battery hybrid vehicles using Pontryagin's Minimal Principle

battery hybrid vehicles using Pontryagin's Minimal Principle

Journal of Power Sources 440 (2019) 227105 Contents lists available at ScienceDirect Journal of Power Sources journal homepage: www.elsevier.com/loc...

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Journal of Power Sources 440 (2019) 227105

Contents lists available at ScienceDirect

Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour

Adaptive energy management strategy for fuel cell/battery hybrid vehicles using Pontryagin’s Minimal Principle Xiyun Li, Yujie Wang, Duo Yang, Zonghai Chen * Department of Automation, University of Science and Technology of China, Hefei, Anhui, 230027, PR China

H I G H L I G H T S

� An adaptive management strategy is proposed for fuel cell/battery hybrid vehicle. � An improved Markov based velocity prediction is proposed. � The particle swarm optimization is adopted to improve the classification accuracy. � The effectiveness of the proposed strategy is verified by simulations. A R T I C L E I N F O

A B S T R A C T

Keywords: Fuel cell hybrid vehicles Energy management strategy Pontryagin’s minimal principle Velocity prediction

The fuel cell hybrid vehicle provides an efficient and low-emission alternative for the internal combustion engine vehicle. The energy management strategy (EMS) commands the power split between the power sources and is crucial to the hybrid vehicles. In this work, we propose an adaptive EMS based on Pontryagin’s Minimal Principle for a fuel cell/battery hybrid vehicle, in which the co-state adaptation is performed by driving cycle prediction. In order to improve the co-state estimation accuracy, an improved Markov based velocity prediction is proposed considering the driving behavior under different driving patterns. Moreover, the driving pattern is recognized online based on a support vector machine method, which is optimized by particle swarm optimization. We build a combined driving cycle to verify the effectiveness of the proposed strategy, simulation results under three cases show that the proposed strategy can foresee the driving behaviors and update the co-state reasonably. Comparing with the rule-based EMS, the proposed strategy achieves a better fuel economy with a hydrogen consumption reduction by 4% and a relatively low average power change rate of fuel cells. Moreover, it achieves a close performance to the offline optimal algorithms.

1. Introduction Issues on energy crisis and climate changing have received wide­ spread attention since the late 20th century. The internal combustion engine vehicles (ICEVs) have primary responsibility for the above issues through consuming large amounts of fossil fuels and emit greenhouse gases such as carbon dioxide, nitrogen oxide, etc. Therefore, using electric sources to replace the internal combustion engine has been extensively studied by automotive manufacturers and researchers. Nowadays, different types of vehicles have emerged such as hybrid electric vehicles (HEVs) and pure electric vehicles (EVs) [1]. The polymer electrolyte membrane fuel cell (PEMFC) is a promising candidate for transportation applications due to the outstanding features which include low operating temperature, high power density, high

efficiency and zero emissions [2,3]. However, using the fuel cell as the sole power source cannot meet the fast power variations due to its slow transient response. Furthermore, the regenerative energy during brakes cannot be absorbed [4]. In order to improve the fuel economy and durability of the power sources, the energy management strategy (EMS) is required to allocate the power between multiple power sources. Most literature deals with energy management problems in HEVs and plug-in HEVs [5–9], some can be applied to fuel cell vehicles but should consider the durability and dynamic characteristics of the fuel cell [10,11]. In general, EMSs can be divided into two categories: rule-based strategies and optimization-based strategies. The rule-based strategies are designed based on engineering experience and are easy to implement [12,13]. Ettihir et al. [12] design an EMS with three modes, which are maximum efficiency mode, maximum power mode and stop mode, and

* Corresponding author. E-mail addresses: [email protected] (X. Li), [email protected] (Y. Wang), [email protected] (D. Yang), [email protected] (Z. Chen). https://doi.org/10.1016/j.jpowsour.2019.227105 Received 5 June 2019; Received in revised form 16 August 2019; Accepted 3 September 2019 Available online 9 September 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.

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switch between the modes only through the battery SOC threshold. But simple deterministic rules often fail to meet the vehicle economic per­ formance. The fuzzy logic control based methods [14–17] and rules extracted from optimal results have been studied [18]. Although rule-based strategies are suitable for online implementation, they lack adaptability and optimality. In recent years, optimization-based strate­ gies have received more attention. Dynamic Programming (DP) algo­ rithm is widely used in energy management problems with the advantage of ensuring a global optimization [19,20]. However, full knowledge of the driving cycle and time-consuming calculation make it impossible to be applied online. Stochastic dynamic programming (SDP) [21] is used to remedy this problem but it is still time-consuming. The model predictive control (MPC) as a local optimization EMS enables planning the control commands on a future time horizon in each time step, which also requires heavy computations [22]. Equivalent Con­ sumption Minimization Strategy (ECMS) as an instantaneous EMS is first proposed by Paganelli et al. in HEVs [23] and is adopted in fuel cell hybrid vehicles (FCHVs) [24–26], which shows good performance in fuel economy and computation. Ref. [27] proves that ECMS is a reali­ zation of Pontryagin’s Minimal Principle (PMP), which provides the possibility to solve global optimization problems. Other EMSs such as reinforcement learning, extreme seeking methods are also developed [28,29]. However, the PMP combines the global optimality and the characteristics of the online application, and thus receives great interest. The PMP introduces a co-state variable, which has the same effect as the equivalent factor (EF) in the ECMS. Their role is to allocate the cost of using electricity and convert it into equivalent fuel consumption. The EF of ECMS or co-state of PMP depends on driving cycles, which limits the online applications. Many researchers have proposed different methods for determining these parameters including driving cycle pre­ diction, driving pattern recognition and feedback of battery SOC [30]. Adjusting the co-state variable through SOC feedback is the most com­ mon method. For a charge sustained EMS, a fixed reference SOC value is required [31,32], and a SOC reference trajectory is needed in a charge consuming EMS [7]. Onori et al. [31] design an adaptive ECMS through SOC feedback, which updates the EF at regular intervals of duration T instead of at each time step. In Ref. [32], a PI controller is used and its parameters are optimized by the genetic algorithm over a set of driving cycles, a fuzzy logic controller is compared and the results show that the PI controller can achieve better performance. A look-up table used for selecting EF is proposed in Refs. [7,8], and in Ref. [33], the co-state is calculated by an approximation model which is relative to effective mean power and effective SOC drop rate. Although there are different update laws, velocity prediction used for updating EF or co-state is the most promising method to achieve near optimality for that they are applied in future driving and should be updated from future informa­ tion. In Ref. [5], an adaptive ECMS combined RBF-NN velocity predic­ tion is constructed for HEVs. The predicted driving information is used for real-time EF adaptation. Simulation results show that the proposed update law has a better fuel economy and a more stable SOC trajectory than SOC feedback. But the over-fitting caused by neural networks will affect the prediction accuracy for different driving conditions. In this work, in response to the above shortcomings, an adaptive PMP combined with improved Markov based velocity prediction is proposed for FCHVs. The contributions can be summarized as follows: first, we apply PMP combined with the velocity prediction in FCHVs and consider the durability of the PEMFCs. Second, an improved Markov based ve­ locity prediction is proposed considering the driving behavior under different driving patterns, which shows a good prediction accuracy compared with the traditional Markov based velocity prediction. Third, the proposed strategy is suitable to complex driving conditions, it can identify the driving patterns online and select the corresponding ve­ locity predictor based on the identification result, and the co-state is searched by the binary algorithm. This paper is organized as follows: In Section 2, the structure of a fuel cell/battery hybrid vehicle and the system model is introduced. In

Section 3, the PMP formulation, the improved Markov based velocity prediction, and the PSO-SVM algorithm are illustrated. In Section 4, a framework of adaptive EMS is presented and simulation results are provided and analyzed. The conclusions are given in Section 5. 2. System structure and model The structure of a fuel cell/battery hybrid vehicle is shown in Fig. 1. The fuel cell system (FCS) is connected to the bus via a unidirectional DC/DC converter whereas the battery is directly connected in order to maintain the bus voltage. The motor uses the electrical energy converted by the DC/AC inverter to drive the vehicle through mechanical connection with the final drive and the wheels. As indicated by the ar­ rows in Fig. 1, both FCS and battery can supply the electric power to the motor, but only the battery can absorb the power during the vehicle brakes. What’s more, it can be charged by the FCS. 2.1. Vehicle model Assuming the vehicle drives on a road, only the longitudinal dy­ namics model is considered. Then the traction force Ft is equal to: Ft ¼ m

dv 1 þ ρCd Af v2 þ mgf cosðαÞ þ mg sinðαÞ dt 2

(1)

where α represents the inclination angle of the road, v represents the speed of the vehicle. The parameters of the vehicle are shown in Table 1. According to the traction force and vehicle speed, the torque on the wheel Tw and the wheel rotation speed ww are then calculated by: Tw ¼ r⋅Ft ww ¼ v=r

(2)

The torque Tm and the rotational speed wm on the motor shaft are given by: � � � Tw ηfd ⋅Rfd ; Tw � 0 � Tm ¼ (3) ηfd ⋅Tw Rfd ; Tw < 0 wm ¼ ww ⋅Rfd The power demand on the DC bus is written as: � Tm ⋅wm =ηm ; Tm � 0 Pbus ¼ ηm ⋅Tm ⋅wm ; Tm < 0

(4)

where ηm is motor efficiency, which can be found by using the efficiency map of the motor. In this research, a 75 kW AC induction motor from ADVISOR is selected. 2.2. Fuel cell system model Before we introduce the model, we need to determine the size of the fuel cell. When we solve a component sizing problem, theoretically, we cannot avoid the optimized energy management problems. L. Xu et al. [34] divided the component sizing problem into three categories:

Fig. 1. Structure of the fuel cell hybrid vehicle. 2

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50 kW FCS is selected. A complete FCS includes a fuel cell stack and other auxiliary com­ ponents such as air compressor, manifolds, humidifier, cooler, etc. A detailed model can be found in Ref. [39]. In this work, the quasi-static model of a 50 kW FCS is considered, assuming the response time is lower than the sample time. In FCS, the net power Pfc is the difference of the stack power Pstack and the auxiliary power Paux consumed by auxil­ iary components as mentioned before:

Table 1 Parameters of the vehicle model. Parameter

Value

Vehicle mass, m (kg) Wheel radius, r (m) Gravity constant, g (m s 2) Rolling resistance coefficient, f Aerodynamic drag coefficient, Cd Air density, ρ (kg m 3) Vehicle frontal area,Af (m2)

2200 0.3 9.8 0.015 0.3 1.209 3

Transmission efficiency,ηfd

0.95

Final drive gear ratio,Rfd

Pfc ¼ Pstack

7

(7)

The hydrogen consumption rate of a fuel cell stack m_ h is given by: m_ h ¼

without considering the EMS, including a simple EMS and optimizing the powertrain parameters and EMS simultaneously. And a bi-level optimization method is applied for component sizing [34], in which the inner loop (DP) is used to solve EMS problem and the outer loop (Pareto optimization) solve the sizing problem. H. Jiang et al. [35] solved the component sizing problem based on two-dimensional algo­ rithms. For simultaneous optimization problems, both control and powertrain parameters need to be optimized simultaneously. Kim et al. [36] proposed a “pseudo-SDP controller” to optimize EMS and compo­ nent sizing simultaneously for FCEVs. In Ref. [37], convex programming is used to simultaneously optimize the hybrid energy storage system dimension and power allocation. And in Ref. [38], a combined optimal sizing and EMS was developed for EVs using a global search method (GSM). Component sizing of FCHV is another interesting topic, in our research, we focus on the energy management problems for a pre-defined component size. The fuel cell sizing in our research is defined by the vehicle performance index, such as maximum speed ability and grade ability. Among them the maximum speed capability is evaluated on a flat road; the climbing ability corresponds to the ability to maintain a constant speed on a specified slope. The fuel cell system is required to provide enough power to satisfy the maximum speed ability and grade ability. The power demand of the maximum speed ability Pv;max and grade ability Pgrad can be calculated by: � � � � Pv;max ¼ mgf þ Cd Af v2max 21:25 ⋅vmax 3600 ηfd (5) � � � � Pgrad ¼ mgf cosðαÞ þ Cd Af v2climb 21:25 þ mg sinðαÞ ⋅vclimb 3600 ηfd

Paux

N⋅Mh ⋅Istack n⋅F

(8)

where N is the number of cells, Mh is the molar mass of hydrogen, n is the transferred electrons, Fis the Faraday’s constant, Istack is the fuel cell stack current. The relationship between FCS net power and hydrogen consumption rate is measured as Fig. 2(a) shows. Given the hydrogen consumption rate, the efficiency of the FCS, defined as the ratio of the net power and the power generated by H2, is calculated by:

ηfcs ¼

Pfc m_ h ⋅LHV

(9)

where LHV is the lower heating value of hydrogen (120 MJ∙kg 1). Fig. 2 (b) shows the relationship between FCS net power and efficiency. 2.3. Lithium-ion battery model In this work, the selected lithium-ion battery has a capacity of 10 Ah and a rated voltage of 3.3 V. 100 batteries are connected in series to maintain the bus voltage around 330 V. The Rint model of the lithiumion battery, which is denoted by open circuit voltage (OCV) Voc in se­ ries with internal resistance R, is widely adopted in energy optimization problems owing to its simplicity and practicality [32]. Relationships between parameters of the Rint model and SOC were obtained by the Hybrid Pulse Power Characterization test (Fig. 3). A single internal resistance R is used to represent the charge and discharge resistance because their values are close as can be seen from Fig. 3. Based on the Rint model, the change rate of the SOC is calculated by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V V 2oc 4R⋅Pb ðtÞ oc I ðtÞ b _ SOCðtÞ ¼ ¼ (10) 2R⋅Qb Qb

(6)

where vmax is the maximum speed on a flat road, vclimb is the climbing speed. In our case, they are 140 km/h and 30 km/h, the inclination angle of the road is 0.2. Then, we get the power demand of the maximum speed ability and grade ability are 47.4 kW and 40.2 kW. Therefore, a

where Ib is the battery current, Qb is the battery capacity and Pb ðtÞ is the

Fig. 2. (a) FCS hydrogen consumption rate curve, (b) FCS efficiency curve. 3

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� Pfc ðtÞ 2 max Pfc;min ; ðPbus ðtÞ

� � Pb;max Þ ηdc ; min Pfc;max ; ðPbus ðtÞ

� �� Pb;min Þ ηdc (16)

In each prediction horizon, the initial state and final state is equal to the setting value: (17)

SOCtk þ1 ¼ SOCtk þHp ¼ SOCref

Without considering the SOC and FCS net output power change rate constraints, we can define the Hamiltonian of the system: � � � H SOCðtÞ; Pfc ðtÞ; λðtÞ ¼ m_ f Pfc ðtÞ þ λðtÞ⋅f SOCðtÞ; Pfc ðtÞ (18) where λðtÞ is called the co-state, but these constraints need to be considered to ensure efficient and stable operation of the system, the limitations are: Fig. 3. Parameters of the battery Rint model.

output power of the battery. Because the OCV and internal resistance of the battery are functions of SOC [40,41], Eq. (10) can be written as: _ SOCðtÞ ¼ gðSOCðtÞ; Pb ðtÞÞ

Pb;max ¼ Vmin ðVoc Pb;min ¼ Vmax ðVoc

Vmin Þ=R Vmax Þ=R

(11)

(12)

where Vmin and Vmax are the minimum allowable voltage and the maximum allowable voltage, respectively.

ΔPfc;fall � ΔPfc ðtÞ � ΔPfc;rise

(20)

L ¼ γ⋅ Pfc ðtÞ

Pfc ðt

�2 1Þ

(22)

where α, β, and γ are tuning parameters. Eq. (18) is redefined as: � � � � H SOCðtÞ; Pfc ðtÞ; λðtÞ ¼ m_ f Pfc ðtÞ þ λðtÞ⋅f SOCðtÞ; Pfc ðtÞ þ S Pfc ðtÞ � þ L Pfc ðtÞ

3. Adaptive energy management strategy formulation For a charge sustained EMS in the fuel cell/battery hybrid vehicle, the general optimal control problem is how to allocate the power be­ tween the battery and fuel cell under the input and state constraints to meet the vehicle power demand, and minimize the hydrogen con­ sumption while maintaining the battery SOC throughout the driving cycles. When the driving cycle and objective function are predefined, the optimal EMS for each set of powertrain parameters is unique. However, the actual driving cycle is often arbitrary, and it is impossible to obtain a global optimal energy management strategy. An effective way to solve this problem is to predict future velocity, and the optimal horizon is the prediction horizon instead of the whole driving cycle.

(23) PMP can find the optimal trajectories of control and state variables to minimize hydrogen consumption by instantaneously providing the following necessary conditions: � � � � ∂H SOC� ðtÞ; P*fc ðtÞ; λ� ðtÞ * _ SOC ðtÞ ¼ (24) ¼ f SOC� ðtÞ; P*fc ðtÞ ∂λ �

*

λ_ ðtÞ ¼

∂H SOC� ðtÞ; P*fc ðtÞ; λ� ðtÞ ∂SOC





¼

∂f SOC� ðtÞ; P*fc ðtÞ

λ* ðtÞ⋅

� � � H SOC* ðtÞ; P*fc ðtÞ; λ� ðtÞ � H SOC* ðtÞ; Pfc ðtÞ; λ� ðtÞ

3.1. The application of Pontryagin’s minimal principle The objective function in the prediction horizon is defined as: Z tk þHp � m_ h Pfc ðtÞ dt J ¼ min (13)

∂SOC

� (25) (26)

where the superscript * represents optimal trajectories. Due to the lim­ itation of SOC in our case (from 0.4 to 0.8), the parameters of the battery Rint model do not change much over such range. Therefore, the function f is independent of SOC in this case. According to Eq. (25), the optimal co-state can be considered as a constant, which guarantees that if the solution of PMP exists, it is a globally optimal solution [43]. For a given co-state, solving Eq. (26) to get the optimal control and applying it to compute state variation in Eq. (24) at each time instant. As a result, the final state at the end of the prediction horizon is decided by the co-state. A binary search method is used to find the optimal co-state which leads to a certain final state at the end of the prediction horizon. Algorithm of finding the optimal co-state at time step tk is summarized as Table 2, where the simple time is 1s, δ is the acceptable error in the final state and Nmax is the maximum number of iterations.

tk þ1

where J is the hydrogen consumption over the prediction horizon, tk is the current time step and Hp is the prediction length. In order to meet the vehicle power demand, Pfc ðtÞ and Pb ðtÞ is governed by: Pbus ðtÞ ¼ Pfc ðtÞ⋅ηdc þ Pb ðtÞ

(19)

where SOCmin and SOCmax are minimum and maximum allowable SOC, Δ Pfc;fall and ΔPfc;rise are minimum and maximum FCS net output power change rate. Through adding penalty functions S and L, which are defined below [42]: 8 SOCðtÞ � SOCmin < α⋅Pfc ðtÞ; SOCðtÞ � SOCmax S ¼ β⋅Pfc ðtÞ; (21) : 0; SOCmin < SOCðtÞ < SOCmax

The limitation of the battery output power is: �

SOCmin � SOCðtÞ � SOCmax

(14)

where ηdc is the DC/DC efficiency, a constant value is considered in this study. As Pbus ðtÞ can be derived from Eq. (1) ~ (4), Eq. (11) can be rewritten as: � _ SOCðtÞ ¼ f SOCðtÞ; Pfc ðtÞ (15) Choosing Pfc ðtÞ as the control variable and SOC as the only state of the system, according to Eq. (12) and the FCS maximum and minimum net power limitation, Pfc ðtÞ is limited by: 4

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interval is usually large. The speed of flowing urban road has increased slightly compared to the crowded urban road. However, due to a large number of intersections, the vehicles will still stop periodically. In this case, the speed of the vehicles can be maintained for a period of time and does not decrease immediately. It can be seen from the sequence in Fig. 4 (c) and (f) that the above features are obvious. As can be seen from Fig. 4, the velocity curves under different patterns vary greatly. If we construct a velocity predictor through a simple combination, it may introduce behavioral characteristics from other driving patterns to the current driving pattern, which will increase the velocity prediction error. It inspires us to build a velocity predictor for each driving pattern to predict future velocity instead of just one velocity predictor. For ease of expression, the formal is named improved Markov based velocity predictor (IMBVP), the latter is named Markov based velocity predictor (MBVP). A comparison is made between IMBVP and MBVP in urban flowing pattern and urban congested pattern to verify our ideas. The highway pattern is not involved for that there is no parking interval and the test set is not well divided. For a fair comparison, part of the UDDS and MANHATTAN driving cycle are not involved in building the Markov velocity predictor. This part is used to test the prediction accuracy under different velocity predictors. A multi-step Markov chain is established for urban flowing pattern and urban congested pattern, respectively. Taking the establishment process in the urban flowing pattern as an example, two standard driving cycles (UDDS and INDIA_URBAN_SAM­ PLE) except the test part are combined to build the Transition Proba­ bility Matrix (TPM). Based on the new driving cycle, the acceleration and velocity are discretized into finite values: � � a 2 a1 ; a2 ; …; aNa (27)

Table 2 Algorithm of finding the optimal co-state. Step 1 Step 2

Record velocity prediction sequence: vp ½tk þ 1 : 1 : tk þ Hp �. Initialization: λmin ¼

1000, λmax ¼ 0, δ ¼ 0:005, Nmax ¼ 15, N ¼ 0.

Step 3

Update: λ ¼ ðλmin þ λmax Þ=2, i ¼ 1, N ¼ N þ 1.

Step 4

(a) According to Eq. (1) ~ (4), calculate the bus demand power Pbus ðtk þ iÞ; (b) According to Eq. (26), find the optimal FCS net output power P*fc ðtk þ iÞ;

(c) Apply P*fc ðtk þiÞ to Eq. (24), calculate the SOC in next time stepSOCðtk þ i þ 1Þ; (d) Next time step: i ¼ i þ 1.

Step 5

If i < Hp , jump to step 4; � � If i � Hp & �SOCðtk þ Hp Þ SOCref � < δ, jump to step 6; If i � Hp & SOCðtk þ Hp Þ < SOCref , then λmax ¼ λ, jump to step 3; If i � Hp & SOCðtk þ Hp Þ > SOCref , then λmin ¼ λ, jump to step 3; If N ¼ Nmax , jump to step 6.

Step 6

Output λ.

3.2. Improved markov based velocity predictor (IMBVP) Different velocity prediction methods have been proposed, such as artificial neural network method, Markov based method, exponentially decreasing model-based method and telematics technique (GPS and navigation system, etc.) based method [44]. Among these methods, Markov based method is applied for predicting future velocity due to its good performance in accuracy and efficiency. In the existing research [6], either multiple sets of actual velocity curves collected on a fixed route are used as data sets to obtain a Markov based velocity predictor for this route, or different types of standard driving cycles are combined to obtain a Markov based velocity predictor for other driving conditions. Fig. 4 shows the six standard driving cycles and they are divided into three driving patterns, namely highway (HWFET, US06_HWY), urban congested (MANHATTAN, NYCC) and urban flowing (UDDS, INDIA_URBAN_SAMPLE). The highway is the road between the main destinations. From Fig. 4(a) and (d), we can see the vehicles maintain high speed driving, smooth running, and no parking interval. The urban road is the road in city, there are many intersections and vehicles. In Fig. 4(b) and (e), the sequences show a crowded city road, the vehicle has low speed (below 12 m/s) and periodically stops, and the stop

� � v 2 v1 ; v2 ; …; vNv

(28)

At time step k, assuming the real acceleration and velocity values are mapped into ai , vl respectively, and the acceleration after n-step is also mapped into aj . Then the n-step transition probability Pnil;j , defined as the probability when acceleration transfer from ai to aj after n-step at the velocity vl has the form:

Fig. 4. Standard driving cycle under different patterns: (a) HWFET, (b) MANHATTAN, (c) UDDS (d) US06_HWY, (e) NYCC, (f) INDIA_URBAN_SAMPLE. 5

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Pnil;j

Journal of Power Sources 440 (2019) 227105

� � ¼ Pr akþn ¼ aj �ak ¼ ai ; vk ¼ vl

where



PNa

n j¼1 Pil;j

(29)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �2 PT Pl va t¼1 k¼1 vp;k RSME ¼ lT

¼ 1, and the n-step transition probability is estimated by

the maximum likelihood method: . Pnil;j ¼ mnil;j mnil

(31)

where T denotes the trip length, l denotes the prediction horizon, vp;k and va are the predicted velocity and the actual velocity, respectively. Table 3 shows the calculation results in the urban flowing test set, and we can get that IMBVP’s RSME is smaller than MBVP in the entire prediction horizon, and prediction accuracy of IMBVP improves up to 3.5% than MBVP in the urban flowing test set. Table 4 shows the results in the urban congested test set, and the improvement is better than the urban flowing test set, which up to 21.7%. The reason is that the total length of the sequence in all three patterns is 6884 s, the total length of the sequence in urban flowing pattern (UDDS and INDIA_URBAN_­ SAMPLE) is 4060 s (59.0%), and the total length of the sequence in urban congested pattern (MANHATTAN, NYCC) is 1689 s (24.5%). Therefore, in our case, the driving behaviors of the highway and urban congested patterns have less impact on the urban flowing pattern, while the urban flowing pattern has a greater impact on other driving patterns, which leads to significant improvement in IMBVP in the urban con­ gested pattern.

(30)

where mnil;j is the number of times that acceleration transfer from ai to aj PNa n after n-step at the velocity vl , and mnil ¼ j¼1 mil;j . According to the

procedure, the TPMs in the urban flowing pattern at velocity 7 m/s is shown in Fig. 5. It can be seen from Fig. 5, the probability distribution is concentrated around the diagonal when the prediction horizon is short, and the longer the predicted horizon, the more dispersed the probability distribution is. This is because the driving behavior becomes more unpredictable as the forecast time increases. The same procedure is used for MBVP, the only difference is that the remaining driving patterns are adopted to build TPMs. In the urban congested pattern, we build the (TPM) in the same way. Fig. 6 shows velocity prediction results for each second of different prediction horizons (5s, 10s and 20s) between IMBVP and MBVP in urban flowing test set and urban congested test set. It is difficult to evaluate the two predictors from the figure because of the similarity of prediction results, but the difference can still be observed. Therefore, the root-mean-squared error (RSME) is used to evaluate the prediction ac­ curacy, which has the form:

3.3. Driving pattern recognition based on PSO-SVM According to Section 3.2, we need to identify the driving patterns online while driving and select the corresponding velocity predictor. The original SVM is to find a separable hyperplane (W, b) with a maximum margin under linearly separable conditions. Then kennel

Fig. 5. TPMs in urban flowing condition at velocity 7 m/s, (a) 1-step, (b) 5-step, (c) 10-step, (d) 20-step. 6

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Fig. 6. Velocity prediction results at different prediction horizons (5s, 10s and 20s): (a) IMBVP in the urban flowing test set, (b) MBVP in the urban flowing test set, (c) IMBVP in the urban congested test set, (d) MBVP in the urban congested test set.

function is introduced to solve nonlinear separable problems. In this work, we use LIBSVM [45] to achieve the classifier and particle swarm optimization (PSO) algorithm is adopted to improve the classification accuracy. The parameters that need to be optimized are penalty factor c and kernel function width b. Evaluation index is the accuracy of classification. When the sample length is close to or exceeds 180 s, it can reflect the current driving condition [46]. Thus, 300 samples of 150 s are generated under each driving condition in Fig. 4. The sample start time T0 is generated by:

Table 3 RSME under different prediction lengths of IMBVP and MBVP in the urban flowing test set. Prediction lengths IMBVP MBVP Improvement

5s

10s

20s

30s

40s

1.385 1.435 3.5%

2.526 2.570 1.7%

3.940 4.058 2.9%

4.915 5.000 1.7%

5.378 5.468 1.6%

T0 ¼ εðT

Table 4 RSME under different prediction lengths of IMBVP and MBVP in the urban congested test set. 5s

10s

20s

30s

40s

0.729 0.849 14.1%

1.451 1.853 21.7%

2.636 3.140 16.1%

3.086 3.599 14.2%

3.068 3.587 14.5%

(32)

where ε is a random number between 0 and 1, T is the cycle length and ΔT is the sample length. Then we extract and normalize the features which are the average speed vmax , the maximum acceleration amax , the minimum acceleration amin and the idle time I [47]. We choose 80% of the samples as the training set and 20% as the test set. After that, we use the PSO-SVM algorithm to build the classifier. The algorithm is described in Table 5.

Prediction lengths IMBVP MBVP Improvement

ΔTÞ

7

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4.2. Simulation under combined driving cycle

Table 5 PSO-SVM algorithm. Step 1

Step 2 Step 3

A combined driving cycle is constructed to verify the effectiveness of the proposed energy management strategy, which consists of a complete HWFET driving cycle, a part of UDDS driving cycle and a part of MANHATTAN driving cycle. The first 150 s of the new driving cycle is called the initialization stage, in which the velocity data is collected for classifier and the initial co-state is initialized according to the standard urban driving cycles [48]. Velocity prediction length plays an important role in fuel economy for the following reasons: (1) As can be seen from Tables 3 and 4, the prediction error becomes larger as the prediction length becomes longer. It will make the co-state found by PMP far away from the real optimal co-state. (2) A short prediction length will limit the usage of the battery because the charge-sustainability is enforced in the prediction length. Therefore, the choice of the velocity prediction length is a compromise process. In this study, we choose the 40s as the velocity prediction length. Three cases are compared based on PMP. The first case, called offline-PMP, has full knowledge of the combined driving cycle and the optimal horizon is the length of the driving cycle. Therefore, the optimal solution of the offline-PMP is a globally optimal solution. The second case is adaptive-PMP with velocity prediction proposed in this study. The third case is adaptive-PMP without velocity prediction, assuming the velocity of the next 40s is known in advance. Fig. 8 shows the results of the three cases. As can be seen from Fig. 8(a), the trained classifier can correctly identify the various driving patterns. The first 766s in the actual velocity profile is highway pattern, which is classified as pattern 1 at each update time step. 767s–972s is the urban flowing pattern and the rest is the urban congested pattern, they are all correctly classified. Then the cor­ responding velocity predictor is selected according to the DPR result and the predicted velocity sequence can basically flow the driving behaviors. Fig. 8(b) shows battery SOC trajectories under different cases. In all three cases, the initial states are started at 0.6 and all final states are around 0.6. In case 1, SOC varies over a wide range because the chargesustainability is enforced at the beginning and end of the driving cycle. However, since charge-sustainability is imposed on each prediction horizon, the SOC varies around 0.6 throughout the driving cycle in cases 2 and 3. The SOC trajectory in case 2 can follow the trajectory in case 3,

Initialization: (a) Initialize parameters: w, c1, c2, maxgen, popsize, k ¼ 0; (b) Randomly generate particle position and velocity: ðx1i ; v1i Þ, i ¼ 1; 2; …; popsize; (c) Initialize individual best position: Pibest ¼ x1i , i ¼ 1; 2; …; popsize; (d) Initialize social best position: Pgbest ¼ argmaxðFðx11 Þ; Fðx12 Þ; …; Fðx1popsize ÞÞ. (a) Next generation: k ¼ k þ 1; i ¼ 1.

(b)

(e) Update speed: vkþ1 ¼ wvki þ c1⋅randðÞ⋅ðpibest i xki Þ;

xki Þ þ c2⋅randðÞ⋅ðPbest

(f) Update position: xkþ1 ¼ xki þ vkþ1 ; i i

(c) Update individual best position: If Fðxkþ1 Þ > FðPibest Þ, Pibest ¼ xkþ1 ; i i (d) Update social best position If Fðxkþ1 Þ > FðPgbest Þ, Pgbest ¼ xkþ1 ; i i (e) Next particle: i ¼ i þ 1.

Step 4

If i < popsize þ 1, jump to step 3; If i ¼ popsize þ 1 and k < maxgen, jump to step 2; If k ¼ maxgen, jump to step 5.

Step 5

Obtain best SVM through optimal c and b.

4. Simulation validation and discussion 4.1. Framework of the adaptive energy management strategy Fig. 7 shows the framework of the adaptive energy management strategy, which consists of two parts, the offline part, and the online part. In the offline part, we first collect driving data for different driving patterns to build the database. In this study, six standard driving cycles in three different patterns are selected. Then we build the corresponding Markov velocity predictor under each driving pattern based on the database (as discussed in Section 3.2). In addition, PSO-SVM is used to build classifier (as discussed in Section 3.3). In the online part, a sliding window with a fixed length of Hl slides every Hp seconds. When t ¼ tk , features extracted from the historical velocity sequence are input to the classifier, and the corresponding velocity predictor is selected according to the obtained driving pattern. After getting the future velocity sequence, the ‘optimal’ co-state can be found by binary search algorithm (details in Section 3.1) and is used for the real driving cycle by online PMP. In the next update time step tk þ Hp , the same procedure is repeated.

Fig. 7. Framework of the adaptive energy management strategy. 8

X. Li et al.

Journal of Power Sources 440 (2019) 227105

Table 6 Fuel economy and the average power change rate of FCS under different EMSs. Different EMSs DP

Offline-PMP

Rule-based

Proposed A-PMP

Final state SOC

0.600

0.599

0.661

0.602

M(g)

233.627

239.888

267.440

243.297

Meq (g)

233.627

239.660

253.502

242.840

1.438

0.483

2.287

0.524

ΔPfcave (kw/s)

As we can see from Table 6, the final SOC under the rule-based EMS is far from the reference value of 0.6, and the remaining three final SOCs are close to or equal to 0.6. DP has the best fuel economy, 233.627 g of hydrogen is consumed and is normalized to 100%. Hydrogen con­ sumption in the proposed A-PMP is close to the Offline-PMP, their hydrogen consumptions are normalized to 96.2% and 97.5%, respec­ tively. Rule-based EMS has the worst fuel economy, which is normalized to 92.2%. According to the normalized results, the proposed A-PMP increases fuel economy by 4% compared to the rule-based EMS and reduces fuel economy by only 1.3% compared to the offline-PMP. The offline-PMP and DP have the same theoretical results because they are all looking for global optimal solutions. In the offline-PMP and proposed A-PMP, the power change rate of FCS is enforced to the optimization problems (as discussed in Section 3.1), which results in a lower ΔPfcave than the other two strategies. When power change rate of FCS is limited, the battery has to be used frequently and the energy depleted by the internal resistance increases, and more hydrogen should be consumed to maintain the total energy demand of the vehicle. 5. Conclusions This paper proposes an adaptive EMS for FCHVs based on the online adaptation of the co-state used for PMP. The adaptation is performed by driving cycle prediction and it is suitable for different driving condi­ tions. A PSO-SVM based driving pattern classifier is built to classify different driving patterns: highway, urban congested, and urban flow­ ing. An improved Markov based velocity predictor is proposed consid­ ering the driving behavior under different driving patterns, each driving pattern corresponding a Markov based velocity predictor instead of only one Markov based velocity predictor. The comparison of IMBVP and MBVP with five prediction horizons (5 s, 10 s, 20 s, 30 s and 40 s) shows that the prediction accuracy of the IMBVP is better and improves up to 3.5% than MBVP. Simulation results validate the effectiveness of the proposed EMS. First, the classifier works well in classifying different driving patterns and the corresponding velocity predictor can foresee the driving be­ haviors. Second, different EMSs are compared, compared with the rulebased EMS, the proposed adaptive EMS reduces hydrogen consumption by 4%, and has a low average power change rate of FCS. Moreover, the performance of the proposed strategy is close to the offline-PMP, which means that it is a near-optimal method.

Fig. 8. (a) velocity prediction and DPR results (b) Comparison of battery SOC trajectories under different cases (c) Comparison of co-state trajectories under different cases.

but more variation is observed. This is because the co-state found in the prediction horizon is the actual optimal co-state in the real driving cycle when the velocity of the next 40s is known in advance. On the contrary, when predicting the velocity in the next 40s, the searched co-state will deviate from the actual optimal one due to the inevitable prediction error. Fig. 8(c) shows the co-state trajectories in the above cases. In case 1, a constant co-state is applied throughout the driving cycle, and the costates in case 2 and 3 are updated every 40s. At some moments, the co-state in case 2 and 3 jumps to zero, which means that the system tends to use the battery as much as possible without using the fuel cell to meet the final value SOC constraint. For example, the vehicle decelerates to stop after 40s when t ¼ 750s, but the SOC at the initial moment is higher than 0.6 and the battery absorbs the braking energy during deceleration. Therefore, the battery SOC increases and the final value SOC cannot reach the reference value. Hence, the co-state search algorithm exits the search after reaching the maximum number of iterations and converges to zero. The difference in the co-states between case 2 and 3 causes the deviation of the final SOC and fuel economy. Four different energy management strategies are compared to eval­ uate their fuel economy and the average power change rate of the FCS, which are dynamic programming, offline-PMP, rule-based and proposed A-PMP. The control rules of the rule-based EMS is extracted from the offline-PMP results [49]. Simulation results are shown in Table 6. For a fair comparison, the difference between the final SOC and the reference SOC is equivalent to the hydrogen consumption saved or consumed, further details can be found in Ref. [33]. Table 6 shows the final state SOC, actual hydrogen consumption M, equivalent hydrogen consump­ tion Meq and the average power change rate of FCS ΔPfcave under different EMSs.

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