Speed profile optimization of catenaryfree electric trains with lithiumion batteries
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Abstract
Catenaryfree operated electric trains, as one of the recent technologies in railway transportation, has opened a new field of research: speed profile optimization and energy optimal operation of catenaryfree operated electric trains. A wellformulated solution for this problem should consider the characteristics of the energy storage device using validated models and methods. This paper discusses the consideration of the lithiumion battery behavior in the problem of speed profile optimization of catenaryfree operated electric trains. We combine the single mass point train model with an electrical battery model and apply a dynamic programming approach to minimize the charge taken from the battery during the catenaryfree operation. The models and the method are validated and evaluated against experimental data gathered from the test runs of an actual batterydriven train tested in Essex, UK. The results show a significant potential in energy saving. Moreover, we show that the optimum speed profiles generated using our approach consume less charge from the battery compared to the previous approaches.
Keywords
Electric train Catenaryfree operation Speed profile optimization Energy efficiency1 Introduction
In the recent years, there has been an increasing interest in catenaryfree operated electric trains equipped with onboard energy storage devices. Catenaryfree operated electric trains can minimize the cost by reducing the maintenance and installation costs of the catenary system and at the same time benefit from the high efficiency of the electric traction system.
There are already light rail vehicles (LRV) such as short distance metro trains or trams with onboard energy storage devices in service (see, e.g., [1]). However, the concept of the catenaryfree operation of medium sized electric multiple units (EMU) with onboard energy storage devices for intercity operations remains mostly in the prototype phase or short distances [2]. An exception is a batterydriven train operated by East Japan Railway Company. In 2014, East Japan Railway Company introduced a battery train equipped with lithiumion batteries; the train is equipped with \({190}\,\hbox {kWh}\) batteries and can go up to \({100}\,\hbox {km/h}\) and run approximately \({20}\,\hbox {km}\) in catenaryfree operation [3].
A major challenge in designing a catenaryfree EMU is the sizing of the batteries, since both the weight of the vehicle and the longer distance that the vehicle needs to run in catenaryfree mode requires batteries with both high energy capacity and high power output. Moreover, a standard regenerative brake system is not always sufficient for charging the batteries, and charging stations are needed. The charging stations should further be capable of charging the batteries in a short time. Fast charging stations for the battery trains are already developed [4], and train units are becoming more energy efficient than before. In spite of the recent developments in designing charging stations and increasing efficiency in train units, energy optimal operation procedures are still crucial for batterydriven trains. According to AlEzee et al. [5], energy management systems are needed to monitor the state of charge and optimize the usage of the batteries during catenaryfree operation. The general focus of this paper is on including the battery characteristic in the problem of energy optimal operation and verifying the results with experimental evaluations.
2 Related work and contributions
For around five decades, studies have been performed on energyefficient train operation for electric and diesel trains (see, e.g., [6, 7, 8, 9, 10, 11, 12]). The proposed solutions cover a wide range of applications from diesel heavy haul trains (e.g., [13]) to multipletrain operation (e.g., [14, 15, 16]). Apart from the methodological research in this field, there has also been research on designing driver advisory systems based on mathematical formulations and optimizations (see e.g., [17]). Hybrid electric trains have also been studied for the speed profile optimization. Miyatake and Matsuda [18] proposed an algorithm based on sequential quadratic programming for energyefficient operation of hybrid electric trains with an onboard electric doublelayered capacitor (EDLC) as the secondary energy source beside the catenary systems. Iannuzzi and Tricoli [19] also proposed an energy management control strategy for a similar train configuration with supercapacitors as the secondary energy source. Sumpavakup et al. [20] presented an approach to minimize the power consumption on the substation through optimizing the use of onboard supercapacitors on the train during the operation. A comprehensive review on different energyefficient train control methods is provided by Scheepmaker et al. [21].
Although the energyefficient train operation is a well studied subject, due to the immaturity of the technology, literature is sparse regarding the energyefficient catenaryfree operation of EMUs. Miyatake and Haga [22] assumed an EDLC as the onboard energy storage for an LRV and solved the energyefficient catenaryfree operation problem using dynamic programming by minimizing the electric power at the capacitors as the objective function. Colak et al. [23] considered the problem as a coast control problem and used particle swarm optimization to find the optimum coasting points for the catenaryfree operation of LRVs with an EDLC as the onboard energy storage device. Li et al. [24] proposed a charging strategy for onboard supercapacitors during the catenaryfree operation to improve the dynamic performance and the reliability of the charging system. Ishino et al. [25] studied the effects of charging and running time on energyefficient operating strategies for an LRV with an onboard EDLC. Miyatake and Ko [26] presented three solutions for energyefficient train control problem based on dynamic programming, gradient method and sequential quadratic programming and argued that they can also be used for the catenaryfree operation; the energy storage device used in the simulation done by Miyatake and Ko is also assumed to be an EDLC.
Most of the research done under the subject of energyefficient catenaryfree operation of electric trains consider an ELDC or a supercapacitor as the energy storage device. EDLC can provide high power and fast charge/discharge time. But on the other hand, it suffers from the extreme voltage drop and low energy density. Hence, it is used for LRVs such as trams with low speed [27]. The few available intercity catenaryfree EMUs use lithiumion batteries or fuel cells as the energy storage device, which have different behavior than ELDCs or supercapacitors [28, 29, 30].
A model of the behavior of the battery can be used to consider the battery characteristics for the problem of speed profile optimization. Battery models are divided in three categories of mathematical, electrochemical, and electrical models [31]. Mathematical models are categorized in two groups of stochastic models (that are mostly based on Markov Chain, e.g., see [32]) and analytical models (e.g., [33]). Electrochemical battery models use electrochemical equations to model the behavior of each cell. Since these types of models are physicsbased, they can provide information on the full dynamic behavior of the battery. But they use a set of partial differential equations, which makes them too complex for fast simulation purposes. According to Fotouhi et al. [34], reducedorder electrochemical models and electrical models are the best choices for the energy management application. Reducedorder electrochemical models are basically a simplification of the complex electrochemical models. Hence, in order to generate such models, an accurate highfidelity electrochemical model needs to be generated, which can be a complex procedure. Electrical battery models on the other hand use different electrical components (e.g., capacitors and resistors) to build an electrical circuit with a behavior similar to the battery. These models provide enough accuracy needed for the battery management application, while avoiding unnecessary complexities of the electrochemical models [34]. Due to their simplicity, electrical models are widely used for battery management applications, such as the application for electrical vehicles (e.g., [35, 36]) and even hybrid propulsion systems for aviation purposes [37, 38]. Electrical models have also been used for other applications such as the modeling of renewable energy systems [39]. An overview on different electrical models is presented by Mousavi and Nikdel [40].
Energyefficient catenaryfree operation of EMUs with lithiumion batteries differs from the case where a supercapacitor or an EDLC is used, as the dynamic behavior of capacitors is different from lithiumion batteries [41]. In the literature, there is no consideration of dynamic battery characteristics for the problem of speed profile optimization. The only exception is the research presented by Noda and Miyatake [27], in which the dynamic characteristics of the lithiumion batteries are considered using a fitted function from the battery supplier’s discharge curve. Noda and Miyatake [27] also note that one of the next steps in the field of energy optimal operation of catenaryfree operated electric trains is to present a relationship between the travel time and energy consumption. Furthermore, all of the solutions for the problem of speed profile optimization of catenaryfree battery trains minimize the energy consumption in kWh on the battery (e.g., DC link) as the objective function, which is calculated based on the mechanical power calculation. But the mechanical power calculation does not provide the best solution as it does not consider the dynamic behavior of the battery (i.e., voltage fluctuation). Most importantly, there are no experimental evaluations of the methods presented in the literature for the speed profile operation during the catenaryfree operation.

We present a method to estimate the state of charge of the battery for the speed profile optimization. For this purpose, we combine the single mass point train model with an electrical battery model. The method is later on used in a dynamic programming approach for the speed profile optimization of batterydriven electric trains during the catenaryfree operation. We minimize the charge taken from the battery instead of the energy consumption in \(\hbox {kWh}\) as the objective and show that our approach will result in more energyefficient speed profiles.

The method and battery models used for the estimation of state of charge are validated against the measurements from the test runs of an actual batterydriven train. We also evaluate the performance of the dynamic programming approach and measure the potential energy saving against the actual test runs of the train. This is the first experimental evaluation of an energy optimal strategy for the catenaryfree operated electric trains against measured values.
Summary of the research presented in the field of energy optimal operation of catenaryfree operated electric trains compared to the current paper
References  Train  Energy  Approach  Experimental evaluation  Objective  

Type  Mass (t)  \(\hbox {Maximum speed}^{*}\) \((\hbox {km/h})\)  Storage  
Miyatake and Haga [22]  LRV  30  30  EDLC, electrical model  Dynamic programming, \(x=(s,v,V)^{\dagger }\)  No  Minimizing power on EDLC (\(V\cdot I)^{\dagger }\) 
Colak et al. [23]  LRV  45  50  EDLC, electrical model  Coast control, particle swarm optimization  No  Minimizing energy on the EDLC (\(C(V_1^2  V_2^2)/2)^{\dagger }\) 
Ishino et al. [25]  LRV  25  40  EDLC, electrical model  Simulation  No  Effects of charging/travel time on energy consumption 
Noda and Miyatake [27]  EMU  80  50  Liion battery, regression model  Dynamic programming, \(x=(s,v)^{\dagger }\)  No  Minimizing mechanical power on batteries (\(F\cdot v)^{\dagger }\) 
Current research  EMU  185  100  Liion battery, two electrical models  Dynamic programming, \(x=(s,v, \text {soc})^{\dagger }\)  Yes  Minimizing charge from the battery (\(I\cdot t)^{\dagger }\) 
The rest of the paper is as it follows: In Sect. 3, we present the optimization approach together with a model and method for estimating the state of charge. We first summarize the dynamic programming approach used for the speed profile optimization of electric trains together with a description of the modifications needed for the problem with the batterydriven trains (Sect. 3.1). Considering the dynamic programming approach, we present a model and an approach for estimating the state of charge for the speed profile optimization in Sect. 3.2. The approach requires a battery model. Two battery models are suggested for this purpose in Sect. 3.3. Section 4 includes the experimental validation and evaluation using measurements from the test runs of an actual batterydriven train. We use a batterydriven train designed by Bombardier Transportation and tested in UK as the experimental case for our evaluations. We first validate the suggested battery models and the method for estimation of the state of charge in Sect. 4.1. Further, we evaluate the results of the dynamic programming approach against the test runs of the batterydriven train in Sect. 4.2. The paper continues with discussions on the method for the estimation of the state of charge and the optimization approach in Sect. 5. Section 5 also includes a discussion on the choice of the objective function. We end the paper with an overall conclusion and a short discussion on the future work in Sect. 6.
3 Optimization and system modeling
In this paper, we use dynamic programming as the optimization technique for the speed profile optimization. The methodology to apply dynamic programming for the speed profile optimization and its application for designing a driver advisory system is presented by Ghaviha et al. [17]. The basics of the same method have been used to solve the problem of speed profile optimization for catenaryfree operated EMUs using the objective function of minimizing energy consumption on the DC link [42]. Ghaviha et al. [42] did not consider the behavior of the battery, nor did they have any experimental evaluation of their results. The same approach is used in this paper as the basis for the speed profile optimization considering the characteristics of the battery. Here, we briefly present the approach introduced by Ghaviha et al. [17, 42] and we further present the modifications needed for the consideration of the battery characteristics. The final approach uses the state of charge of the battery as an input variable. Hence, we present a method to estimate the state of charge for our application, which represents a propulsion system model for the speed profile optimization of battery trains. Finally, we present two battery models suitable for our application.
3.1 Optimization technique
For the dynamic programming approach in this research, we assume travel time as the horizon and total charge from the battery as the cost function. In other words, the aim is to minimize the charge taken from the battery over the travel time.
In the previous research in the literature, the transition cost was assumed to be the energy consumption on the DC link in kWh. In this paper, in order to consider the characteristics of the battery, we assume the transition cost to be the charge taken from the battery in Ah.
3.2 State of charge estimation and the propulsion system model
3.3 Battery models
As presented in Sect. 2, electrical battery models provide a balance between simplicity and accuracy needed for transportation application [34, 40]. In this section, two electrical battery models are suggested and validated, and the suitable one is chosen for the speed profile optimization using dynamic programming.
The generic characteristic of these models is one of the main reasons for the wide application of them, since both battery models can be applied to different batteries with different chemistries [34]. Moreover, the design variables of both models can be easily calculated using the manufacturers’ data sheet [46]. The simple battery model has been used before for different applications such as leadacid battery modeling (see, e.g., [47]) and for the application of wind power generation (see, e.g., [48]). The generic battery model has also been used in different applications such as photovoltaic power generation (e.g., [39]), wind energy generation (e.g., [49]) and DC microgrid (e.g., [50]).
To choose the right model for the speed profile optimization, we need to validate and compare both models for the estimation of the state of charge during operation.
4 Validation and evaluation
In this section, we validate and evaluate both battery models and the method for the estimation of the state of charge and choose the suitable battery model. Further, we apply the dynamic programming approach presented in Sect. 3 using the chosen battery model and present the results compared to the experiments with an actual battery train.
The IPEMU train was operated in two modes: the catenary mode and the catenaryfree mode. During the catenary mode, the train would run under the overhead lines and use the power grid for both driving and charging the batteries. In the catenaryfree mode, the train would operate without the connection to the power grid and by using only the batteries. Regenerative brake system charges the batteries during the catenaryfree mode. The operation mode targeted in this research paper is the catenaryfree mode.
4.1 Battery models validation
MAPE and Rsquared values for voltage estimation
–  Generic battery model  Simplified battery model 

Rsquared  0.95  0.93 
MAPE (%)  0.63  0.77 
Comparing both battery models using Fig. 5, it can be seen that the simplified battery model is underestimating the voltage value. This is because in the simple battery model, the battery voltage is assumed to be constant with respect to the state of charge. Despite the underestimation, there is still a low deviation of \(0.77\%\) in average for the voltage estimation (Table 2). This issue is improved in the generic battery model, as the state of charge is one of the considered variables in this model.
MAPE and Rsquared values for state of charge estimation
–  Generic battery model  Simplified battery model 

Rsquared  0.99  0.99 
MAPE (%)  0.11  0.17 
The results of the state of charge estimation validation show that both battery models can estimate the state of charge with minor error (MAPE in Table 3 and Fig. 7). It is also understood from the value of Rsquared and the parity plots in Fig. 8 that both models provide values with minor deviation from the measured values.
The validation results in general show that both battery models function with similar accuracy for our application in modeling of the state of charge. Apart from the accuracy, there is also the factor of simplicity of the model. The final goal is to use the battery and train models in a dynamic programming approach to minimize the charge used from the battery. Although DP is known to be suitable for speed profile optimization of trains and batterydriven trains (see, e.g., [27]), it is also wellknown that it suffers from the curse of dimensionality [17]. Therefore, it is important to keep the calculations as simple as possible. Moreover, there is a problem of generating the coefficients for the generic battery model (Eqs. (10) and (11)) from the discharge curve. To generate the coefficients, three points from the discharge curve need to be selected [45, 46]. In our experience, the points need to be selected with a high accuracy to have a precise model. In light of this, considering the overall simplicity of the simplified battery model over the generic model (both the equations and the coefficients), and the similar accuracy of both models in estimating the state of charge, we can conclude that the simplified battery model is suitable for our optimization application.
4.2 Experimental evaluation
The batterydriven EMU was tested for a limited number of test runs on a certain line section in UK. We apply the presented dynamic programming approach using the simplified battery model to find the optimum speed profiles for two trips.
5 Discussion
5.1 Battery models validation
The validation is done against the results from two of the longest journeys a batterydriven train has ever done in the catenaryfree mode. The journey consists of constant load phases and multiple charge/discharge phases in which the peak charge and discharge powers were observed. In other words, the full scope of the operation is included in the validation experiments. The validation experiments cover a range of state of charge from 85% to 25%. The performance of the lithiumion batteries are relatively stable between around 95% and 5% of the state of charge, which is also the recommended operating condition from most of the battery suppliers [44]. Therefore, the validation is acceptable, even though it was done for a specific range of state of charge. This can also be understood from the results of the validation. Both battery models result in a similar accuracy, which shows that in the IPEMU train, the performance of the battery is relatively stable with respect to the changes in state of charge. Moreover, the batteries used for the application of catenaryfree operation are sized with the consideration of redundancy for the extreme cases. During the test runs of the IPEMU train for instance, the state of charge below 25% was not observed.
The method used in this paper to measure the state of charge is based on the Coulomb counting method, which measures the state of charge according to the integral of the current [52]. This method, although accurate, is sensitive to the initial value of the state of charge. The final goal with speed profile optimization in this paper is to implement it in a form of a driver advisory system onboard the train (such as the one presented Ghaviha et al. [17]). There are already battery systems available with battery modules that take advantage of sophisticated electrochemical battery models to monitor the state of charge with high accuracy. Such systems are usually provided by the battery supplier. A driver advisory system based on the approach presented here will be used together with such battery management system onboard the train and will take the initial value of state of charge from it.
Apart from the state of charge and current, two other variables of temperature and state of health can also affect the voltage of a battery [34]. Although the models presented in this paper are sufficient for the speed profile optimization, having a more detailed battery model can lead to a more accurate voltage modeling. For instance, if needed, the behavior seen from time \({140}\,\hbox {s}\) to \({230}\,\hbox {s}\) in Fig. 4 (or the behavior seen in Fig. 5 around \({720}\,\hbox {V}\) to \({740}\,\hbox {V}\) in the measured value) can be modeled with a more detailed battery model.
5.2 Selection of the objective function
We minimize the charge taken from the battery (in Ah) instead of the conventional approach of minimizing the energy consumption (in kWh) in the form of mechanical power. Charge in Ah times voltage is the energy consumption in kWh. Since the voltage is not constant during operation, measuring of the charge in A h will have a better estimation of the energy taken from the battery and the capacity left.
Comparison between the optimization using charge minimization as the objective function and power consumption minimization as the objective function
Point no.  Time (s)  Optimum charge consumed on the battery with charge in Ah as the objective (Ah)  Optimum charge consumed on the battery with energy consumption in kWh as the objective (Ah) 

1  380  28.6784  28.7595 
2  375  28.8727  28.9267 
3  370  28.9993  29.1427 
4  365  29.1773  29.2429 
5  360  29.3331  29.4101 
6  355  29.5022  29.5136 
7  350  29.6528  29.7465 
8  345  29.8098  29.8290 
9  340  29.8791  30.0549 
10  335  30.1080  30.1587 
11  330  30.4443  30.4574 
12  325  31.2518  31.3757 
13  320  32.0509  32.1410 
14  315  34.3438  34.3506 
15  310  36.4492  36.4579 
Comparison between the optimization using charge minimization as the objective function and power consumption minimization as the objective function with higher internal resistance
Point no.  Time (s)  Optimum charge consumed on the battery with charge in Ah as the objective (Ah)  Optimum charge consumed on the battery with energy consumption in kWh as the objective (Ah) 

1  380  29.8649  30.2438 
2  375  30.1684  30.4801 
3  370  30.3799  30.8610 
4  365  30.6022  31.0282 
5  360  30.9530  31.2961 
6  355  31.1128  31.4805 
7  350  31.3831  31.8490 
8  345  31.6732  31.8930 
9  340  31.7513  32.2436 
10  335  32.1331  32.3396 
11  330  32.6684  32.7130 
12  325  33.5626  34.0049 
13  320  34.4636  34.8746 
14  315  36.9956  37.3597 
15  310  39.4786  39.5495 
5.3 Speed profile optimization
As the validation results in Sect. 5 showed, the simple battery model provided enough accuracy for our case. The reason is the fact that in case of the IPEMU train, the battery voltage has minor fluctuations with respect to the state of charge, which can also be understood from the validation results. In case of batteries with a voltage more sensitive to the state of charge, the generic battery model or other similar models can be used instead.
The voltage of the battery used for the IPEMU project, however, is relatively stable with respect to the state of charge. Moreover, the designed motor converter module in the IPEMU train can handle the voltage fluctuations of the designed battery and keep the same tractive effort curve. Therefore, one tractive effort curve is used in this research. Furthermore, dynamic programming is known to be suitable for handling the constraints on state variables [17]. In the dynamic programming approach used in this paper, state of charge is assumed as a state variable. Therefore, the constraints related to the state of charge can be handled by this approach (such as the constraint on the battery capacity, i.e., Eq. (5)). This also includes the constraint resulting from different tractive effort curves in different states of charge, which can happen in other battery types or train configurations.
Application of dynamic programming for the speed profile optimization using the methods presented in this paper shows the potential of 31.6% reduction in charge consumption from the battery (Fig. 9). It is important to note that reduction in charge consumption can be less in practice. Such algorithms for the speed profile optimization are usually implemented in a form of a driver advisory system, which gives instructions to the driver (see e.g., [17]). The drivers, however, may not always follow the instructions (e.g., due to the lack of trust or difficulty of the provided instructions), which will result in a lower energy efficiency. Moreover, we considered the problem of energyefficient train operation for a single train. In order to maximize the effectiveness of the solution in practice, the whole network of trains and timetables should be considered. Consideration of the whole network of trains will result in a much more complex optimization problem, which was out of the scope of the work presented in this article.
6 Conclusion and future work
In this paper, we studied the consideration of battery characteristics in the problem of speed profile optimization of batterydriven electric trains during catenaryfree operation. This was done by combining the single mass point train model and an electrical battery model. We further applied a dynamic programming approach to minimize the charge taken from the battery and discussed the behavior of the battery in the speed profile optimization using this approach. Moreover, the models and the results are validated and evaluated against the test runs of an actual battery train.
We estimated the state of charge for the speed profile optimization using the single mass point train model and an electrical battery model. We studied two battery models for this purpose: the simple battery model and the generic battery model. We showed that for the case of the IPEMU train, the simple battery model can provide the same accuracy as the generic battery model. However, the generic battery model or other similar battery models (models following the function of \(V_{\text {bat}} = U({\text {soc}}, I_{\text {bat}})\)) can be used for other similar cases where the behavior of the battery is more dependent on the state of charge. There are catenaryfree light rail vehicles currently in service with EDLCs onboard (see e.g., [1]) and there are also reports on catenaryfree EMUs with onboard fuel cell [28]. The verification and the approach presented in this paper were specific for the case of lithiumion batteries, but both fuel cell and EDLC technologies have the potential of benefiting from the same methodology for speed profile optimization. There are similar models for these type of energy storage devices available in the literature (e.g., the fuel cell model proposed by Njoya et al. [53]). Further studies are, however, needed to verify the applicability of the approach for such technologies.
We used charge taken from the battery as the objective function instead of the energy consumption in kWh. The results of a comparison between the two approaches show that, in general, the application of the battery charge as the objective function provides more energyefficient speed profiles in comparison with the method with the energy consumption in kWh as the objective. Moreover, the dynamic programming approach used in this paper considers the state of charge as a state variable. Having the state of charge as a state variable facilitates handling different constraints that come from the specific behavior of the batteries in different states of charge. This is particularly important as it provides the opportunity to apply the same optimization approach for the energy storage devices with a behavior more dependent on the state of charge.
The experimental evaluation of our approach using the test runs of an actual batterydriven train shows significant potential in saving energy consumption from the batteries. This concludes that the approach presented here can be used as a basis for designing a driver advisory system for catenaryfree operated electric trains. Further research is needed to investigate the barriers and challenges for the implementation of the mathematical solution in the form of a driver advisory system onboard the train.
This paper provided the first experimental evaluation of a speed profile optimization approach for catenaryfree operated electric trains. Further studies in this field require more experimentation and test runs with new batterydriven trains with different energy storage technologies.
Notes
Acknowledgements
This research was funded by VINNOVA (Sweden's Innovation Agency) Grant Numbers 201404319 and 201201277. Authors would like to thank Martin Joborn from Linköping University for his help, guidance and discussion on the train modeling.
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