 rad/s to 0 rad/s in 3 s.

Parameters of electrical machine, grid and proposed drive

Parameter | Value |
---|---|

Nominal power | 37.3 kW |

Nominal voltage and frequency | 460 V, 60 Hz |

Grid voltage | 367 V |

Grid inductor | 2 mH |

DC link capacitor | 1.6 mF |

Bidirectional converter inductor | 3.3 mH |

Unidirectional converter inductor | 33 mH |

Ultra-capacitor | 21.27 F |

Series resistance | 0.1 Ω |

Battery voltage | 24 V |

Figure 5b shows torque curve during cycle. When the IEM accelerates, torque is positive and when the IEM brake, torque is negative. Figure 5c shows DC link voltage during this cycle. When the IEM accelerates at 5 s, *V*_{dc} drops and when IEM brakes at 11 s, *V*_{dc} increases.

The main idea of the proposed control system is to store regenerative energy in ultra-capacitor and battery. Besides that, whenever *V*_{dc} drops down, ultra-capacitor will supply DC link capacitor with its charged energy. The proposed system must work accurately based on flowchart shown in Fig. 3 and track the reference values of state variables, \( I_{{L_{1} }}^{*} = 1{\text{ A}} \) (buck), \( I_{{L_{1} }}^{*} = 20{\text{ A}}\, \)(boost), \( I_{{L_{2} }}^{*} = 2{\text{ A}} \), \( V_{dc}^{*} = 500{\text{ V}} \) and \( V_{uc}^{*} = 30{\text{ V}} \).

*δ*

_{1}= −60,

*δ*

_{2}= −0.2,

*δ*

_{3}= −5,

*δ*

_{4}= −0.1. After applying the proposed system, the extra energy resulted by braking process must be stored in ultra-capacitor. Also, DC link voltage drop resulted by acceleration process is compensated by energy stored in ultra-capacitor. As seen, at 11 s,

*V*

_{dc}rises during the braking of IEM. Before applying the proposed system, in this moment

*V*

_{dc}reaches 1800 V but after applying the proposed system,

*V*

_{dc}is limited near to 500 V. Energy management system decides whenever

*V*

_{dc}is higher than

*V*

_{dc,max}and

*SOC*

_{uc}is lower than

*SOC*

_{uc,max}, and then the bidirectional converter operated in buck mode. Other modes are basically applicable according to the requirements of the system and the energy management system designed in the previous section. As shown in Fig. 6, in the 5–7 s, DC link voltage drops, ultra-capacitor charges the DC link capacitor.

*V*

_{dc}, and therefore the SOC of ultra-capacitor is always alternating (decreasing or increasing). However, since battery is operating only when unidirectional converter is activated, its SOC is sometime constant and the other times increasing.

As seen in Fig. 1b, there is an auxiliary load that uses the energy stored in battery. Due to different reference currents of ultra-capacitors in modes 1 and 2, whenever the system operates in mode 2, *SOC*_{uc} rises sharply and whenever system works in mode 1, *SOC*_{uc} falls slowly. On the other hand, the incline of *SOC*_{b} is always constant because reference current of battery is always 2 A and doesn’t change. Because of very high amount of regenerative energy that is generated while braking, whenever the IEM enters braking mode, the SOC of ultra-capacitor sees the biggest changes. It is also observed that all of ultra-capacitor energy that is consumed by DC link capacitor and battery before 11 s, is compensated after 11 s due to braking process.

*V*

_{dc}increases (and

*SOC*

_{uc}is lower than

*SOC*

_{uc,max}), bidirectional converter works in buck mode, ultra-capacitor gets charged and battery voltage remains constant. Whenever

*V*

_{dc}decreases, ultra-capacitor voltage decreases to charge the DC link capacitor and compensate voltage drop in DC link. In mode 3, whenever DC link voltage is in admissible range and

*SOC*

_{uc}is higher that

*SOC*

_{uc,max}, battery gets charged by the energy stored in ultra-capacitor. Whenever battery is being charged, a 0.9 V growth in battery voltage can be seen.

## 4 Conclusion

In this paper, a topology for saving regenerative braking energy in ultra-capacitor and battery was proposed. The proposed circuit is also able to compensate DC link voltage drop, while IEM needs to accelerate. The topology is based on cascade structure of DC/DC converters. The controller is designed based on Lyapunov stability theorem that guarantees system’s global stability. Simulation results validate that the proposed system works properly and all reference values are accurately tracked. It is also shown that energy management algorithm works in coordination with the controller. The proposed system can be implemented in all devices and facilities equipped with induction machines in order to store additional energy and consequently reducing the total costs.

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