A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle

Applied Energy xxx (2016) xxx–xxx

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle Shouyang Wei a, Yuan Zou a,⇑, Fengchun Sun a, Onder Christopher b a Beijing Collaborative Innovation Center for Electric Vehicle, National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China b Institute for Dynamic Systems and Control, Department of Mechanical and Process Engineering, ETH Zurich, Zurich 8092, Switzerland

h i g h l i g h t s
A serial hybrid propulsion system with two motors for tracked vehicle is modeled.
A DC-DC converter is equipped to enabling a more flexible power distribution.
The Radau pseudospectral method is used to solve optimal energy control problem.
The Radau pseudospectral method and dynamic programming are compared.
RPM offers the higher computation efficiency and better fuel economy than DP.

a r t i c l e

i n f o

Article history: Received 14 March 2016 Received in revised form 12 May 2016 Accepted 9 July 2016 Available online xxxx Keywords: Optimal control Radau pseudospectral method Hybrid electric vehicle Tracked vehicle

a b s t r a c t This study explored the feasibility of using the Radau pseudospectral method (RPM) to optimize the energy management strategy for a hybrid tracked vehicle. The engine–generator set and the battery pack of the serial hybrid tracked vehicle were modeled and validated through the bench test. A DC-DC converter was equipped between the battery pack and the DC bus in this hybrid powertrain, which increased the flexibility of energy distribution between the engine–generator set and the battery. It was simplified as a voltage regulator in the hybrid powertrain model. The power demand during the vehicle operation was calculated according to the vehicle dynamics and driving schedules. The optimal control problem was formulated to minimize the fuel consumption through regulating the power distribution properly between the engine–generator set and battery pack during a typical driving schedule. The RPM was applied to transform the optimal control problem to a finite-dimensional constrained nonlinear programming problem. A comparison of the solutions from RPM and dynamic programming showed that the former offers the higher computation efficiency and better fuel economy. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction To cope with the global energy shortage and satisfy the growing performance requirement, the powertrains in the machinery and military tracked vehicles have been increasingly hybridized. The hybrid electric drive configuration has potential to improve the fuel economy and the maneuverability, and bring more reliable onboard electricity supply; also, the additional stealth operation mode can be fulfilled when the internal combustion engine switches off. The typical powertrain configurations for mature hybrid tracked vehicles on the market include the electric drive–hydraulic steering ⇑ Corresponding author at: No. 5 Zhongguancun South Street, Haidian District, Beijing 100081, China. E-mail address: [email protected] (Y. Zou).

system of the Caterpillar D7E bulldozer, the multimode hybrid electric-mechanical transmission from General Motors, and the E-X-Drive system from BAE Systems, etc. [1–5]. In this study, a serial dual-motor drive configuration is used in a hybrid tracked vehicle to realize a relative flexible package under the constraints of the component power density and installation space inside the vehicle, as shown in Fig. 1. In this configuration, the two electric motors drive the two sprockets separately. A diesel engine– generator set (EGS) and a traction battery pack supply power to the two motors. A DC-DC converter between the DC bus and the lithium-ion battery pack regulates the voltage of the DC bus, enabling the output power of the battery pack controllable. Therefore, through optimizing the power distribution between the engine-generator set and the battery pack, the engine can be controlled to operate with greater efficiency in order to improve the

http://dx.doi.org/10.1016/j.apenergy.2016.07.020 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Wei S et al. A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.07.020

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Sprocket 1 Gear reduction 1 AC DC Bus

EM1

Generator /Rectifier

AC EM2

Integrated power electronic unit

Gear box

DC

Gear reduction 2

+

Diesel engine

Li-Ion Battery

DC/DC Converter

Sprocket 2

Fig. 1. Dual-motor drive configuration of a hybrid tracked vehicle.

fuel economy. In addition, the DC-DC converter can prevent overdischarging and overcharging of the battery to improve the system reliability. An integrated power electronic unit works as the inverter, and it is responsible for regulating the speed and torque of the two motors. Differing from the previous studies [6,7], the DC-DC converter in this study makes the energy utilization more flexible, however it increases the complexity of energy distribution control. The design of a controller with an optimal energy management strategy to regulate the power distribution efficiently is challenging. Numerous articles focusing on this topic mainly use two typical methods: dynamic programming (DP) and Pontryagin’s minimum principle (PMP). Based on Bellman’s principle of optimality [8], the DP method yields a global optimal solution generally used as a benchmark to evaluate the controller design or to be reproduced and extracted to generate a set of rules for realtime control [9–11]. However, this method involves high computational load, especially when there is a relatively high number of state and control variables, and/or these variables are discretized on a fine grid [8]. When the probabilistic statistics of the power request over a period of time is considered, the stochastic dynamic programming (SDP) algorithm can be applied to find the optimal control map to minimize the cost expectation [12]. The PMP method is used to formulate the analytical necessary condition equation, which defines the mathematical relationship among the state variables, the Hamiltonian, and co-states. The extremal solution is obtained by computing and minimizing the Hamiltonian function at each instant. It is reasonable to consider the variation of the co-state with respect to time negligible, due to the fact that the battery efficiency is almost constant in this application, which makes the necessary conditions of optimality provided by PMP also sufficient [13–15]. The nonlinear programming (NLP) method, which theoretically differs from the DP and PMP methods, has been used to solve the optimal energy management problem for hybrid electric vehicles [16]. When the state and control variables are discretized into the finite grid in a time span, the differential equations and constraints are reduced to a set of the algebraic equations, and the optimal solution is directly extracted by solving a large-scale NLP problem with the help of sophisticated software packages, such as GPOPS-II or PSOPT [17]. The convex optimization method is a special NLP method. In the method, the optimal control problem is reformulated as a convex optimization problem, and the developed solver, such as CVX [18], is applied to find the solution. Because of the convex property, the computation efficiency is substantially improved. However, the necessary convex property for the reformulated optimal control problem often leads to relaxation

of the constraints or model changes and affects the solution accuracy [19,20]. A few studies have explored the possibility of applying NLP to the optimization of the energy management strategy for hybrid electric vehicles [16,21]; however, such investigations are barely sufficient or comprehensive, partly because of the lack of general methodologies and tools to achieve direct transcription of the optimal control problem to an NLP problem. As typical direct methods categorized as NLP methods, pseudospectral methods have been increasingly used in the numerical solution to optimal control problems for various dynamic systems [22–24]. The pseudospectral methods are a class of direct collocation methods in which the optimal control problem is transcribed to an NLP problem by parameterizing the state and control variables through global interpolation polynomials, and collocating the differentialalgebraic equations at the nodes obtained from a Gaussian quadrature. They are potentially computationally more efficient than DP methods because they exploit well developed NLP codes, and the problem can be reformulated more flexibly compared with the use of optimality conditions or the convex method. In practice, three types of collocation schemes, namely Legendre–Gauss, Lege ndre–Gauss–Radau, and Legendre–Gauss–Lobatto schemes, are often used, and the Radau pseudospectral method (RPM) outperforms other methods in terms of computation convergence and accuracy in some cases [25]. Compared with the energy management investigations based on the principle of optimality for the hybrid wheeled vehicle, pseudospectral method–based energy management studies on the hybrid tracked vehicle are scarce. DP and stochastic DP methods have been applied to find the optimal control strategy for the serial hybrid electric tracked vehicle without a DC-DC converter between the battery and the DC bus [6,7]. The existence of the DC-DC converter provides a more flexible power distribution but also leads to greater complexity and challenge because the excessive state variables lead to the ‘‘curse of dimensionality” in DP and PMP, considerably increasing the computation time. In this study, the RPM is applied to solve the optimal energy management problem for the hybrid tracked vehicle. The optimality and effectiveness of the RPM were validated through the comparison with the DP method and the rapid convergence and high computation efficiency is observed. The remaining of the paper is organized as follows. The modeling of the power demand and the hybrid powertrain is discussed in Section 2, including the power balancing, the diesel engine–generator rectifier, and the lithium-ion battery with a DC-DC converter. This section also formulates the two-state and two-control optimal energy management problem to meet the power request history calculated from a field driving schedule. Section 3 describes the RPM-based transcription from the optimal

Please cite this article in press as: Wei S et al. A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.07.020

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generators and rectifiers have been investigated in the previous study [26], with the leverage of equivalent electric circuit illustrated in Fig. 2. The output voltage and the electromagnetic torque of the generator are calculated as

control problem to an NLP problem. The results of the RPM and DP are compared and merits of the RPM are discussed in Section 4. Section 5 concludes the paper.

(

2. Modeling of hybrid tracked vehicle and formulation of optimal energy management problem

U g ¼ K e xg  K x xg I g

ð1Þ

T m ¼ K e Ig  K x I2g

2.1. Modeling of the hybrid tracked vehicle

where Ug and Ig are the output voltage and current of the generator; Ke is the electromotive force coefficient; Kx is the electrical resistance coefficient; xg is the generator speed with the unit rad/s; Tm is the torque of the generator. The dynamics of the diesel engine and the generator are described as

A systematic control-oriented model is established for the hybrid tracked vehicle to formulate the optimal energy management problem. The EGS, battery, and power request models are described as follows [7].

8 < T eng g

2.1.1. Modeling of the EGS In the EGS, a diesel engine and a permanent magnetic synchronous AC generator are used to generate electricity. The engine outputs 300 kW peak power and 2000 Nm peak torque. The top speed of the engine is 2000 r/min. The generator outputs 270 kW rated power within the speed range from 2500 r/min to 3100 r/min and 960 Nm rated torque within the speed range from 0 to 2500 r/min. The gearbox with a fixed gear ratio matched the speed of the engine in the feasible range for the generator. A three-phase full-wave uncontrolled rectifier is used to transform the AC voltage to DC voltage. Models of permanent magnetic synchronous AC

:

ieg

eg

pi  T m ¼ 30 eg





Je i2eg

þ Jg

dneng dt

ð2Þ

neng ¼ ng =ie-g

where Teng is the engine torque; ie-g is the gear ratio between the engine and the generator; ge-g is the average efficiency from the engine to the generator; Je and Jg are the moments of inertia for the engine and generator, respectively; neng and ng are the rotational speeds of the engine and generator with the unit of r/min. The model and its parameters were validated through the dynamic test bench as shown in Fig. 3. The simulated Ug, neng, and Tm are predicted accurately by the model. 2.1.2. Modeling the battery and DC-DC converter The lithium-ion battery pack is modeled as the voltage source and the internal resistance by the following equation:

8 > :

 bat

¼

VðSOCÞ  Ibat  Rint

ch ðSOCÞðI bat

VðSOCÞ  Ibat  Rint dis ðSOCÞðIbat < 0Þ   R 1 Ibat dt  100% SOC ¼ 1  3600C

ð3Þ

where U bat is the battery output voltage; VðSOCÞ is the open circuit voltage; Ibat is the battery current; Rint ch ðSOCÞ and Rint dis ðSOCÞ are the internal resistance during charging ðIbat > 0Þ and discharging ðIbat < 0Þ, respectively; C is the battery capacity. The values of VðSOCÞ, Rint ch ðSOCÞ, and Rint dis ðSOCÞ were obtained through the

Fig. 2. Equivalent electric circuit of the EGS.

1000

800 Ug (tested)

800

600

600

400

400

200

200 2

3

4

5

6

7

8

9

10

11

Ig (A)

Ug (simulated)

Load current

Ug (V)

> 0Þ

0 12

t (s) 1000

5000 neng (simulated)

Tm (tested)

neng (tested)

800

4000

600

3000

400

2000

200

1000

0 2

3

4

5

6

7

8

9

10

11

neng (rpm)

Tm (Nm)

Tm (simulated)

0 12

t (s) Fig. 3. The simulation and experiment results for the validation of the engine–generator model.

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0.8

480

Rint_ch

V(SOC)

0.6

460

0.4

440

V(SOC) (V)

Rint_ch , Rint_dis (Ω)

average value, 0.8; the turning radius R can be calculated as follows when the track slippage is negligible:

500

1

0

0.2

0.4

0.6

0.8

1

420

SOC Fig. 4. Battery open-circuit voltage and internal resistance obtained through experiments.

experiment, as shown in Fig. 4. The capacity of the battery pack is 100 Ah, with the rated terminal voltage 480 V. The DC-DC converter is used to protect the battery from abuse and facilitate a more flexible energy distribution between the engine–generator and the battery pack. In this study, the DC-DC converter serves as a buck converter and regulates the DC-bus voltage in the proper range. The DC-DC converter is modeled by the average efficiency as

U bat Ibat gdc

signðIdc Þ

¼ U dc Idc

ð4Þ

where gdc is the average efficiency of the DC-DC converter and Udc and Idc are the output voltage and electric current, respectively. The DC-DC converter is capable of regulating the DC bus voltage and the electric current distribution between the generator and the battery and offers considerable flexibility in the control strategy design. 2.1.3. Modeling the power request The power balance is always maintained as follows:

Preq ¼ U dc ðIg þ Idc Þ

ð5Þ

The tracked vehicle dynamics equations are formulated based on Wong’s vehicle-terrain theory [27]. When only the longitudinal/lateral/yaw motions are considered, the governing equations are

8   h i 2 Iz r 2 R >  x_ i < T i  iF0rigr  BiM0rgr ¼ mr 2 2 ðRB=2Þ i g i0 gi BðRB=2Þ i i   h0 i i 2 > Iz r2 R : T o  iF rogr þ BiMrgr ¼ mr þ i2 g BðRþB=2Þ x_ o 0 0 i2 g ðRþB=2Þ o

o

0 o

ð6Þ

0 o

where Ti, To are the torques of the inside and outside drive motors respectively; Fri, Fro are the rolling resistance of the two tracks; Mr is the resisting yaw moment from the ground; B is the tread of the vehicle; Iz and m are the yaw moment of inertial and the mass of the vehicle, respectively; r is the radius of the sprocket; i0 is the fixed gear ratio between motors and sprockets; go;i is the efficiency from motor shafts to tracks; R is the turning radius of the vehicle; xi and xo are the rotational speeds of the inside and outside spockets, respectively. Considering a steady-state turning, the resisting yaw moment from the ground is calculated as

Mr ¼

B xo þ xi 2 xo  xi

lt mgl 4

ð7Þ

where lt is the coefficient of lateral resistance; g is 9.81 m/s2; l is the contact length of the track on ground. On the basis of empirical results, lt was found to be [28]

lt ¼ lmax  ð0:925 þ 0:15  R=BÞ1

ð8Þ

where lmax is the maximum value of the coefficient of lateral resistance, which is dependent on the terrain type; here it is taken as the

ð9Þ

The rotational speeds of the outside and inside sprockets, xo and xi can be calculated as

xo;i ¼

Rint_dis

0.2

R¼

30v o;i i0 pr

ð10Þ

where v o;i is the speed of the two tracks. An implicit assumption in Eq. (13) is that the track slippage is ignored. The rolling resistance acting on the two tracks are

F ri ¼ F ro ¼ 0:5f r mg

ð11Þ

where fr is the coefficient of motion resistance of the vehicle in the longitudinal direction. The efficiency from motor shafts to tracks go;i is calculated as

go;i ¼ 0:941ð0:97  0:003v o;i Þ

ð12Þ

The electric power Preq requested by the two motors varies as the driving modes changes, especially during braking. Two kinds of electric braking are adopted to reduce mechanical friction brake: resistor braking and regenerative braking. To recuperate the braking energy as much as possible, regenerative braking is applied to prior to resistor braking. However, resistor braking is used of the DC bus voltage Udc is higher than a threshold value Uthr for safe operation of the electronic devices. Preq is calculated from Eq. (13), where Preq is positive when the electric power outputs and negative when the electric power is recuperated during braking. It must be noted that in this vehicle, the regenerative braking only happens when Udc < Uthr; otherwise the resistor braking is triggered, and the electric power will be consumed by the resistorheating. In the latter case, Preq remains positive whenever the electric motors drive or brake.

Preq

8 sgnðT o Þ sgnðT i Þ < T o xo gmot þT xg U dc < U thr i i mot ¼ sgnðT o Þ sgnðT Þ : T o xo gmot þ T i xi gmot i U dc P U thr

ð13Þ

where gmot is the efficiency of the motors and integrated power electronic unit. Because this study is focused on the optimal control algorithm for the hybrid powertrain, it is assumed that the motors and associated power electronics are only power-conversion devices with the constant average efficiency. Each motor outputs 160 kW rated power within the speed range from 2000 r/min to 5500 r/min and 1200 Nm rated torque within the speed range from 0 to 2000 r/min. The values of the key parameters are summarized in Table 1.

Table 1 The values of the key parameters. Parameter

Value

Unit

Vehicle mass m Vehicle tread B Contact length of the track on ground l Electromotive force coefficient Ke Electrical resistance coefficient Kx Gear ratio between engine and generator ie-g Battery capacity Cbatt Engine inertial Je Generator inertial Jg Efficiency from engine to generator ge-g

15,000 2.55 3.57 1.64 0.00037 1.58 100 3.6 2.2 0.9

kg m m Vsrad2 NmA2 / Ah kgm2 kgm2 /

0.97 0.9

/ /

Efficiency of the DC-DC converter gdc Efficiency of motor and integrated power electronics

gmot

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A set of track speeds recorded in a field test is used as the typical driving schedule for the design and evaluation of the energy distribution strategy, shown in Fig. 5. The schedule consists of a 990s continuous drive and typical operations for tracked vehicles, such as accelerating, decelerating, and steering. 2.2. Formulating the optimal energy management problem for the hybrid tracked vehicle

applied to calculate Teng. The function f represents the system dynamics in Eqs. (1)–(13). The values of the variables n_ eng ðtÞ, Teng(t), Udc(t), neng(t), Ibat ðtÞ, and Ig ðtÞ are constrained in the feasible range. To maintain the energy balance of the battery pack, at the end of the driving schedule, SOCðt f Þ is set to be equal to the initial value. Eqs. (14) and (15) represent the typical nonlinear constrained optimal control problem in a specific time span. 3. RPM-based numerical optimization

For the hybrid tracked vehicle, energy management refers to the design of a high-level control strategy, which determines the power flows from two or more energy converters to achieve the overall control objectives, such as minimization of the vehicle fuel consumption, minimization of engine emissions, and improving drivability. In this study, the objective of the optimal energy management problem for the hybrid powertrain is to minimize the fuel consumption under the system performance constraints. This problem is formulated as a typical optimal control problem as follows

t¼

 Z minu J ¼

Eq. (14) is reformulated as

tf

The formulated optimal control problem was solved by directly transforming the continuous-time problem to an NLP problem. The Legendre–Gauss–Radau collocation schemes are used for the approximation of the all continuous signals because they show stiff decay and algebraic stability [29,30]. The time interval is transformed from [t0, tf] to [1, 1] via the affine transformation

Fðneng ; T eng Þdt

ð14Þ

t0

ð16Þ



Z tf  t0 1 minuðsÞ J ¼ Fðneng ðsÞ; T eng ðsÞÞds 2 1

subject to

8_ x ¼ f ðx; u; P req Þ > > > > jn_ eng ðtÞj < Dn0 > > > > > > 0 < T eng < T eng max ðneng Þ > > > n eng idle < neng < neng max > > > > > Ibat max char < Ibat < Ibat max disch > > > > > 0 < Ig < Ig max > > : jSOCðtf Þ  SOCðt0 Þj < DSOC

ð15Þ

where the instantaneous fuel consumption rate F is determined by the engine speed ne and the engine torque Teng, which are determined from the brake specific fuel consumption (bsfc) map obtained from the engine bench test, shown in Fig. 8. The integral operation yields the objective J that is to be minimized, namely the cumulative fuel consumption. Furthermore, x ¼ ½neng ; SOC is the state variable and u ¼ ½th; U dc  is the control variable. Here, the variable th 2 ½0; 1 is the throttle opening percentage, which is

Track Velocity (km/h)

tf  t0 t þt sþ f 0 2 2

ð17Þ

and Eq. (15) is also defined with respect to the variable s 2 ½1; 1. The Legendre–Gauss–Radau collocation points lie in the half-open interval s 2 ð1; 1. Assuming that N collocation points exist and PN(s) is the Nth-degree Legendre polynomial, the Legendre–Gauss– Radau points are the roots of PN1(s) + PN(s). When all Legendre– Gauss–Radau points are determined, the state variables and control variables at the points are approximated by Lagrange interpolation polynomials. The first-order differential of the state variables at the Legendre–Gauss–Radau points can be obtained by differentiating the approximation, thereby establishing an equal constraint between the Lagrange interpolation polynomials and the system state equation. Finally, the optimal control problem is transformed to a standard NLP problem. Additional details are available in the related literature [31–33]. When the Legendre–Gauss–Radau points sk 2 ½1; 1Þ (k = 1, 2, . . ., N  1) are chosen, the grid of N points along the time axis is mapped as t k 2 ½t0 ; t f Þ, yielding N  1 intervals of the length hk = tk+1  tk (k = 1, 2, . . ., N  1). In particular, sN ¼ 1 and

40

Track 1

Track 2

30 20 10 0

0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

t (s)

Preq (kW)

200 100 0 -100 -200

0

100

200

300

400

500

t (s) Fig. 5. Reference cycle for the hybrid tracked vehicle.

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t N ¼ tf when k ¼ N. The state variables are discretized in ti and approximated as

xi ¼

N X Lðt i ÞXðti Þ

problem [34]. The first-order derivative (i.e., the Jacobian matrix of the objective and the constraints), is constructed using a custom code [35], where the partial derivatives of the model functions are calculated using forward finite differentials.

ð18Þ

i¼0

4. Results and discussion

where Lðt i Þ denotes the Lagrange polynomials at ti and X(ti) represents the state value at ti. The state differential is calculated as N N X X _ i ÞXðti Þ ¼ x_ i ¼ Lðt Dðti ÞXðti Þ i¼0

RPM was first applied to optimize the power distribution between the engine–generator set and the battery by regulating the engine torque and DC voltage. To validate RPM, DP was applied to solve the same problem. Fig. 6 shows the trajectories of the state variables (SOC and neng) and control variables (Udc and th). The similar trends of the dynamics of SOC and neng are found. However, the differences between the state variables obtained from the RPM and DP are observed. It is found that RPM regulates the variable th and Udc more frequently, and the fluctuation range of the state variable neng of RPM is smaller. The engine maintains the rotational speed of approximate 1200 rpm. Fig. 7 shows the correlation between Udc and neng. There are three evident distribution areas for RPM: when neng is lower than 1200 rpm, Udc increases linearly with neng; when neng is approximately 1200 rpm, Udc fluctuates in the range of 314–321 V; and when neng is higher than 1200 rpm, Udc drops to 300 V. The engine speed is higher than 1200 rpm most of the time for DP. Although the working points of the engine in DP is more scattered than that in RPM, the linear relationship between Udc and neng can still be observed in some local ranges. When neng is approximately 1200 rpm, Udc lies between 320 and 325 V, and when neng is higher than 1220 rpm, Udc changes in the range of 300–315 V. The engine operation areas in RPM and DP are shown in Fig. 8. A slightly better fuel economy is achieved in RPM. Although the

ð19Þ

i¼0

where Dðti Þ is an N  ðN þ 1Þ matrix, called the Radau pseudospectral differentiation matrix and has the property D0  D1:N ¼ 1. Here, D0 represents the first column and D1:N is a square matrix containing all the remaining columns. The optimal control problem expressed by Eqs. (14) and (15) is finally transformed to a finitedimensional NLP problem as follows:

(

min

U i ;i¼1;2;...N1;t f

) N1 tf  t0 X J¼ wk  FðXk ; Uk ; sk ; t 0 ; t f Þ 2 k¼1

ð20Þ

subject to N X tf  t0 Dk;i Xi  fðXk ; Uk ; sk ; t 0 ; t f Þ ¼ 0; 2 i¼1

k ¼ 1; 2; . . . N  1

Eðx0 ; xN ; t 0 ; tf Þ ¼ 0;

ð21Þ ð22Þ

cðXk ; Uk ; sk ; t 0 ; t f Þ 6 0;

t 2 ½t0 ; t f ;

k ¼ 1; 2; . . . N  1

ð23Þ

where wk is the weight coefficient in the Gauss integration, and can be calculated further. The NLP solver SNOPT is used to solve the

SOC

0.75

SOC of RPM

0.73

SOC of DP

0.71 0.69 0.67

0

100

200

300

400

500

600

700

800

900

1000

800

900

1000

800

900

1000

800

900

1000

t (s)

n

eng

(rpm)

1500

n

1400

eng

n

of RPM

eng

of DP

1300 1200 1100

0

100

200

300

400

500

600

700

U

dc

(V)

t (s) 400 380 360 340 320 300

U of RPM

U of DP

dc

0

100

200

300

400

500

dc

600

700

t (s) 1

th of RPM

th

0.75

th of DP

0.5 0.25 0

0

100

200

300

400

500

600

700

t (s) Fig. 6. Trajectories of the optimal states and controls from RPM and DP.

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7

engine operates in the same area in both methods, the distribution of the operation points is different when the specific area is magnified, as shown in Fig. 9. The operation points are clustered more densely in RPM than that of DP. It may be concluded that the RPM refines the grids adaptively and provides a more accurate solution compared with the DP. The latter method discretizes the states and controls at fixed time steps, approximates the cost function in the backward calculations and the state variables in the forward searching process. The accuracy and computation efficiency of RPM is found to outperform DP. Table 2 lists the computation durations and the fuel consumption from the two methods. For DP, when the state and control variables are discretized into 31  31 grids, the solving process takes 4 h; when the state and control variables are discretized into 501  41 and 31  41 grids respectively, the solving process takes 112 h. The fuel consumption for this two different discrete density levels are 2816 g and 2788 g, respectively. By contrast, RPM takes 3.7 h and the fuel consumption is 2771 g, yielding the better fuel economy and accurate results with less computational time-consuming. It should be noted that, unlike DP, the hp-adaptive mesh refinement technique [35,36] is adopted in our RPM algorithm. The number of mesh intervals in regions of the trajectory and the order of the Lagrange polynomials are adjusted automatically depending on the equations of optimal control problem, increasing the convergence rate and solving accuracy. Clearly, RPM is more effective for solving the proposed two-state twocontrol optimal energy management problem; and it offers more advantages than DP especially in the case of problem with a higher number of state and control variables. The theoretical optimal results obtained by RPM can be used to benchmarking the onboard real-time control strategies, or to extract rule-based strategies, instead of by DP, in order to considerably reducing the offline computational load.

Fig. 7. Correlation between Udc and neng.

Fig. 8. Comparison of the engine operation area.

5. Conclusions

Fig. 9. Local comparison of the engine operation area. Table 2 Fuel consumption and computation durations.

RPM was applied to solve the optimal energy management problem for hybrid tracked vehicles. The power demand and hybrid powertrain system was modeled, and the energy management problem was formulated as a constrained optimal control problem including the two state variables and two control variables. The optimal control problem was transformed into a finitedimensional NLP problem by using the Legendre–Gauss–Radau pseudospectral scheme. The effectiveness of RPM was demonstrated through the comparison with DP. It is found that RPM provides slightly more accurate results at the less computation cost. The results indicate that RPM is a promising candidate to solve the optimal energy management of the hybrid electric vehicles. It can be used to facilitate the development of control strategies and shorten developing periods for the hybrid powertrains, which promote the penetration of hybrid vehicles in cities, especially for hybrid tracked machinery and passenger cars in civilian fields that have high potential to reduce energy consumption and emissions. Acknowledgments

Method

Fuel consumption/gram

Computation duration/hour

RPM DP

2771 2816 (States: 31  31; Control: 31  31; Time step: 0.1 s) 2788 (States: 501  41; Control: 31  41; Time step: 0.1 s)

3.7 4

112

Note: Computer parameters: IntelÒ CoreTM i5-3450 CPU @ 3.10 GHz, 8G RAM, with 64-bit Windows 7 OS.

This research was conducted under the support from Joint Research Center of New Energy Vehicle Dynamic System and Control, jointly set up by the Beijing Institute of Technology, Beijing, and the Institute of Dynamic Systems and Control, ETH Zürich. This research is partly supported by National Natural Science Foundation, China under Grant 51375044, Defense Basic Research under Grant B2220102013 and University Talent Introduction 111 Project B12022. The substantial contribution from Dr. Asprion Jonas and Ms. Dongge Li is acknowledged.

Please cite this article in press as: Wei S et al. A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.07.020

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Please cite this article in press as: Wei S et al. A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.07.020