Qi Jiang

O Béthoux

F Ossart

E Berthelot

C Marchand

A COMPARISON OF REAL-TIME ENERGY MANAGEMENT STRATEGIES OF FC/SC HYBRID POWER SOURCE: STATISTICAL ANALYSIS USING RANDOM CYCLES

Keywords: Fuel Cell System, on-line Energy Management, Adaptive Equivalent Consumption Minimization strategy, Stochastic Dynamic Programming, Markov chain, statistical analysis

This paper presents a comparative study of two promising real-time energy management strategies for fuel cell electric vehicle applications: adaptive equivalent consumption minimization strategy (A-ECMS), and stochastic dynamic programming (SDP). An off-line algorithm -classic dynamic programming -provides reference results. On-line and off-line strategies are tested both in simulation and using a dedicated test bench completely consistent with an electric scooter powertrain.

The hybrid power source combines a fuel cell, a supercapacitor pack and two related power converters. The system model is carefully calibrated using experimental data. This allows meaningful identification of parameters of the various strategies. The model data is determined using a motorcycle certification driving cycle.

The robustness of each strategy is then analyzed using a large number of random driving cycles. Experimental and simulation results

show that a specific SDP approach, based on Markov chain modeling, has the best overall performance in real-world driving conditions. It achieves the minimum average hydrogen consumption while respecting the state-sustaining constraint. Conversely, the A-ECMS results lack robustness and show poor performance indexes when facing unknown real world power demand profile.

In conclusion, the present results indicate SDP is an interesting approach for future hybrid source energy allocation.

Introduction

With increasing climate change concern, public attention tends to focus on the environmental impacts of fossil fuel use, especially after the DieselGate emissions scandal [START_REF] Mihankhah | Contamination and cancer risk assessment of polycyclic aromatic hydrocarbons (PAHs) in urban dust from different land-uses in the most populated city of Iran[END_REF]. New legislative measures promote clean energy technologies and specifically electric vehicles [START_REF] Maesano | Impacts on human mortality due to reductions in PM10 concentrations through different traffic scenarios in Paris, France[END_REF][START_REF] Guo | Does air pollution stimulate electric vehicle sales? Empirical evidence from twenty major cities in China[END_REF]. In comparison with grid charged battery electric vehicles, Fuel Cell Electric Vehicles (FCEVs) permit fast refueling and extended driving range within strict integral mass constraints. This advantage means hydrogen energy seems to be the new trend of both public and individual zero-emission transportation methods [START_REF] Liu | The impact of fuel cell vehicle deployment on road transport greenhouse gas emissions: The China case[END_REF][START_REF] Chang | Impact of urban development on residents' public transportation travel energy consumption in China: An analysis of hydrogen fuel cell vehicles alternatives[END_REF][START_REF] Liu | Comparison of well-to-wheels energy use and emissions of a hydrogen fuel cell electric vehicle relative to a conventional gasoline-powered internal combustion engine vehicle[END_REF]. In this context, large car manufacturers, like Toyota, Honda and Hyundai have been already manufacturing and selling FCEVs for several years [START_REF] Nonobe | Development of the fuel cell vehicle mirai[END_REF].

A FCEV commonly contains two power sources, a Fuel Cell system (FC system) made up of an H2 tank and a Fuel Cell (FC) combined with an Energy Storage System (ESS) such as a supercapacitor (SC) pack or a rechargeable battery pack [START_REF] Li | Energy management and economic analysis for a fuel cell supercapacitor excavator[END_REF][START_REF] Changizian | Performance optimization of hybrid hydrogen fuel cell-electric vehicles in real driving cycles[END_REF][START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF][START_REF] Fragiacomo | Fuel cell hybrid powertrains for use in Southern Italian railways[END_REF][START_REF] Fu | A hierarchical energy management strategy for fuel cell/battery/supercapacitor hybrid electric vehicles[END_REF]. Currently, SCs are often chosen because they have a higher specific power than batteries and a longer lifetime in terms of the number of charge/discharge cycles [START_REF] Fu | A hierarchical energy management strategy for fuel cell/battery/supercapacitor hybrid electric vehicles[END_REF][START_REF] Kaya | Two new control strategies: For hydrogen fuel saving and extend the life cycle in the hydrogen fuel cell vehicles[END_REF].

The combination of the FC system and the ESS constitutes a hybrid source and the dual source power sharing gives a degree of freedom which can be utilized to improve the system management. It permits the main energy source, the FC system, to operate more efficiently and the vehicle kinetic energy to be recovered during braking into the ESS. However, the fuel saving and the related vehicle autonomy increase strongly depend on the embedded Energy Management Strategy (EMS) that controls in real-time the power sharing between the two sources (FC system and ESS) to fulfill the power requirement of the drivetrain [START_REF] Wu | A robust online energy management strategy for fuel cell/battery hybrid electric vehicles[END_REF][START_REF] Rezk | Comparison among various energy management strategies for reducing hydrogen consumption in a hybrid fuel cell/supercapacitor/battery system[END_REF][START_REF] Zheng | Fuel economy evaluation of fuel cell hybrid vehicles based on equivalent fuel consumption[END_REF][START_REF] Muñoz | Energy management control design for fuel cell hybrid electric vehicles using neural networks[END_REF].

While finding the optimal solution for this sharing, can be achieved off-line using Pontryagin's Minimum Principle (PMP) [START_REF] Kim | Optimal Control of Hybrid Electric Vehicles Based on Pontryagin's Minimum Principle[END_REF][START_REF] Xu | Application of Pontryagin's Minimal Principle to the energy management strategy of plugin fuel cell electric vehicles[END_REF][START_REF] Ettihir | Design of an adaptive EMS for fuel cell vehicles[END_REF] or Dynamic Programming (DP) [START_REF] Teng | A comprehensive review of energy management optimization strategies for fuel cell passenger vehicle[END_REF][START_REF] Fletcher | An Energy Management Strategy to concurrently optimise fuel consumption & PEM fuel cell lifetime in a hybrid vehicle[END_REF][START_REF] Cheng | Optimal warm-up control strategy of the PEMFC system on a city bus aimed at improving efficiency[END_REF][START_REF] Fares | Dynamic programming technique for optimizing fuel cell hybrid vehicles[END_REF], the design of an adequate on-line EMS for real-world applications remains a challenge and an open question. Off-line optimization requires a complete knowledge of the vehicle driving conditions which cannot be obtained in advance in the real world. This issue has had considerable research interest and the related literature is very rich. The present article aims at assessing two prominent on-line EMS systems dedicated to supervising the on-board electrical power distribution to the FCEV powertrain. The first approach is based on the equivalent consumption concept, derived from PMP [START_REF] Han | Hierarchical energy management for PV/hydrogen/battery island DC microgrid[END_REF][START_REF] Odeim | Power management optimization of fuel cell/battery hybrid vehicles with experimental validation[END_REF]. This approach is popular because of its simplicity and ease of implementation. The present paper focuses on the Adaptative Equivalent Consumption Minimization Strategy (A-ECMS) proposed in [START_REF] Han | Hierarchical energy management for PV/hydrogen/battery island DC microgrid[END_REF][START_REF] Odeim | Power management optimization of fuel cell/battery hybrid vehicles with experimental validation[END_REF][START_REF] Onori | Adaptive equivalent consumption minimization strategy for hybrid electric vehicles[END_REF][START_REF] Li | Online adaptive equivalent consumption minimization strategy for fuel cell hybrid electric vehicle considering power sources degradation[END_REF]. The second method under study is Stochastic Dynamic Programming (SDP) [START_REF] Chan-Chiao | System-level model and stochastic optimal control for a PEM uel cell hybrid vehicle[END_REF][START_REF] Johannesson | Assessing the Potential of Predictive Control for Hybrid Vehicle Powertrains Using Stochastic Dynamic Programming[END_REF][START_REF] Jiao | SDP policy iteration-based energy management strategy using traffic information for commuter hybrid electric vehicles[END_REF][START_REF] Liu | Modeling and Control of a Power-Split Hybrid Vehicle[END_REF][START_REF] Jiang | Comparative Study of Real-Time HEV Energy Management Strategies[END_REF], which generalizes DP in the case where random processes are involved, as in real-life driving. Despite its complexity, SDP is appealing because it has a clear theoretical background and can take into account statistical information for future driving cycles. This paper focuses exclusively on the EMS [START_REF] Ou | Optimized power management based on adaptive-PMP algorithm for a stationary PEM fuel cell/battery hybrid system[END_REF][START_REF] Liu | An optimal method of the energy consumption for fuel cell hybrid tram[END_REF] and not on the system itself [START_REF] Marx | On the sizing and energy management of an hybrid multistack fuel cell -Battery system for automotive applications[END_REF][START_REF] Hocksun Kwan | Parameter sizing and stability analysis of a highway fuel cell electric bus power system using a multi-objective optimization approach[END_REF]. Thus, all EMS strategies are applied to the same FC/SC hybrid system, in the same context, and are then tested both in a simulation environment and on the corresponding experimental setup. The on-line EMS results are compared to DP, which provides reference results. Two indicators are defined in order to quantify the consumption performance on one hand, and the state of energy (SOE) sustaining performances on the other hand.

To obtain a relevant assessment of the EMS performance, this study uses a large number of driving cycles representative of real world operational conditions, and not only certification type cycles as it is usually done. To limit operating expenses, the vehicle's owner is basically interested in minimizing his fuel consumption not for a single standard route, but for a global use, including multiple itineraries. For instance, commuting from home to work is a typical use which shows daily differences due to changing traffic conditions. This naturally leads to base the EMS assessment on a large statistical study. To the best of our knowledge, this statistical approach, closely taking account of real world use, is novel and differs from earlier studies focusing only on certification driving cycles. Hence, this paper provides an improved understanding of the EMS performance and permits to offer suggestions for possible improvements.

The main highlights of the present contribution are:

1. A statistical assessment of two pure on-line EMS regarding real-life use based on a very large number of trials (thousand).

To this end, a Markov chain model is used to generate random but realistic powertrain power profiles.

2. The development of a new methodology to carry out this comprehensive study. Pure DP provides an absolute reference, against which each real-time EMS is compared. Two performance indexes are also derived from the analysis of the customer expectations.

3. The results clearly show that real-life performances may strongly differ from the certification cycle one. From that perspective, SDP proves to be a very attractive on-line EMS.

4. More generally, any on-line EMS approach should be evaluated using such statistical analysis in order to significantly account for the future user experience.

The remainder of the paper is organized as follows. Section 2 presents the FC/SC hybrid system and its related model. In Section 3, the energy management problem is stated. A short review of existing real time strategies is given and the tested ones are presented. Section 4 describes the model experimental calibration and validates the DP benchmark result using a certification cycle.

Section 5 presents the performance criteria and discusses the performances of the tested EMS on a statistical basis. Section 6 concludes the paper and provides some perspectives for future work.

FC/SC hybrid system description and modeling

Our goal is to carry out a full-size test, representative of the real use of a FC/SC hybrid system. The rated power of the FC system available in our lab is 800 W. Hence, we have chosen to conduct the present study with data corresponding to a low power vehicle, namely a light urban scooter. Such vehicles are promising candidates for FC applications [START_REF] Changizian | Performance optimization of hybrid hydrogen fuel cell-electric vehicles in real driving cycles[END_REF] in the city driving context, in which pollution effects have a high impact in densely populated areas [40] [41]. This type of scooter operates at low speed (< 50 km/h) and the urban part of the WLTC certification cycle dedicated to light vehicles provides a speed profile representative of its use. We have verified that the corresponding power remains within the limits of our experimental system (see section 2.D).

2.A. System presentation

The present work deals with the FC/SC hybrid system shown in Fig. 1, suitable for powering a full-hybrid urban electric scooter.

The considered architecture combines a FC system rated for the powertrain power demand and a SC pack sized for braking energy recovery. The scooter traction power is fed by a DC electric bus, connected to the FC through a unidirectional boost converter. The SC pack is connected in parallel through a bidirectional converter, so that it serves as an energy buffer between the FC and the scooter wheel power.

Since this study focuses exclusively on EMS comparative assessment, the experimental test bench includes neither the electric motor drive nor the mechanical transmission. The power load connected to the DC bus directly emulates the powertrain demand while neglecting the various losses in the inverter, the motor and the transmission. Hence, the DC bus load is directly the scooter drive wheel power demand Pwheel, calculated using the motion equation of the scooter and the driving cycle, as explained in section 2.D. A bi-directional power load emulates the scooter traction power: in acceleration and constant speed phases, the hybrid source supplies power to the load, while this returns power back to the FC/SC source during braking.

2.B. Component modeling 1) Fuel cell system

The main power source of the system is a 800 W proton exchange membrane FC. As it is operating at low frequency thanks to the energy storage system assistance, the FC is modeled using its static V-I characteristic [START_REF] Hilairet | A passivity-based controller for coordination of converters in a fuel cell system[END_REF]. The hydrogen chemical power 𝑃𝑃 𝐻𝐻 2 is expressed as a function of the FC system output electric power 𝑃𝑃 𝐹𝐹𝐹𝐹 by fitting experimental data with a fourth-order polynomial.

Then the hydrogen consumption rate is determined by [START_REF] Mihankhah | Contamination and cancer risk assessment of polycyclic aromatic hydrocarbons (PAHs) in urban dust from different land-uses in the most populated city of Iran[END_REF], where 𝐻𝐻𝐻𝐻𝐻𝐻 𝐻𝐻 2 is the high heating value of hydrogen.

𝑚𝑚̇𝐻𝐻 2 = 𝑃𝑃 𝐻𝐻 2 (𝑃𝑃 𝐹𝐹𝐹𝐹 ) 𝐻𝐻𝐻𝐻𝐻𝐻 𝐻𝐻 2 (1)
Fig. 2 shows the measured FC system efficiency and the modeled one. The FC system model gives a maximum efficiency for an output power around 𝑃𝑃 𝐹𝐹𝐹𝐹,𝑜𝑜𝑜𝑜𝑜𝑜 = 300 𝑊𝑊. test data and model curve.

2) Supercapacitor stack

The SC pack is modeled as a RC circuit in parallel with an internal leakage current source which represents the losses due to the active cell balancing technology used in SC pack [START_REF] Linzen | Analysis and evaluation of charge-balancing circuits on performance, reliability, and lifetime of supercapacitor systems[END_REF], as shown in Fig. 3. For a given output current ISC and output voltage VSC, the internal capacitor is charged (PSC_int< 0) or discharged (PSC_int> 0) according to formula [START_REF] Maesano | Impacts on human mortality due to reductions in PM10 concentrations through different traffic scenarios in Paris, France[END_REF].

𝑃𝑃 𝑆𝑆𝐹𝐹_𝑖𝑖𝑖𝑖𝑜𝑜 = 𝑅𝑅 𝑆𝑆𝐹𝐹 �𝐼𝐼 𝑆𝑆𝐹𝐹 + 𝐼𝐼 𝑓𝑓_𝐹𝐹𝐹𝐹 � 2 + 𝐻𝐻 𝑆𝑆𝐹𝐹 �𝐼𝐼 𝑆𝑆𝐹𝐹 + 𝐼𝐼 𝑓𝑓_𝐹𝐹𝐹𝐹 � (2)
The leakage current 𝐼𝐼 𝑓𝑓_𝐹𝐹𝐹𝐹 is expressed as a quadratic function of VSC obtained by fitting experimental data measured in no load condition.

Fig. 3. SC pack model.

3) Interface converters

The FC/SC hybrid system needs two DC/DC converters, both modeled as ideal converters connected in parallel with leakage current sources that represent the converter internal losses. These losses are due to conduction and switching losses in the semiconductor components, Joule losses in smoothing inductors and current ripples both in the filter inductance and DC bus filter capacitor. The total losses are accurately measured using an opposition method described in [START_REF] Forest | Use of opposition method in the test of high-power electronic converters[END_REF].

The output current of the FC converter can be expressed by [START_REF] Guo | Does air pollution stimulate electric vehicle sales? Empirical evidence from twenty major cities in China[END_REF], where 𝐼𝐼 𝑓𝑓_𝑐𝑐𝑜𝑜𝑖𝑖𝑛𝑛 is the leakage current modeling power losses. It is obtained by fitting the experimental data with a quadratic function of IFC.

𝐼𝐼 𝐹𝐹𝐹𝐹_𝑐𝑐𝑜𝑜𝑖𝑖𝑛𝑛 = 𝐻𝐻 𝐹𝐹𝐹𝐹 • 𝐼𝐼 𝐹𝐹𝐹𝐹 𝐻𝐻 𝐵𝐵𝐵𝐵𝐵𝐵 -𝐼𝐼 𝑓𝑓_𝑐𝑐𝑜𝑜𝑖𝑖𝑛𝑛 (3)
V SC I SC C SC R SC I f

2.C. Power demand

The instantaneous power demand 𝑃𝑃 𝑊𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐹𝐹. 𝑣𝑣 depends on the vehicle speed and acceleration according to Newton's law (4):

𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = [0.5 𝜌𝜌 𝑎𝑎𝑖𝑖𝑎𝑎 𝐴𝐴𝐶𝐶 𝑑𝑑 𝑣𝑣 2 + 𝜇𝜇 𝑎𝑎 𝑚𝑚𝑚𝑚 + 𝑚𝑚𝑚𝑚]. 𝑣𝑣 (4)
where ρair is air density; A is the vehicle reference area; Cd is the drag coefficient; μr is the rolling resistance coefficient; m is the vehicle mass; g is the gravitational acceleration; 𝑣𝑣 and 𝑚𝑚 are the vehicle velocity and acceleration, respectively.

The present study is based on a standard driving cycle: the Worldwide Harmonized Light Vehicles Test Cycles (WLTC), dedicated to low power motorcycles [START_REF] Benmouna | Experimental study of energy management of FC/SC hybrid system using the Passivity Based Control[END_REF], and shown in Fig. 4.The current report uses it as a calibration driving profile. Its statistical characteristics are used to generate various random profiles for sensitivity analyses. @ WLTC regulated at its 50 V constant set point by acting on the SC power (𝑃𝑃 𝑆𝑆𝐹𝐹 * ). Two low-level controllers (green boxes) enable controlling the FC system and SC pack powers by acting on the DC/DC converter duty cycles (𝑑𝑑 𝐹𝐹𝐹𝐹 and 𝑑𝑑 𝑆𝑆𝐹𝐹 ).

2.D. Overview of the FC/SC hybrid system and its overall control system

At all times a wide range of (PFC, PSC) couples can provide the powertrain demand, leading to a control allocation problem. The overall control system continually controls the power split between the FC system and the SC pack. To arbitrate between all possible solutions, it utilizes both the SC pack voltage, enabling to estimate the SC SOE, and the load power as inputs. This information is used by the EMS to compute its output, namely the instantaneous FC output power. Simultaneously, the low level controllers manage the voltage and current in the different parts of the system. Specifically, the SC power is controlled to achieve DC bus regulation, i.e. the combined sources permanently provide the powertrain required power [START_REF] Benmouna | Experimental study of energy management of FC/SC hybrid system using the Passivity Based Control[END_REF].

Fig. 5. Overview of the FC/SC hybrid system and its overall control system

Online energy management methods studied

3.A. Optimization problem statement

As depicted in Fig. 1, the powertrain demand is supplied by the DC bus, which is powered by two electrical sources, the FC system and the SC pack. This means that there are an infinite number of possibilities as to how to split the power between the sources and provide the instantaneous required power. This degree of freedom can be used to optimize a criterion. Clearly, a relevant criterion would be the fuel consumption during a defined route.

In this context, the EMS aims at minimizing the core cost criterion given by equation ( 5). In equation ( 5), 𝑡𝑡 0 and 𝑡𝑡 𝑓𝑓 are the initial Bidirectional load

H 2 tank PEMFCS i FC d FC p FC control p SC control d SC SC V Bus V SC i SC V Bus control w = P Load V * Bus p * SC i Load m H2 u * = p * FC V FC Energy Management Strategy x min x max u min u max V SC
and final moments while the control variable 𝑢𝑢(𝑡𝑡) is the electric power 𝑃𝑃 𝐹𝐹𝐹𝐹 (𝑡𝑡) = 𝐻𝐻 𝐹𝐹𝐹𝐹 (𝑡𝑡) × 𝐼𝐼 𝐹𝐹𝐹𝐹 (𝑡𝑡) delivered by the FC system. The SC power is then defined to ensure the instantaneous DC balance. The whole system dynamics is modelled by the state equation [START_REF] Liu | Comparison of well-to-wheels energy use and emissions of a hydrogen fuel cell electric vehicle relative to a conventional gasoline-powered internal combustion engine vehicle[END_REF]. In this equation, the unique state variable is the energy stored in the SC pack, denoted by 𝑥𝑥(𝑡𝑡) and referred to as state of energy (SOE), whereas 𝑤𝑤(𝑡𝑡) stands for the powertrain demand 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 which acts on the DC bus as an external perturbation. This perturbation is known in advance in the case of off-line tests, using certification driving profile for example. In real-life cases, 𝑤𝑤(𝑡𝑡)

is unknown but some of its characteristics can be estimated if route and traffic conditions forecast are available. This hypothesis is perfectly credible considering the rapid progresses of advanced driver-assistance systems in actual and future connected vehicles.

As well as optimizing such criteria, the EMS has to ensure the integrity of each of the system components: the different operating variables must remain within the specified limits of the devices. This results in the three inequality constraints listed in equations ( 7) to [START_REF] Changizian | Performance optimization of hybrid hydrogen fuel cell-electric vehicles in real driving cycles[END_REF].

Finally, the difference between the final and the initial SC SOE has a great impact both on the global hydrogen consumption and on the capability to efficiently enable a new driving mission. To simplify consumption comparison between different EMS algorithms and to enable an easy sequence of driving cycles, the present study considers the standard charge sustaining strategy described by equation [START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF].

To sum up, the EMS optimization problem is formalized by the set of six equations ( 5) to [START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF].

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑚𝑚𝑀𝑀𝑀𝑀𝑀𝑀 𝐽𝐽 = � 𝑚𝑚̇𝐻𝐻 2 �𝑢𝑢(𝑡𝑡)�. 𝑑𝑑𝑡𝑡 𝑜𝑜 𝑓𝑓 𝑜𝑜 0 (5)
𝑥𝑥(𝑡𝑡) = 𝑓𝑓�𝑢𝑢(𝑡𝑡), 𝑥𝑥(𝑡𝑡), 𝑤𝑤(𝑡𝑡)� (6)

𝑥𝑥 𝑚𝑚𝑖𝑖𝑖𝑖 ≤ 𝑥𝑥(𝑡𝑡) ≤ 𝑥𝑥 𝑚𝑚𝑎𝑎𝑚𝑚 (7)
𝑥𝑥̇𝑚𝑚 𝑖𝑖𝑖𝑖 ≤ 𝑥𝑥(𝑡𝑡) ≤ 𝑥𝑥̇𝑚𝑚 𝑎𝑎𝑚𝑚 (8)

𝑢𝑢 𝑚𝑚𝑖𝑖𝑖𝑖 ≤ 𝑢𝑢(𝑡𝑡) ≤ 𝑢𝑢 𝑚𝑚𝑎𝑎𝑚𝑚 (9)
𝑥𝑥�𝑡𝑡 𝑓𝑓 � = 𝑥𝑥(𝑡𝑡 0 ) = 𝑥𝑥 𝑎𝑎𝑒𝑒𝑓𝑓 (10)

3.B. Energy management strategies review

The optimization problem is different in nature, depending on whether the perturbation 𝑤𝑤(𝑡𝑡) is known ahead of time or not.

The first approach applies to situations where the driving cycle is known, typically for system design or system and EMS certification. This problem is addressed by two methods that have been developed and experimented in many dynamic systems for which various combinations of devices permit to ensure the requested output trajectory. The first one was created by the Russian mathematician Pontryaguin [START_REF] Kim | Optimal Control of Hybrid Electric Vehicles Based on Pontryagin's Minimum Principle[END_REF][START_REF] Xu | Application of Pontryagin's Minimal Principle to the energy management strategy of plugin fuel cell electric vehicles[END_REF]. It uses Lagrange multipliers, referred to as co-state 𝑝𝑝(𝑡𝑡), to integrate the system dynamics in the cost criterion minimization. Then the PMP is used, relying on the Hamiltonian minimization. The second method, DP combines mathematics and computer science [START_REF] Teng | A comprehensive review of energy management optimization strategies for fuel cell passenger vehicle[END_REF][START_REF] Fletcher | An Energy Management Strategy to concurrently optimise fuel consumption & PEM fuel cell lifetime in a hybrid vehicle[END_REF][START_REF] Cheng | Optimal warm-up control strategy of the PEMFC system on a city bus aimed at improving efficiency[END_REF][START_REF] Fares | Dynamic programming technique for optimizing fuel cell hybrid vehicles[END_REF][START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]. This is a typical sampled approach both in time and in state. In this discrete framework, the DP technique consists in both exploring a search tree and limiting the combinatorial explosion by subdividing the initial problem in many smaller problems. This approach intrinsically enables to directly use the state and control constraints defined in equations ( 7)- [START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF]. Despite a higher computation cost than PMP, DP is used in this study in order to establish a benchmark for each of the experiments (see section 5.A).

The second approach deals with real-life operation where the mission profile is clearly unknown in advance. In this much more challenging configuration, the EMS still has to define the best control law permitting to minimize the cost criterion defined by equation ( 5). This is still an open problem, although many authors have proposed sub-optimal strategies. The major real-time EMS can be divided into three categories: rule-based strategies, instantaneous optimization of an equivalent fuel consumption accounting for the ESS power flux and global optimization based on dynamic programming.

Rule-based strategies are attractive due to their relative simplicity of implementation. Some authors propose heuristic rules that define the FC system power according to the load power and the quantity of energy available in the ESS [START_REF] Yun | Energy management for fuel cell hybrid vehicles based on a stiffness coefficient model[END_REF][START_REF] Ferodldi | Energy Management Strategies based on efficiency map for Fuel Cell Hybrid Vehicles[END_REF][START_REF] Kaya | Two new control strategies: For hydrogen fuel saving and extend the life cycle in the hydrogen fuel cell vehicles[END_REF] including those using fuzzy logic [START_REF] Shen | Variable structure battery-based fuel cell hybrid power system and its incremental fuzzy logic energy management strategy[END_REF][START_REF] Geng | Simulation research on a novel control strategy for fuel cell extended-range vehicles[END_REF]. However, evaluated on engineering common sense, it seems these strategies cannot guarantee an optimal solution. Furthermore, they require long calibration periods which cannot easily be transferred from one system to another.

For those reasons, rule-based strategies are not considered in this comparative work.

The second approach is based on the equivalent consumption concept, derived from PMP [START_REF] Han | Hierarchical energy management for PV/hydrogen/battery island DC microgrid[END_REF][START_REF] Odeim | Power management optimization of fuel cell/battery hybrid vehicles with experimental validation[END_REF]. The equivalent consumption minimization strategy (ECMS) minimizes an instantaneous equivalent fuel consumption. The key to this strategy is an equivalence factor, representing the equivalent hydrogen consumption of the SC pack power. For an off-line optimization, its instantaneous optimal value can be evaluated by PMP [START_REF] Kim | Optimal Control of Hybrid Electric Vehicles Based on Pontryagin's Minimum Principle[END_REF][START_REF] Xu | Application of Pontryagin's Minimal Principle to the energy management strategy of plugin fuel cell electric vehicles[END_REF][START_REF] Ettihir | Design of an adaptive EMS for fuel cell vehicles[END_REF]. The latter is not applicable to real world systems since the driving cycle cannot be known in advance. To overcome this difficulty and allow useful but sub-optimal behavior, several pragmatic methods have been developed to adapt the equivalence factor. These are mainly based on ESS evolution [START_REF] Han | Hierarchical energy management for PV/hydrogen/battery island DC microgrid[END_REF][START_REF] Odeim | Power management optimization of fuel cell/battery hybrid vehicles with experimental validation[END_REF][START_REF] Onori | Adaptive equivalent consumption minimization strategy for hybrid electric vehicles[END_REF][START_REF] Li | Online adaptive equivalent consumption minimization strategy for fuel cell hybrid electric vehicle considering power sources degradation[END_REF][START_REF] García | Viability study of a FC-battery-SC tramway controlled by equivalent consumption minimization strategy[END_REF][START_REF] Zhao | Metaheuristic-based energy management strategies for fuel cell emergency power unit in electrical aircraft[END_REF]. The current work focuses on the popular technique developed by Onori et al. [START_REF] Onori | Adaptive equivalent consumption minimization strategy for hybrid electric vehicles[END_REF], known as Adaptive Equivalent Consumption Minimization Strategy (A-ECMS).

DP is not only able to give the optimal strategy as a benchmark but also has the capability to deal with the on-line optimization problem by integrating a statistical model of the real-world driving conditions. In this instance, the strategy is then called stochastic dynamic programming (SDP) [START_REF] Chan-Chiao | System-level model and stochastic optimal control for a PEM uel cell hybrid vehicle[END_REF][START_REF] Johannesson | Assessing the Potential of Predictive Control for Hybrid Vehicle Powertrains Using Stochastic Dynamic Programming[END_REF][START_REF] Jiao | SDP policy iteration-based energy management strategy using traffic information for commuter hybrid electric vehicles[END_REF][START_REF] Liu | Modeling and Control of a Power-Split Hybrid Vehicle[END_REF][START_REF] Jiang | Comparative Study of Real-Time HEV Energy Management Strategies[END_REF]. The perturbation 𝑤𝑤(t) modeling the driving cycle is considered as a random variable with a given probability law. The driving cycle prediction can be carried out either by a normal distribution of the vehicle speed as in previous work [START_REF] Jiang | Comparative Study of Real-Time HEV Energy Management Strategies[END_REF], or closer to reality, by Markov chains [START_REF] Chan-Chiao | System-level model and stochastic optimal control for a PEM uel cell hybrid vehicle[END_REF][START_REF] Liu | Modeling and Control of a Power-Split Hybrid Vehicle[END_REF][START_REF] Lee | Stochastic self-optimizing power management for fuel cell hybrid scooters of different sized components[END_REF]. SDP can be applied to on-line problem, provided that statistical knowledge of the load power profile is available. This method adds complexity because at each time step and each state level many perturbation values may occur, weighted by their occurrence probability. On the other hand, it provides a framework to account for such statistical knowledge. SDP enables the creation of a control law which is optimal in an average sense, i.e. when many profiles are successively operated.

The present work aims at assessing both A-ECMS and SDP online EMS by evaluating their performances over a large number of random driving cycles with identical statistical characteristics. The idea is to establish if the complexity of SDP is worth it when facing real and hence random conditions.

3.C. A-ECMS strategy

A-ECMS is a continuous time technique derived from the Lagrange multipliers concept applied to equations ( 5) to [START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF]. At all times 𝑡𝑡, the optimal control denoted 𝑢𝑢 * (𝑡𝑡) is obtained by minimizing the sum of the actual hydrogen consumption 𝑚𝑚̇𝐻𝐻 2 and a hydrogen cost 𝑝𝑝(𝑡𝑡) relative to the use of storage, as stated by equation [START_REF] Fragiacomo | Fuel cell hybrid powertrains for use in Southern Italian railways[END_REF].

∀𝑡𝑡, 𝑢𝑢 * (𝑡𝑡) = arg min 𝐵𝐵 �𝑚𝑚̇𝐻𝐻 2 (𝑢𝑢) + 𝑝𝑝(𝑡𝑡). 𝑥𝑥(𝑢𝑢, 𝑥𝑥(𝑡𝑡), 𝑤𝑤(𝑡𝑡))� (11)
The parameter 𝑝𝑝(𝑡𝑡) represents the equivalent hydrogen cost of the power provided by the supercapacitor. It controls the power split between the FC system and the SC pack, and hence the SC SOE during the whole trip. A-ECMS relies on a feedback loop that controls the value of 𝑝𝑝 so as to keep the SC SOE around a reference value 𝑥𝑥 𝑎𝑎𝑒𝑒𝑓𝑓 defined by the user. In the present paper, the popular Onori's algorithm [START_REF] Onori | Real-Time Markov Chain Driver Model for Tracked Vehicles[END_REF] is used: it combines a second-order low-pass filter and a proportional term representing a charge sustaining constraint. The value of the equivalent cost 𝑝𝑝 is periodically adjusted according to equation [START_REF] Fu | A hierarchical energy management strategy for fuel cell/battery/supercapacitor hybrid electric vehicles[END_REF]. Clearly, real A-ECMS implementation in a microprocessor needs to use a discrete time technique. In this article the sampling time 𝑇𝑇 𝑆𝑆 is fixed at 1 s both for A-ECMS and SDP methods.

𝑝𝑝 𝑘𝑘 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑝𝑝 𝑘𝑘 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 -1 + 𝑝𝑝 𝑘𝑘 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 -2 2 + 𝐾𝐾 𝑜𝑜 . �𝑥𝑥 𝑎𝑎𝑒𝑒𝑓𝑓 -𝑥𝑥 𝑘𝑘 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 -1 � (12)
A-ECMS has four parameters:

• the feedback sampling period of 𝑝𝑝, denoted 𝑇𝑇 𝐴𝐴𝐴𝐴𝐹𝐹𝐴𝐴𝑆𝑆 = 𝑁𝑁 𝐴𝐴𝐴𝐴𝐹𝐹𝐴𝐴𝑆𝑆 . 𝑇𝑇 𝑆𝑆 , which is a multiple of the basic sampling period 𝑇𝑇 𝑆𝑆 and enables tuning the filter cut-off frequency. Note that 𝑘𝑘 𝑘𝑘 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is the index of the sub-sampled variables,

• the proportional gain 𝐾𝐾 𝑜𝑜 ,

• the two initial conditions of the filter 𝑝𝑝 0 and 𝑝𝑝 1 .

Equation ( 11) alone cannot account for the operational state constraints [START_REF] Nonobe | Development of the fuel cell vehicle mirai[END_REF]. Hence, the two heuristic rules ( 13) and ( 14)

supplement it. 𝑀𝑀𝑓𝑓 𝑥𝑥(𝑡𝑡) > 𝑥𝑥 𝑚𝑚𝑎𝑎𝑚𝑚 , 𝑢𝑢 * (𝑡𝑡) = 0 (13)
𝑀𝑀𝑓𝑓 𝑥𝑥(𝑡𝑡) < 𝑥𝑥 𝑚𝑚𝑖𝑖𝑖𝑖 , 𝑢𝑢 * (𝑡𝑡) = 𝑢𝑢 𝑚𝑚𝑎𝑎𝑚𝑚 (14)

3.D. SDP strategy 1) SDP Principle

In contrast to A-ECMS, SDP is a discrete-time process, with a sampling period denoted 𝑇𝑇 𝑆𝑆 hereafter. Within the formal framework of DP, SDP takes into account the random disturbance𝑤𝑤(𝑡𝑡) as a random variable modeled using statistical information.

Even if it is subject to uncertainty, the forthcoming itinerary can be analyzed using standard navigation aid systems. Extracting certain route macroscopic characteristics may help to take relevant decisions at all time. The optimization formal framework is the following:

Minimize 𝐽𝐽 = ∑ 𝐸𝐸 𝑤𝑤 𝑘𝑘 �𝑚𝑚̇𝐻𝐻 2 (𝑢𝑢 𝑘𝑘 )�. 𝑇𝑇 𝑆𝑆 𝑁𝑁-1 𝑘𝑘=0 (15)
𝑥𝑥 𝑘𝑘+1 = 𝑥𝑥 𝑘𝑘 + 𝑓𝑓(𝑢𝑢 𝑘𝑘 , 𝑥𝑥 𝑘𝑘 , 𝑤𝑤 𝑘𝑘 ). 𝑇𝑇 𝑆𝑆 [START_REF] Zheng | Fuel economy evaluation of fuel cell hybrid vehicles based on equivalent fuel consumption[END_REF])

𝑥𝑥 0 = 𝑥𝑥 𝑁𝑁 = 𝑥𝑥 𝑎𝑎𝑒𝑒𝑓𝑓 (17)
𝑥𝑥 𝑚𝑚𝑖𝑖𝑖𝑖 ≤ 𝑥𝑥 𝑘𝑘 ≤ 𝑥𝑥 𝑚𝑚𝑎𝑎𝑚𝑚 (18)

∆𝑥𝑥 𝑚𝑚𝑖𝑖𝑖𝑖 ≤ 𝑥𝑥 𝑘𝑘 -𝑥𝑥 𝑘𝑘-1 ≤ ∆𝑥𝑥 𝑚𝑚𝑎𝑎𝑚𝑚 ( 19
)
𝑢𝑢 𝑚𝑚𝑖𝑖𝑖𝑖 ≤ 𝑢𝑢 𝑘𝑘 ≤ 𝑢𝑢 𝑚𝑚𝑎𝑎𝑚𝑚 (20)
Where 0, 𝑘𝑘 and 𝑁𝑁 are the initial, current and final time steps indexes, respectively.𝑤𝑤 𝑘𝑘 is a random power variable described by a probability law. SDP specifically addresses the cost 𝐽𝐽 as a mathematical expectation, as set by [START_REF] Kaya | Two new control strategies: For hydrogen fuel saving and extend the life cycle in the hydrogen fuel cell vehicles[END_REF], where 𝐸𝐸 𝑤𝑤 𝑘𝑘

Conversely, the forward procedure simply consists of choosing at each time 𝑡𝑡 𝑘𝑘 the optimal control policy 𝑢𝑢 𝑘𝑘 * corresponding to the current state 𝑥𝑥 𝑘𝑘 and given by the command matrix 𝑢𝑢 𝑘𝑘 * (𝑥𝑥) previously stored. This step can be easily implemented in real-time and only needs to read the database previously established offline. The single limitation is the downloaded data volume of the 𝑢𝑢 𝑘𝑘 * (𝑥𝑥) look-up table.

2) Driving cycle modeling SDP provides an interesting framework, but the quality of the results relies on the quality of the random process model. In the present work, two methods are tested.

The first one models the driving cycle by a random wheel power. An empirical probability law is calculated using the 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 histogram of the selected reference cycle. Fig. 6 shows the probability law obtained for the WLTC cycle, and to be used for any driving cycle with identical statistical properties. This model is very easy to implement in the SDP algorithm, but its accuracy is limited, since the power values at successive time steps are assumed to be independent. The second model allows a richer statistical description of the driving cycle. It is based on two-dimensional Markov chains [START_REF] Onori | Real-Time Markov Chain Driver Model for Tracked Vehicles[END_REF][START_REF] Mcdonough | Modeling of vehicle driving conditions using transition probability models[END_REF]. As in [START_REF] Liu | Modeling and Control of a Power-Split Hybrid Vehicle[END_REF], the power demand 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 at a given time step 𝑡𝑡 𝑘𝑘+1 is assumed to depend on both 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 and the scooter speed 𝑣𝑣 𝑘𝑘 at 𝑡𝑡 𝑘𝑘 .The probability that 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘 is equal to a certain value 𝑃𝑃 𝑒𝑒 at time tk+1, knowing that at time 𝑡𝑡 𝑘𝑘 the speed was 𝑣𝑣 𝑖𝑖 and the power was 𝑃𝑃 𝑗𝑗 is determined by the maximum-likelihood estimation [START_REF] Teng | A comprehensive review of energy management optimization strategies for fuel cell passenger vehicle[END_REF]:

Ρ�𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘+1 = 𝑃𝑃 𝑒𝑒 �𝑣𝑣 𝑘𝑘 = 𝑣𝑣 𝑖𝑖 , 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘 = 𝑃𝑃 𝑗𝑗 � = 𝑝𝑝 𝑖𝑖𝑗𝑗,𝑒𝑒 = 𝑚𝑚 𝑖𝑖𝑗𝑗,𝑒𝑒 𝑚𝑚 𝑖𝑖𝑗𝑗 (21)
where 𝑚𝑚 𝑖𝑖𝑗𝑗,𝑒𝑒 is the number of times the transition from 𝑃𝑃 𝑒𝑒 to 𝑃𝑃 𝑒𝑒 has occurred at 𝑣𝑣 𝑖𝑖 and 𝑚𝑚 𝑖𝑖𝑗𝑗 is the total number of transitions that are initiated from 𝑃𝑃 𝑒𝑒 at 𝑣𝑣 𝑖𝑖 . A reference driving cycle is used to calculate transitions matrices at different speeds, and Fig. 7 shows the result obtained with the WLTC scooter cycle at the specific 6 𝑚𝑚. 𝑠𝑠 -1 speed.

Experimental setup and model calibration

As described above, the control laws related to either the off-line strategy (i.e. DP technique) or the various online approaches (i.e. A-ECMS and both SDP methods) require the best possible knowledge of the FC/SC hybrid system behavior. In order to realize a relevant study, a proper validation of the proposed model outlined in Section 2 is therefore required.

For this purpose, an experimental system has been implemented in the laboratory (Fig. 8). The hardware part is composed of a 800 W FC system, a 32 V/50 kJ SC pack and two identical 20 kHz buck-boost converters combined with voltage and current sensors. This electrical system is operated from a DS1103 DSPACE real-time control system. It ensures EMS as well as safety functions, fast loops to drive the power converters, and slow loops to achieve a constant DC bus voltage, as described in section 2-D. All programs are developed using Matlab/Simulink software and then compiled and loaded into the DSPACE system, which permits simple and fast prototyping. variables; the setup is operated using the WLTC cycle and using the DP approach (EMS block in Fig. 5).Throughout the ten minute operation, the control board imposes the FC system power while the inner loops ensure at all times the accurate balance of the DC bus voltage (bus voltage control and source power control blocks in Fig. 5). As a consequence, the SC pack energy fluctuates while respecting its requirements (see equations ( 16) and ( 17)). The same experiment is replicated in pure simulation. Both trials exhibit close results as comparative Fig 9 shows. It synchronously depicts the instantaneous hydrogen consumption 𝑚𝑚̇𝐻𝐻 2 (𝑡𝑡) and the SC SOE 𝑥𝑥(𝑡𝑡).This leads to the conclusion both that the model is well calibrated and that the system management as well as the DP computation are correctly designed and implemented. This last point is particularly important since DP method is used as the benchmark of all following tests.

EMS assessment results

In the present section, the three studied real-time EMS are implemented and assessed: A-ECMS and two versions of SDP, referred to as "Classic-SDP" and "Markov-SDP" according to the 𝑤𝑤(𝑡𝑡) statistical model used. The former is based on modeling the driving cycle by a simple power demand probability density, whereas the latter relies on a finer statistical knowledge of the load profile modeled as a Markov process (see section 3.D.2). The assessment criteria are carefully defined and justified. Subsequently, the three EMS are calibrated using the WLTC certification driving cycle and their performances are compared in this context. Finally, the EMS are operated with a large set of random cycles generated by a Markov chain process, in order to evaluate the robustness of the various approaches facing real-life situations.

5.A. Performance criteria

This study focuses on the EMS strategies comparison for a given power architecture. To this end, two performance criteria are defined: the over-consumption with respect to an ideal EMS (namely DP) and the difference between the final and initial states.

Indeed, as stated in equation ( 5), the primary goal of any strategy is to achieve the lowest fuel consumption on a specified trip, but ] the control of the final SC SOE is also an important point in order to insure that the vehicle will be ready for the trip to follow.

The optimization problem has been stated with the charge-sustaining constraint expressed by equation [START_REF] Wu | Hybrid fuel cell and battery propulsion system modelling and multi-objective optimisation for a coastal ferry[END_REF]. This constraint insures that the FC system consumption exactly and solely corresponds to the energy needed for traction, with no energy stored or coming from the SC pack. This condition is compulsory if one wants to compare the consumptions of different off-line strategies.

In the case of real-time strategy, the final SC SOE cannot be exactly controlled: each strategy ends up with its own final SOE, making it impossible to directly compare their fuel consumption.

Hence, to allow the comparative investigation, the proposed novel approach defines an over-consumption criterion that compares the actual fuel consumption of a given EMS with the optimal one, obtained using DP for the same initial and final SC SOE. This ideal consumption is computed afterward, when the final SC SOE of the tested EMS is known. The over-consumption criteria ∆𝐽𝐽 (%) is defined by equation [START_REF] Fletcher | An Energy Management Strategy to concurrently optimise fuel consumption & PEM fuel cell lifetime in a hybrid vehicle[END_REF], where 𝐽𝐽 𝐻𝐻 2 (𝑅𝑅𝑇𝑇) denotes the consumption obtained by the tested real-time strategy (RT = A-ECMS or SDP) and 𝐽𝐽 𝐻𝐻 2 (𝐷𝐷𝑃𝑃) denotes the optimal consumption, calculated afterward by DP for the same initial and final SC states of charge. By definition, this factor is positive.

∆𝐽𝐽 (%) = 𝐽𝐽 𝐻𝐻 2 (𝑅𝑅𝑇𝑇) -𝐽𝐽 𝐻𝐻 2 (𝐷𝐷𝑃𝑃) 𝐽𝐽 𝐻𝐻 2 (𝐷𝐷𝑃𝑃) × 100% (22)
The over-consumption with respect to an ideal EMS is the first key criterion. On the other hand, it is also crucial that the on-line EMS should be able to respect the charge-sustaining constraint as well as possible. For this reason, the second criteria, denoted ∆𝑥𝑥, intends to assess in relative terms the EMS capacity to achieve the final state objective. ∆𝑥𝑥 (%) is defined as the SOE difference factor, given by equation [START_REF] Cheng | Optimal warm-up control strategy of the PEMFC system on a city bus aimed at improving efficiency[END_REF].

∆𝑥𝑥 (%) = 𝑥𝑥�𝑡𝑡 𝑓𝑓 � -𝑥𝑥(𝑡𝑡 𝑖𝑖 ) 𝑥𝑥(𝑡𝑡 𝑖𝑖 ) × 100% (23)
The ∆𝑥𝑥 criterion quantifies how well the charge-sustaining constraint is respected. A state-difference close to zero indicates that the strategy is likely to perform well when applied to a longer mission (e.g. several cycles in a row).

5.B. Tuning the EMS using the WLTC driving cycle

The first test to assess the online EMS performances is to tune them using a given cycle (WLTC in the present case) and compare their performance criteria in these conditions. For the A-ECMS, the tuning phase consists in determining the four parameters that allow the best performance for the considered cycle, using a genetic optimization algorithm [START_REF] Linzen | Analysis and evaluation of charge-balancing circuits on performance, reliability, and lifetime of supercapacitor systems[END_REF][START_REF] Pajak | A multiobjective optimization of a catalyst distribution in a methane/steam reforming reactor using a genetic algorithm[END_REF]. For SDP, the WLTC data are used to build empirical probability laws. This well-defined tuning procedure is an interesting feature of SDP. An intermediate initial SOE 𝑥𝑥 𝑎𝑎𝑒𝑒𝑓𝑓 is chosen, which gives the EMS a great degree of freedom, to either charge or discharge the SC pack. The influence of this choice was discussed in a previous work [START_REF] Jiang | A-ECMS and SDP Energy Management Algorithms Applied to a Fuel Cell Electric Scooter[END_REF].

Fig. 10-a) shows the SC state evolution 𝑥𝑥(𝑡𝑡) obtained with the tested real-time control strategies and the reference one (i.e. optimal off-line DP) while Fig. 10-b) plots the four FC system powers in accordance with the SC state evolution. Table 1 reports the proposed performance indexes of the three EMS.

∆𝑥𝑥 (%) 2 0 -9
The three methods perform rather well; among them Markov-SDP seems to reach lower performances than the two others.

Indeed, A-ECMS and Classic-SDP results are close to the DP benchmark ones. Both have a 1% over-consumption and a final SOE very close to the initial one, 2% and 0% respectively. This in fact indicates that the parameters of the A-ECMS method, namely the gain 𝐾𝐾 𝑜𝑜 and the initial co-state values 𝑝𝑝 0 and 𝑝𝑝 1 , have been carefully adjusted to address the specific standard certification driving profile. This setting can be seen to be relevant for this calibration. However, its robustness with regard to real-life load cycle uncertainties has to be assessed.

Conversely, as a global optimization method, SDP methods use a statistical approach to predict the upcoming driver demand and calculate an average optimal control law based on this information and the actual SC SOE. Hence, these strategies minimize the hydrogen consumption in an average sense, rather than on some particular cycle. In the case of SDP, the certification cycle, which represents a typical motorcycle use is used to derive statistical characteristics. The so-called "classic" statistical modeling, based simply on a probability density whatever the actual speed 𝑣𝑣 𝑘𝑘 or power demand 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘 , seems to have the best performance among the three methods. On the other hand, Markov-SDP establishes in a more realistic way how the future power demand 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘+1 may evolve using the current information, which should make sense. For instance, an instantaneous high power demand 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘 at low speed 𝑣𝑣 𝑘𝑘 indicates a strong acceleration of the driver, making a future negative power 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑘𝑘+1 quite unlikely. Based only on the WLTC calibration profile, the Markov-SDP over-consumption factor (+2%) is an acceptable outcome, while its SOE difference factor is, as an absolute value, higher (9%) than the two other competitors, and especially the other statistical based method, Classic-SDP (0%).

In summary, these preliminary results are best interpreted as both indicating a correct implementation of the three methods and a precise setting of their parameters. On the basis of this first analysis, it appears that an additional sensitivity analysis is the only way to assess the comparative performances regarding real scooter utilization. The latter is subject to large uncertainties. Typically, the end user requires both to make a good return on investment, which is a long-term goal, and to minimize surprises and problems regarding the vehicle range, which is a crucial mid-term aim. This is why this second analysis requires a large number of tests. To speed up the process, it is performed using the validated simulation tools.

5.C. Robustness assessment by statistical analysis

The main disturbance affecting the real-time EMS results is the power demand uncertainty. It is necessary that the embedded EMS deals with those variations in the real-world driving conditions and that the fuel consumption remains low in any conditions.

In the present section, statistical tests are conducted in order to assess the EMS robustness.

The stationary Markov model described in section 3.D.2 is exploited to generate random cycles with statistical characteristics identical to the WLTC [START_REF] Marx | On the sizing and energy management of an hybrid multistack fuel cell -Battery system for automotive applications[END_REF]. One thousand random cycles are generated and the three EMS are applied to each of them, the setting parameters calculated in the previous section being unchanged [START_REF] Hocksun Kwan | Parameter sizing and stability analysis of a highway fuel cell electric bus power system using a multi-objective optimization approach[END_REF]. DP is applied to all generated profiles enabling to compare the optimal FC system power profile with those defined in real-time. The resulting performance criteria are calculated and the corresponding histograms are shown in Fig. 11 and Fig. 12. Their mean value (Mean) and standard deviation (Std dev) are reported in Table 2; the former is representative of return on investment concern, while the latter is a good indicator of the risks of a temporary deterioration in the vehicle's range. It should be noted that this number of cycles was chosen after checking that it is sufficient to obtain significant results and draw conclusion on the sensitivity analysis, as it will be shown thereafter.

The over-consumption criterion shows significant disparities among the three tested EMS. Although the A-ECMS results in an average over-consumption (4%) a little higher (1.5) than the Markov-SDP one (2.6 %), its standard deviation is significantly higher.

In a meaningful way, using the Markov-SDP improves this performance index by a 3.2 factor; this point is specifically relevant concerning the user experience. Indeed, even if rare, a fuel breakdown has a highly negative impact on the way the motorcycle is perceived. This good over-consumption performance represents the first benefit of the Markov-SDP approach.

The second main weakness of the A-ECMS is the drift of the final state. The parameters calculated using the WLTC cycle do not enable a good final state control for other driving cycles, although they are of similar nature. The A-ECMS ∆𝑥𝑥 histogram (Fig. 12)

presents a large number of occurrences around ∆𝑥𝑥 = 55%. This indicates that the SOE often drifts to the upper limit of the SC pack capacity. From the perspective of the control of final SC SOE, SDP approaches prove to be much better than A-ECMS strategy. As for the previous criterion, Markov-SDP exhibits the best ∆𝑥𝑥 characteristics with a 5.2 times better mean value and a 7.5 times better standard deviation value than those of A-ECMS.

SDP performance relies on the adequacy of the vehicle power demand probability distribution to predict the future driving conditions. That makes Markov-SDP the most robust real-time strategy in the present context. It is important to underline that the statistical study is based on a number of random cycles large enough to be statistically significant. This point is proven by Fig. 13(a) and (b), which represent the mean and standard deviation values calculated for an increasing number of random cycles. These statistical parameter tend to be stable after four hundred cycles, for both the over-

Conclusion and perspectives

This study focuses on methods that achieve a control allocation regarding a complete use cycle in hybrid power systems. Firstly, this requires a dynamic system with redundant equipment, hence allowing to adjust in real-time their operating points in order to increase the performance of the overall system. Secondly, this optimization only makes sense when considering a long time period.

The use case under consideration is a FC/SC electric scooter. In this case, the powertrain power can be delivered, at any time, either by the FC system or the SC pack. The objective is to minimize the hydrogen fuel consumption on a chosen route. In contrast to many previous works, the present paper considers pure on-line EMS strategies (namely A-ECMS, Classic-SDP and Markov-SDP) and assesses their behavior regarding real-world use. This requires the development of a new methodology, which can then be extended to other use-cases. The main contributions of this work are the following.

First, for a fair comparison, each EMS is carefully adjusted using the dedicated certification driving cycle. This requires a preliminary modeling work of the global system under study. The EMS under test are implemented on the experimental setup and shown to be working properly.

Secondly, after each test, an optimal consumption is calculated using a DP optimization applied to exactly the same conditions: identical load profile, but also equal initial and final SC states of charge. This provides an absolute reference, against which each real-time EMS is compared.

Thirdly, two performance indexes have been derived from the analysis of the user's expectations. The first index deals with the return on investment closely linked to the global consumption reduction. The second index addresses the EMS capacity to efficiently undertake the subsequent driving cycles.

Fourth, the assessment of each EMS is carried out by simulation based on verified modeling. It enables dealing with a large number of tests, which is mandatory for real life operation is concerned. The results are analyzed in terms of mean and standard deviation of the over-consumption and final SOE criteria. This gives an accurate and comprehensive view of the statistical analysis.

The average value provides an overall outline, while the standard deviation indicates the probability of extreme undesired events, such as a breakdown due to a lack of fuel.

Finally, the statistical analysis, as conducted on one thousand random cycles with identical statistical characteristics shows that

Fig. 1 .

 1 Fig. 1. Over-actuated system under study : FC/SC EV scooter powertrain supply.

Fig. 2 .

 2 Fig. 2. FC system efficiency versus electric output power:

Fig. 4 .

 4 Fig. 4. WLTC speed profile (a) and related scooter traction power𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 (b).

Fig 5

 5 Fig 5 shows the FC/SC hybrid system and its overall control system, subdivided into a power and an energy management control parts. These two parts are organized in a hierarchical way. At the upper level, the EMS (purple box) monitors the SC pack state and generates the FC system power set point (𝑃𝑃 𝐹𝐹𝐹𝐹 * ) according to the embedded algorithm, while matching the FC system and SC pack constraints. The EMS algorithms are described in Section 3. Simultaneously the overall control system provides the load power demand (𝑃𝑃 𝑒𝑒𝑜𝑜𝑎𝑎𝑑𝑑 * = 𝑃𝑃 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 *

Fig. 6 .

 6 Fig. 6. Pwheel histogram and corresponding empirical probability law calculated for the WLTC scooter cycle.

Fig. 7 .

 7 Fig. 7. Transition matrix for 𝑣𝑣 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = 6 [𝑚𝑚. 𝑠𝑠 -1 ].

Fig. 8 .

 8 Fig. 8. Laboratory experimental setup of the FC/SC hybrid system

Fig. 9 .

 9 Fig. 9. Comparison between simulation and experimental results: SC stored energy and FC system H2 mass-meter flow.

Fig. 10 .

 10 Fig. 10. Online EMS and DP simulation results: a) SC pack energy profile. b) FC system power profile.

Fig. 11 .

 11 Fig. 11. Over-consumption ∆𝐽𝐽 histogram for 1000 random cycles.

Fig. 12 .

 12 Fig. 12. ∆𝑥𝑥 histogram for 1000 random cycles.

Fig. 13 .

 13 Fig. 13. EMS performance criteria mean and standard deviation (MD and SD) versus number of tested cycles a) over-consumption factor ∆𝐽𝐽 b) SOE difference factor ∆𝑥𝑥

  [•] represents the mathematical expectation with respect to 𝑤𝑤 𝑘𝑘 probability law. SDP consists in a recursive process (21)-[START_REF] Fletcher | An Energy Management Strategy to concurrently optimise fuel consumption & PEM fuel cell lifetime in a hybrid vehicle[END_REF] where 𝑢𝑢 𝑘𝑘 * (𝑥𝑥) denotes the optimal control at time 𝑡𝑡 𝑘𝑘 as a function of the current system state 𝑥𝑥 𝑘𝑘 = 𝑥𝑥 and the cost-to-go function 𝐽𝐽 𝑘𝑘 (𝑥𝑥) which is also defined at each time step 𝑡𝑡 𝑘𝑘 . 𝐽𝐽 𝑘𝑘 (𝑥𝑥) represents the minimum average cost from a given state 𝑥𝑥 at time 𝑡𝑡 𝑘𝑘 to the final state 𝑥𝑥 𝑁𝑁 . It is expressed by equation (21), showing that it is computed backwards similarly to the optimal control value. The whole backward process from final state to initial one is computed off-line and the command matrix 𝑢𝑢 𝑘𝑘 * (𝑥𝑥) is stored for latter use. This part of the algorithm uses significant processing power since it considers any power in any state at any time.

	𝐽𝐽 𝑘𝑘 (𝑥𝑥) = min 𝐵𝐵	�𝐸𝐸 𝑤𝑤 𝑘𝑘 �𝑃𝑃 𝑓𝑓𝐵𝐵𝑒𝑒𝑒𝑒 (𝑢𝑢)	(21)
		+ 𝐽𝐽 𝑘𝑘+1 (𝑥𝑥 + 𝑓𝑓(𝑢𝑢, 𝑤𝑤 𝑘𝑘 ). 𝑇𝑇 𝑆𝑆 )��
	𝑢𝑢 𝑘𝑘 * (𝑥𝑥) = argmin

𝐵𝐵

�𝐸𝐸 𝑤𝑤 𝑘𝑘 �𝑃𝑃 𝑓𝑓𝐵𝐵𝑒𝑒𝑒𝑒 (𝑢𝑢) + 𝐽𝐽 𝑘𝑘+1 (𝑥𝑥 + 𝑓𝑓(𝑢𝑢, 𝑤𝑤 𝑘𝑘 ). 𝑇𝑇 𝑆𝑆 )��

TABLE 2 .

 2 Statistical analysis of the performance criteria

			Classic-	Markov-
		A-ECMS
			SDP	SDP
	Mean	4.3	3.1	2.6
	∆𝐽𝐽 (%)
	Std dev.	3.5	2.1	1.1
	Mean	36.7	18.1	-7.1
	∆𝑥𝑥 (%)
	Std dev.	24.9	17.7	3.3

SDP methods give better results for both criteria used. Specifically Markov-SDP achieves the best performances: well centered results with the lowest standard deviations. Conversely, the A-ECMS proves very sensitive to its tuning parameters. Facing multiple varied cycles, it induces higher fuel consumption and very poor final SOE control.

In sum, the performance of the SDP behaviors make their relative complexity worth the effort. Among these methods, Markov-SDP is by far the most promising approach. On the contrary, the A-ECMS ease of implementation seems to be the single argument in favor of this technique. Future works should tackle both the issue of Markov-SDP off-line complexity and its need for need for on-line data storage resources.


Dr Yu Cao

Siew Li

Eric Jun

Hao Cheung

Dr Shawn Yohanes Siew

Dr Changjian Li

Dr Yan Liu

Dr Chi Sin Tang

Dr Manohar Lal

Prof Guanyu S Chen

Dr Karim Dogheche

Dr Ping Yang

Steven Pennycook

Andrew Thye

Shen Wee

Soojin Chua

Elhadj Dogheche

Thirumalai Venkatesan

Aaron Danner
email: adanner@nus.edu.sg

Dr S L Tan

Dr E J H Cheung

Prof T Chua

Prof. T A Venkatesan

Danner

Prof A T S Wee

Prof. S Pennycook

Prof E Dogheche

A barium titanate-on-oxide insulator optoelectronics platform

bulk lithium niobate is widely used in commercial modulators, and recent development in its thin-film form (lithium niobate-on-insulator) has attracted considerable attention, because its excellent optical mode confinement enables a high level of integration of photonic devices. An alternative material, barium titanate, has an electro-optic effect which is in principle 50 times stronger than that of lithium niobate. However, it is challenging to fabricate devices in barium titanate compared to lithium niobate, because a phase transition at 120 °C causes cracks in bulk barium titanate, and attempts at creating thin-film barium titanate-oninsulator with substrates such as magnesium oxide typically end up with degraded crystal quality. Here, we introduce a barium titanate-on-oxide insulator platform with a family of alternative low-index substrates that concurrently allow both high crystal quality and excellent vertical mode confinement. The grown single-crystal barium titanate thin film with a well-aligned c-axis orientation is ideal for electrode placement due to crystal symmetry considerations. Furthermore, we fabricated waveguides on barium titanate-on-dysprosium scandate as example devices and verified low optical propagation loss. The structure is stable at higher temperatures and does not suffer from the 120 °C phase Electro-optic modulators are among the most important building blocks in optical communication networks. Lithium niobate, for example, has traditionally been widely used to fabricate high-speed optical modulators due to its large Pockels effect. Another material, barium titanate, nominally has a 50 times stronger r-parameter and would ordinarily be a more attractive material choice for such modulators or other applications. In practice, barium titanate thin films for optical waveguide devices are usually grown on magnesium oxide due to its low refractive index, allowing vertical mode confinement. However, the crystal quality is normally degraded. Here, a group of scandate-based substrates with small lattice mismatch and low refractive index compared to that of barium titanate is identified, thus concurrently satisfying high crystal quality and vertical optical mode confinement. This work provides a platform for nonlinear on-chip optoelectronics and can be promising for waveguide-based optical devices such as Mach-Zehnder modulators, wavelength division multiplexing, and quantum optics-on-chip.

Introduction

Electro-optic modulators form an integral part of the telecommunications ecosystem, encoding binary information for transmission along fiber optic cables. Modulator speed greatly influences Internet data rates, and such modulation is achieved in fabricated optoelectronic devices using materials with an electro-optic effect, or Pockels effect. Among these materials, transition; this makes barium titanate-on-insulator an attractive option for future optoelectronics applications.

Single-crystal barium titanate (BTO, or BaTiO 3 ) has excellent textbook electro-optic properties, making it an attractive material for potential fabrication of nonlinear optical devices (r 51 = r 42 = 1640 pm V -1 in BTO vs r 33 = 30 pm V -1 in lithium niobate). [START_REF] Boyd | Nonlinear Optics[END_REF][START_REF] Johnston | [END_REF] Bulk BTO single crystals are expensive, and are difficult to process in such a way as to achieve good index contrast. However, a BTO thin film hypothetically grown on a substrate with a lower refractive index can be etched into single mode waveguides with good optical mode confinement within the BTO layer. BTO films (n o = 2.254, n e = 2.178) grown on magnesium oxide (MgO), which has a refractive index of 1.7, [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF] have previously been used as a platform for fabricating electro-optic modulators. [START_REF] Petraru | [END_REF][5][6][7][8][9][10][11] However, due to the large lattice mismatch between MgO and BTO and the hygroscopic nature of MgO resulting in a poor surface, the crystal quality of BTO grown on MgO is normally degraded, and BTO thin films end up polycrystalline, [12,13] multi-domain, [7,14] or single crystal but with poor crystal quality. [9,11] The electro-optic effect in polycrystalline barium titanate grown on MgO is already equivalent to that in single-crystal lithium niobate, [12,13,15] and the r-parameter in poor single crystal or multi-domain barium titanate films can be several times larger. [7,11,14] Thus one would expect that with improved crystal quality of grown single-crystal BTO thin films the electro-optic effect would be further enhanced. The observed reduction of the r-parameter in BTO films on MgO compared to the bulk value can be partly attributed to mis-orientation of domains, which causes averaging out of the strong r 42 and r 51 matrix elements with much weaker r-values along incorrect crystal or field orientations, which cannot be avoided in multi-domain BTO. Aside from MgO, strontium titanate (STO, or SrTiO 3 ) has previously shown to be a superb substrate for high-quality single-crystal BTO growth, but its refractive index is unfortunately larger than that of BTO, making it unsuitable for creating optical slabs. An appropriate substrate choice needs to concurrently ensure both good crystal quality and optical confinement. Dysprosium scandate (DSO, or DyScO 3 ) has both of these qualities-a small lattice mismatch (≈1%) and a lower refractive index, making it a perfect substrate for BTOthin-film-based optical devices. Here, we use dysprosium scandate to grow high quality single-crystalline c-axis aligned BTO films. We characterize the films by X-ray diffractometry (XRD) over a wide range of temperatures showing no phase transition up to 700 °C. We then fabricate single-mode waveguides and verify low loss. In this letter we also introduce other substrate candidates aside from DSO which can potentially support this barium titanate platform and we verify their optical properties.

Results and Discussion

First, growth quality of a typical 850 nm-thick BTO film grown on a 25 mm 2 substrate by pulsed laser deposition (PLD) is presented in Figure 1, illustrating a smooth surface and high single crystallinity. Figure 1a shows a smooth surface of the BTO thin film that has root-mean-square (RMS) roughness of 0.754 nm, and this smooth surface helps reduce light scattering in the resultant waveguides. Figure 1b shows the BTO films only have (001) and (002) peaks, indicating that the film has a single (001)/c-axis orientation, and the rocking curve measured at the BTO (002) peak has a full-width at half-maximum (FWHM) of 0.075 (Figure 1c), indicating the c-axes are very well aligned. High crystal quality is illustrated by the scanning transmission electron microscopy (STEM) image in Figure 1d. The atomically sharp interface between BTO and DSO also reduces the light scattering in the resultant waveguides. For waveguide applications, BTO a few hundred nanometers thick is required to achieve mode confinement, and thus the BTO films at such thickness are relaxed from their fully strained state (Figure 1e,f).

The ordinary and extraordinary refractive indices (respectively n o and n e ) of BTO film on DSO are measured with the prism coupling method as a function of wavelength, [16] as shown in Figure 2a, and subsequently fitted with the Sellmeier equation. n o follows the dispersion trend nicely; at one wavelength (984 nm) n e has a discrepancy and is not included. The discrepancy results from the fact that the TM polarization is more difficult to couple into the slab waveguide than the TE polarization as an inherent result of the Fresnel Law, and missing modes can result in an incorrect estimate of the index. The trends are otherwise likely highly accurate.

It is worth noting that for DSO (110), the in-plane lattice constants are not equal (a ≠ b), thus the lattice of grown BTO films seems to follow the substrate at an early growth stage, causing in-plane anisotropy. As BTO films thicken, relaxation is expected to reduce this difference. In any case, refractive indices for both directions were measured at 637 nm wavelength and only a small difference of 0.009 was noted.

Bulk BTO has a Curie temperature (T C ) of 125 °C which would ordinarily present potential process challenges, [17] but luckily the T C in thin films is typically higher due to strain induced by the substrate. For a fully strained (50 nm) BTOon-DSO film, the T C is known from the literature to increase to 680 °C, [18] but such a thin BTO layer would be unsuitable for optoelectronics purposes. An appropriate thickness would be a few hundred nanometers, where the strain on the film is relaxed, with the T C expected to be possibly lower than that of a fully strained BTO but higher than in the bulk. Figure 2b presents refractive indices at elevated temperatures for an 850 nmthick BTO film on DSO. Both n o and n e increase linearly and do not coincide up to 200 °C, indicating that the T C is at least higher than 200 °C. Note that the birefringence is almost unchanged as the temperature increases, maintaining a stable optical property. The temperature range of index measurement is limited by the instrument.

A hypothetically low phase-transition temperature of 125 °C would normally create difficulties during fabrication, because the structural phase transition which occurs due to heating (during etching, for example) can easily cause BTO to crack. However, BTO in the thin-film form is much more stable due to the increased T C as atoms in the film are held firmly in place by strain mediation from the substrate (i.e., substrate clamping). We measured the lattice constants of a 500 nm-thick BTO film on DSO as a function of temperature (during heating and cooling). The results (Figure 3) show that the smooth change of lattice constants during cooling and heating up to a temperature of 700 °C indicate no clear phase transitions or cracking, thus permitting all ordinary lithography, etching, and other fabrication processes to be executed.

We then fabricated waveguides for evaluation of optical propagation loss. Loss in the straight waveguides is characterized by the Fabry-Perot method. [19] For the Fabry-Perot method measurement (Figure S1, Supporting Information), a 0.7 µm-wide straight ridge waveguide is fabricated (800 nm BTO thin film with 170 nm etch depth), and Figure 4a,b shows the smooth surface and side wall of the etched waveguide. The transmission spectrum is subsequently measured and a low loss of about 2 dB cm -1 is obtained at wavelengths near 1550 nm (Figure 4c). We also measured the loss of this waveguide by the camera method which shows commensurate results (Figure S2, Supporting Information). The measured propagation loss in the BTO waveguides is comparable to that in ultralow-loss lithium niobate-on-insulator waveguides. [15,[20][21][22][23][24] Recently, a significantly large r-value of 923 ± 215 pm V -1 was achieved in multi-domain structured BTO grown on buffered silicon. [25] As in lithium niobate, a specific electrode orientation is needed in a-c domain BTO, bringing limitations for certain applications. In contrast, for our c-axis oriented BTO film, r 51 and r 42 are both in-plane and expected to be equal, therefore bringing the convenience of positioning co-planar electrodes along waveguides in any direction, potentially following curves, ring resonators, etc., while still achieving the maximum electrooptic effect.

The reported a-axis/or a-c domain in ref. [25] has a thin layer of BTO at the bottom with a high index strip loaded on top. In such a configuration, most of the power of the optical field is confined in the Si or SiN x in such way that only a small fraction of the optical field overlaps the BTO and the modulation of refractive index in such a region is effectively performing a modulation of the whole device. In our BTO thin film, with a larger thickness of over 500 nm, the etched waveguide presented in the inset of Figure 4c has 92.0% of the optical field power distributed in the BTO crystal. Even though a satisfactory modulation effect has been achieved in Reference 25, there are still possible improvements. First, a-axis BTO thin film has a-c domain formation in-plane, where the c-axis is distributed inplane with an angle to the desired field direction. Second, the effective planar electro-optic effect is not uniform in different directions; c-axis BTO would be preferred so that r is consistent in-plane, because r 51 = r 42 as induced by crystal symmetry, which would provide more convenience in device design.

Other than the DSO substrate, we have also explored other possible substrates for BTO which have good lattice match and low refractive index. The scandate family, with elements neighboring dysprosium, would have similar lattice parameters and refractive indices. We have thus grown BTO films on several scandate substrates, including TbScO 3 (TSO), GdScO 3 (GSO), SmScO 3 (SSO), NdScO 3 (NSO), and managed to achieve highquality single-crystal BTO films. Additionally, although the lattice mismatch in La 0.3 Sr 0.7 Al 0.65 Ta 0.35 O 3 (LSAT) and LaSrAlO 4 (LSAO) would nominally cause strain much larger compared with other substrates tested, these also turned out to be suitable for growth. To further confirm the optical confinement in BTO thin film grown on these substrates, refractive indices of these samples as shown in Figure 5a,b are measured by the prism coupling method. The results are presented in Figure 5f. Although some of the TM modes did not couple to the waveguide properly, leading to a few spurious index values, the trends are otherwise likely highly accurate. Due to the relaxed nature of the film, the indices do not show a clear trend as a function of strain (Figure S3, Supporting Information).

The indices of BTO and their substrates are utilized to perform cross-sectional mode analysis with a typical waveguide structure (Figure 5g), and optical field confinement within the BTO is evaluated (Figure 5h). The power confinement within BTO (PC BTO ) is defined as the optical power distributed in the BTO divided by the total power in the optical mode

BTO BTO PC Sds Sds A A = ∫ ∫ ∞ ( 1
)
where S is the Poynting vector, and A BTO and A ∞ correspond to the BTO area and mode area, respectively. [26] The results show that all TE modes are well confined, while for TM modes, BTO on LSAO, LSAT, and NSO do not show good confinement. These differences are caused by differing substrate and film indices; for example n e of BTO-on-TSO is a bit smaller than n e of BTO-on-DSO while the refractive index of TSO is a bit larger than that of DSO, thus BTO-on-DSO has proportionally more optical power distributed in the BTO layer than in the case of BTO-on-TSO. Therefore, for applications where only the TE mode is used, both platforms are fine, but for applications where the TM mode is used, BTO-on-DSO is a better choice.

Conclusion

We have demonstrated high-quality single-crystal thin-film BTO-on-DSO as an alternative optoelectronics platform. We characterized the refractive indices on BTO thin film as functions of wavelength and temperature, and show that such a platform can withstand the high temperatures encountered during fabrication processes. The fabricated BTO-on-DSO waveguides showed a low optical loss of around 2 dB cm -1 . We proposed other candidate substrates and explored structural and optical properties of BTO on these substrates. This study provides a rich choice for high quality c-axis single-crystal BTO optoelectronics platforms, where easy access to the in plane isotropic properties can be beneficial for the design of photonic and optoelectronic devices. This will be advantageous to multiple fabrication and device application studies to come based on this platform.

Experimental Section

Sample Preparation: BTO films were grown by PLD with a KrF laser (248 nm wavelength). Laser energy density is fixed at 1.5 J cm -2 and repetition rate is 5 Hz. Deposition was performed at a 650 °C substrate temperature and under 10 mTorr oxygen pressure, followed by postannealing in an oxygen-rich environment (200 Torr) at 520 °C for 30 min.

Device Fabrication: Optical waveguides were fabricated by electron beam lithography followed by argon-ion milling or inductively coupled plasma (ICP) etching. First a thin layer of Cr was deposited on the BTO thin-film samples, followed by spin coating of photoresist. Then waveguides were patterned by lithography followed by development and subsequently Cr was etched away with Cr etchant. Finally, samples were etched by argon-ion milling or ICP etching (with Ar rather than a standard process gas). End facets of waveguides were first diced and subsequently polished by either focused ion beam (FIB) milling or broad ion beam polishing.

Structural Characterization: XRD was performed in the XDD (X-ray Diffractometry and Development) beamline at Singapore Synchrotron Light Source. The X-ray wavelength was 0.15405 nm. The lattice constants of the BTO thin film were measured from room temperature to 700 °C in reciprocal vector space, with the lattice constant of DSO substrate at room temperature as a reference. Surface topography was characterized by atomic force microscopy (AFM, Park). STEM samples were prepared by a FIB system (FEI Versa 3D) with 30 kV Ga ions, followed by a low-voltage (2 kV) cleaning step. STEM imaging was performed using a JEM-ARM200F (JEOL) microscope equipped with an ASCOR aberration corrector, a cold-field emission gun, and a Gatan Quantum ER spectrometer, operated at 200 kV.

Refractive Indices: Refractive indices of BTO films were measured by the prism coupling method using the Metricon system at the Université Polytechnique Hauts-de-France. During the measurement, TE polarized light was first coupled to obtain n o and film thickness t, and then TM polarized light was used to calculate n e with known n o and t. Refractive indices of substrates were characterized by spectroscopic ellipsometry (J. A. Woollam).

Waveguide Propagation Loss: Propagation loss was measured with both the camera method and the Fabry-Perot method. The laser input and output were directed from a Keysight Lightwave measurement system by optical fiber.

Simulation: Cross-sectional mode analyses were performed by COMSOL Multiphysics.

Figure 1 .

 1 Figure 1. Crystal quality of 850 nm BTO film on DSO. a) AFM morphology image showing small roughness at the BTO surface of BTO on DSO, with RMS roughness of 0.754 nm. b) XRD 2θ-ω coupled scan showing that the BTO film is purely c-axis single crystal. The dash-dot lines in blue and green label the Bragg angles for the c plane and the a plane in bulk material, respectively. c) A small FWHM of 0.075° in the rocking curve of the BTO (002) Bragg peak verifies the high crystal quality of the BTO film. d) High-resolution STEM cross-sectional image of BTO on DSO showing the high crystal quality and the BTO's c-axis being orthogonal to the interface. e,f) XRD reciprocal space mapping (RSM) of BTO (-103) and (013) plane showing the relaxed nature of the BTO film. The red dashed lines label the fully strained lattice constant, indicating that the BTO film is relaxed at its thickness.

Figure 2 .

 2 Figure 2. Optical properties of 850 nm-thick BTO-on-DSO. a) Refractive indices of BTO film as function of wavelength. n o , n e are measured at 450, 521, 637, 984, 1308, 1548 nm and fitted with the Sellmeier equation. The measurements are performed at room temperature. b) Measured (n o , n e ) as a function of temperature are linearly fitted indicating phase stability of BTO films up to 200 °C. The difference between n o and n e plotted in the inset indicates a stable birefringence property.

  Figure 5a,b show XRD spectra for (≈500 nmthick) BTO films grown on various substrates in comparison with BTO-on-STO and BTO-on-MgO. The spectra clearly indicate that the films are all c-oriented, and the crystal quality is very high as indicated in Figure 5c by the small FWHM of the rocking curve at the BTO (002) Bragg peak. Other than crystallinity of BTO, we checked other important factors, such as index contrast. Refractive indices of the substrates were measured by spectroscopic ellipsometry, as shown in Figure 5d,e. Compared with indices n o and n e of BTO on DSO, the indices of the other substrates tested are all lower.

Figure 3 .

 3 Figure 3. Lattice constant of BTO on DSO as a function of temperature. Lattice constants a, b, c, and unit cell volume V (normalized to V 1/3 ) of 500 nm-thick BTO-on-DSO as a function of temperature measured by XRD RSM during heating (solid symbols) and cooling (hollow symbols).

Figure 4 .

 4 Figure 4. Propagation loss of BTO waveguide. a) Optical microscopy and b) helium-ion microscopy image of 0.7 µm-wide BTO ridge waveguide. c) Propagation loss characterized by the Fabry-Perot method. The inset shows the cross-sectional TE mode simulated by COMSOL.

Figure 5 .

 5 Figure 5. Equivalent substrate alternatives (LSAO, LSAT, DSO, TSO, GSO, SSO, NSO) for the BTO electro-optic platform. a) XRD 2θ-ω coupled scans for BTO films grown on different substrates, all showing single c-axis orientation except for BTO on MgO which shows a tiny portion with (110) orientation, indicated by the black arrow. BTO on STO and MgO are plotted for reference. b) Out-of-plane lattice constant converted from (a) near the BTO (002) peak showing a clearer single c-axis orientation and different strain states. Note that intensity in (a,b) is on a logarithmic scale. c) Comparison of alignment of crystal orientation along the z-axis in the rocking curve at the BTO (002) Bragg peak showing superior crystallinity of BTO films on the proposed substrates compared to MgO. d,e) Real and imaginary refractive indices of the proposed substrates measured by ellipsometry. Refractive indices of BTO film on DSO are plotted as a reference showing that these substrates have lower indices than BTO film. f) Refractive indices of BTO films grown on different substrates as a function of wavelength. n o , n e measured at 450, 521, 637, 984, 1308, 1548 nm are fitted with the Sellmeier equation. g,h) Comparison of optical field power confinement in BTO for TE and TM modes for BTO grown on different substrates with an example etched waveguide cross section (as shown schematically in (g)).

Acknowledgements

This work was supported by the Singapore National Research Foundation (NRF) under the Competitive Research Programs (CRP Grant No. NRF-CRP15-201501). Y.C, E.D, and A.D. would like to acknowledge Hassan Ahmad for his help in the characterization of optical properties. P.Y. was supported by SSLS via NUS Core Support C-380-003-003-001. The authors would also like to acknowledge the Singapore Synchrotron Light Source (SSLS) for providing the facilities necessary for conducting the research. The SSLS is a National Research Infrastructure under the National Research Foundation Singapore.