Energy Management Optimization for a Hybrid ... - ScienceDirect

2 downloads 0 Views 361KB Size Report
bInstitute for Dynamic Systems and Control, ETH Zurich, Zurich, 8092, Switzerland ... In this study, the RPM is applied to solve the optimal energy management.
Available online at www.sciencedirect.com

ScienceDirect Energy Procedia 88 (2016) 957 – 963

CUE2015-Applied Energy Symposium and Summit 2015: Low carbon cities and urban energy systems

Energy Management Optimization for a Hybrid Tracked Vehicle Using the Radau Pseudospectral Method Shouyang Weia, Yuan Zoua, *, Fengchun Suna, Onder Christopherb a

National Engineering Laborotory for Electric Vehicles, Beijing Institute of Technology, Beijing, 100081, China b Institute for Dynamic Systems and Control, ETH Zurich, Zurich, 8092, Switzerland

Abstract 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. 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 finitedimensional 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. © 2016 by Elsevier Ltd. This an openLtd. access article under the CC BY-NC-ND license © 2015Published The Authors. Published by is Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of CUE Peer-review under responsibility of the organizing committee of CUE 2015

Keywords: Optimal control; Radau pseudospectral method; Hybrid electric vehicle; Tracked vehicle

Nomenclature DP

Dynamic programming

NLP

Nonlinear programming

RPM

Radau pseudospectral method

* Corresponding author. Tel.: 8610-68944115; fax: 8610-68944115. E-mail address: [email protected].

1876-6102 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of CUE 2015 doi:10.1016/j.egypro.2016.06.119

958

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963

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. 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. 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 fuel economy. Differing from the previous studies [1,2], the DC-DC converter in this study makes the energy utilization more flexible, however it increases the complexity of energy distribution control.

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

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 differential-algebraic equations at the nodes obtained from a Gaussian quadrature. In this study, the RPM is applied to solve the optimal energy management problem. The rapid convergence and high computation efficiency of RPM is observed. 2. Modeling of Hybrid Tracked Vehicle and Formulation of Optimal Energy Management Problem 2.1. Modeling the Powertrain of a Hybrid Tracked Vehicle The output voltage of the generator–rectifier set, Ug, can be expressed as U g KeZm - KxZm I g

(1)

where Ke is the electromotive force coefficient, Zm is the generator speed, Kx is the electrical resistance coefficient, and Ig is the output current. The electromagnetic torque Tm is calculated as Tm Ke I g  Kx I g2 (2) The dynamics of the diesel engine and generator can be modeled as §J · dneng Teng S ie-g ¨ 2 e  J g ¸  Tm (3) ¸ dt ie-g 30 ¨© ie-g ¹ where Teng is the engine torque, ie-g is the gear ratio between the engine and the generator, Je and Jg are the inertia of the engine and generator, respectively, and neng is the engine speed. The lithium-ion battery pack was modelled as the voltage source and the internal resistance as follows

959

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963

­V (SOC)  I bat ˜ Rint_ch (SOC) (I bat ! 0) ° (4) ® ° ¯V (SOC)  I bat ˜ Rint_dis (SOC) (I bat  0) 1 (5) SOC (1  I bat dt ) u 100% 3600C ³ where Ubat is the battery output voltage, V(SOC) is the open circuit voltage, Ibat is the battery current, Rint_ch(SOC) and Rint_dis(SOC) are the internal resistance during charging and discharging, respectively, and C is the battery capacity. The values of V(SOC), Rint_ch(SOC) and Rint_dis(SOC) were tested through the experiment. The DC-DC converter is modelled by the average efficiency as follows (6) U bat I batK sign( Idc-dc ) U dc-dc I dc-dc where K is the average efficiency of the DC-DC converter and Udc-dc and Idc-dc 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. The power balance is (7) Preq U dc-dc ( I g  I dc-dc ) U bat

The power request Preq is calculated using Wong’s vehicle-terrain theory, Preq Rtot ua  M rZz  M bZb

(8)

where Rtot is the total motion resistance, ua is the speed of the center of gravity of the tracked vehicle, Mr is the moment of turning resistance, Zz is the angular speed about the z axis, Mb is the frictional braking torque in the steering system, and is the relative angular speed of the frictional elements. The parameter Rtot is calculated as (9) Rtot f r mg where fr is the rolling resistance coefficient, normally decided by the terrain type, m is the total mass of the tracked vehicle, and g equals to 9.81 m/s2. The parameter Mr is calculated from the expression P t mgl (10) Mr 4 l is the length of track on ground. On the basis of empirical results, P t was found to be (11) Pt Pmax ˜ (0.925  0.15 ˜ R / B)1 where P max is the maximum value of the coefficient of lateral resistance, which is dependent on the terrain type, B is the tread of the vehicle, and R is the turning radius, calculated as B vo  vi (12) R 2 vo  vi where vo and vi are the speeds of the two tracks. The angular speed about the z axis is calculated as (vo  vi ) Zz (13) 2B 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. 2.

960

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963

Fig. 2 Reference cycle for the hybrid tracked vehicle

2.2. Formulating the Optimal Energy Management Problem for the Hybrid Tracked Vehicle 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, tf ­ ½ ° ° (14) min ® J ³ F ( neng , Teng )dt ¾ u t ° ° 0 ¯ ¿ subject to x f ( x , u, Preq ) ­ ° neng (t )  'n0 ° ° ° 0  Teng  Teng_max neng °° U dc_min  U dc  U dc_max (15) ® ° neng_idle  neng  neng_max ° ° I bat_max_char  I bat  I bat_max_disch ° 0  I g  I g_max ° | SOC( t °¯ f )  SOC(t0 ) | 'SOC where the instantaneous fuel consumption rate F is determined by ne and Teng. The integral operation yields the objective J that is to be minimized. Furthermore, x [ neng , SOC] is the state variable and u [th, U dc ] is the control variables. Here, the variable th[0,1] is the throttle opening percentage, which is applied to calculate Teng. The function f represents the system dynamics in Equations (1)–(13). 3. RPM-Based Numerical Optimization 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 [3,4]. The optimal control problem expressed by Equations (14) and (15) is finally transformed to a finitedimensional NLP problem as follows: t f  t0 N 1 ­ ½ J wk ˜ F ( Xk , U k ,W k ; t0 , t f ) ¾ min (20) ® ¦ U i ,i 1,2,... N 1;t f 2 k1 ¯ ¿ subject to

961

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963 N

¦D i 1

k ,i

Xi 

t f  t0 2

f ( Xk , U k ,W k ; t0 , t f ) 0, k 1,2,... N  1

(21)

E(x0 , x N , t0 , t f ) 0,

(22)

(23) c( Xk , Uk ,W k ; t0 , t f ) d 0, t [t0 , t f ], k 1,2,...N  1 where wk is the weight coefficient in the Gauss integration, and can be calculated further. The NLP solver SNOPT is used to solve the problem [5]. The first-order derivative (i.e., the Jacobian matrix of the objective and the constraints), is constructed using a custom code [6], where the partial derivatives of the model functions are calculated using forward finite differentials. 4. Results and discussion To validate RPM, DP was applied to solve the same problem. Fig. 3 shows the trajectories of the state variables and control variables. 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. 3 Trajectories of the optimal states and controls from RPM and DP

The engine operation areas in RPM and DP are shown in Fig. 4. A slightly better fuel economy is achieved in RPM. Although the 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. 5. It may be concluded that the RPM refines the grids adaptively and provides a more accurate solution compared with the DP.

962

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963

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, shown in table 1. Clearly, RPM is more effective for solving the proposed two-state two-control 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.

Fig.4 Comparison of the engine operation area

Fig. 5 Local comparison of the engine operation area

Table 1 Fuel consumption and computation durations. Method

Fuel consumption /gram

Computation duration /h

RPM

2771

3.7

2816 DP

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

4

112

5. Conclusion 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 finite-dimensional 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 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. In the next step, the practical implementations of RPM will be discussed. Acknowledgements 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

Shouyang Wei et al. / Energy Procedia 88 (2016) 957 – 963

Natural Science Foundation, China under Grant 51375044, Defense Basic Research under Grant B20102013 and University Talent Introduction 111 Project B12022. The substantial contribution from Dr. Asprion Jonas and Ms. Dongge Li is acknowledged. References [1] Zou, Yuan, et al. Optimal energy management strategy for hybrid electric tracked vehicles. International Journal of Vehicle Design 58.2-4 2012: 307-324. [2] Zou, Yuan, et al. Combined optimal sizing and control for a hybrid tracked vehicle. Energies 5.11 2012: 4697-4710. [3] Hairer, Ernst, and Gerhard Wanner. Stiff differential equations solved by Radau methods. Journal of Computational and Applied Mathematics 111.1 1999: 93-111. [4] Butcher, John Charles. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, Ltd, 2005. [5] Gill, Philip E., Walter Murray, and Michael A. Saunders. SNOPT: An SQP for large-scale constrained optimization. SIAM journal on optimization 12.4 2002: 979-1006. [6] Asprion, Jonas, Oscar Chinellato, and Lino Guzzella. Optimal Control Of Diesel Engines: Numerical Methods, Applications, And Experimental Validation. Mathematical Problems in Engineering 2014 2014: 1-21.

Biography Dr. Yuan Zou is currently an associate professor with the National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology. His research interests include new energy vehicle system dynamics and control, especially ground vehicle hybrid propulsion modeling and optimal control and the battery storage modeling and estimation.

963