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Torque control strategy incorporating charge torque and optimization for fuel consumption and emissions reduction in parallel hybrid electric vehicles

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Abstract

This paper proposes a torque control strategy to charge the battery of parallel hybrid electric vehicles (HEVs), in which the charge torque by an internal combustion engine (ICE) is newly introduced. In our previous paper, only regenerative braking was used to charge the battery. However, this is not sufficient for CO2 and NOx emissions reduction, and thus, a more effective energy management system (EMS) should be developed. To maintain the battery state of charge (SOC), a target SOC is also introduced in the charge torque. In addition, a CHARGE driving condition where the charge torque by the ICE is generated is newly proposed. By introducing the CHARGE driving, the ICE driving region on torque-engine speed plane was reduced. As a result, a greater reduction in CO2 and NOx emissions is expected. Based on the EMS proposed in our previous papers, an EMS incorporating the CHARGE driving is developed. Through typical driving cycles, the validity of the proposed EMS algorithm incorporating the CHARGE driving is examined.

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Abbreviations

A-ECMS:

Adaptive-equivalent consumption minimization strategy

BSFC:

Brake specific fuel consumption

EM:

Electric motor

EMS:

Energy management system

FLC:

Fuzzy logic control

HEV:

Hybrid electric vehicle

ICE:

Internal combustion engine

JC08:

Japan chassis 08

LHD:

Latin hypercube design

NEDC:

New European driving cycle

RBF:

Radial basis function

SAO:

Sequential approximate optimization

SOC:

State of charge

WLTC:

Worldwide harmonized light duty driving test cycle

SOCmin :

Lower bound of the battery SOC

SOCmax :

Upper bound of the battery SOC

SOC(t):

Battery SOC at time t

SOC T :

Target SOC

ω ICE :

ICE speed

ω ICE max :

Maximum ICE speed

T D REQ(t):

Driving torque request at time t

T ICE T(t):

Target torque of the ICE at time t

T EM T(t):

Target torque of the EM at time t

T ICE C(ω ICE ):

ICE torque control function

T CH(t):

Charge torque at time t

ω ICE L :

Lower bound of the ICE speed (Design variable)

ω ICE U :

Upper bound of the ICE speed (Design variable)

ω ICE V :

ICE speed betweenω ICE L andω ICE U (Design variable)

T(ω ICE L):

Torque at ω ICE L (Design variable)

T(ω ICE U):

Torque at ω ICE U (Design variable)

T(ω ICE V):

Torque at ω ICE V (Design variable)

T SW :

Switch torque from the EM driving to the ICE driving (Design variable)

T coef CH :

Coefficient of charge torque (Design variable)

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Correspondence to Satoshi Kitayama.

Appendices

Appendix I Sequential approximate optimization with radial basis function network

1.1 A.1 Radial basis function network and width in the Gaussian Kernel

The RBF network is a three-layer feed-forward network. Given the training data expressed by {x j , y j } (j = 1, 2, ⋯, m), where m represents the number of sampling points, the output of the network (response surface) is given by

$$ \widehat{y}(x)={\displaystyle {\sum}_{j=1}^m{w}_jK\left(x,{x}_j\right)} $$
(A1)

where m denotes the number of sampling points, K(x, x j ) is the j-th basis function, and w j denotes the weight of the j-th basis function. The following Gaussian kernel is generally used as the basis function:

$$ K\left(x,{x}_j\right)= \exp \left(-\frac{{\left(x-{x}_j\right)}^T\left(x-{x}_j\right)}{r_j^2}\right) $$
(A2)

In (A2), x j represents the j-th sampling point, and r j is the width of the j-th basis function. The response y j is calculated at the sampling point x j . The learning of RBF network is usually accomplished by solving

$$ E={\displaystyle {\sum}_{j=1}^m{\left({y}_j-\widehat{y}\left({x}_j\right)\right)}^2}+{\displaystyle {\sum}_{j=1}^m{\lambda}_j{w}_j^2}\to \min $$
(A3)

where the second term is introduced for the purpose of the regularization. It is recommended that λ j in (A3) is sufficient small value (e.g. λ j =1.0 × 10−2). Thus, the learning of RBF network is equivalent to finding the weight vector w. The necessary condition of (A3) result in the following equation.

$$ w={\left({H}^TH+\varLambda \right)}^{-1}{H}^Ty $$
(A4)

where H, Λ and y are given as follows:

$$ H=\left[\begin{array}{cccc}\hfill K\left({x}_1,{x}_1\right)\hfill & \hfill K\left({\mathbf{x}}_1,{x}_2\right)\hfill & \hfill \cdots \hfill & \hfill K\left({x}_1,{x}_m\right)\hfill \\ {}\hfill K\left({x}_2,{x}_1\right)\hfill & \hfill K\left({x}_2,{x}_2\right)\hfill & \hfill \cdots \hfill & \hfill K\left({x}_2,{x}_m\right)\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill K\left({x}_m,{x}_1\right)\hfill & \hfill K\left({x}_m,{x}_2\right)\hfill & \hfill \cdots \hfill & \hfill K\left({x}_m,{x}_m\right)\hfill \end{array}\right],\;\varLambda =\left[\begin{array}{cccc}\hfill {\lambda}_1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\lambda}_2\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\lambda}_m\hfill \end{array}\right] $$
(A5)
$$ y={\left({y}_1,{y}_2,\cdots, {y}_m\right)}^T $$
(A6)

The width in the Gaussian kernel plays an important role for good approximation. The first author of this paper has proposed the following simple estimate of the width (Kitayama et al. 2011):

$$ \begin{array}{cc}\hfill {r}_j=\frac{d_{j, \max }}{\sqrt{n}\sqrt[n]{m-1}}\hfill & \hfill j=1,2,\cdots, m\hfill \end{array} $$
(A7)

where r j denotes the width of the j-th Gaussian kernel, and d j,max denotes the maximum distance between the j-th sampling point and the other sampling points. (A7) is applied to each Gaussian kernel individually, and can deal with the non-uniform distribution of sampling points.

1.2 A.2 Density function using RBF network

In the SAO, it is important to find out the unexplored region for global approximation. In order to find out the unexplored region with the RBF network, we have developed a function called the density function (Kitayama et al. 2011). The procedure to construct the density function is summarized as follows:

(D-STEP1) The following vector y D is prepared at the sampling points.

$$ {y}^D={\left(1,1,\cdots, 1\right)}_{m\times 1}^T $$
(A8)

(D-STEP2) The weight vector w D of the density function D(x) is calculated as follows:

$$ {w}^D={\left({H}^TH+\varLambda \right)}^{-1}{H}^T{y}^D $$
(A9)

(D-STEP3) The density function D(x) is minimized.

$$ D(x)={\displaystyle {\sum}_{j=1}^m{w}_j^DK\left(x,{x}_j\right)}\to \min $$
(A10)

(D-STEP4) The point minimizing D(x) is taken as the new sampling point.

Figure 13 shows an illustrative example in one dimension. The black dots denote the sampling points. It is found from Fig. 13 that local minima are generated around the unexplored region. The RBF network is basically the interpolation between sampling points: therefore, points A and B in Fig. 13 are the lower and upper bounds of the design variables of the density function.

Fig. 13
figure 13

Illustrative example of density function

Appendix II Detailed numerical procedure for response surface by RBF network

The source code of the SAO using the RBF network has been developed by Excel VBA. The development of response surface by the RBF network is a crucial part, and we describe this part in this appendix. For better understanding, the main program of Excel VBA is provided with the explanation. The scaling technique, called the adaptive scaling, is used for numerical accuracy. The crucial point is that the objective functions are scaled by using the scaling coefficient of the design variables. Refer to Ref. (Kitayama et al. 2011) for the adaptive scaling technique. The main program is listed in Fig. 14, in which the following symbols are used:

Fig. 14
figure 14

Main program for response surface by the RBF network

  • nsp (integer) : the number of sampling points.

  • ndv (integer) : the number of design variables.

  • sfact (real) : the scaling coefficient used for the design variables. The initial value is set to 0.1, and this is updated. The details can be found in Ref. (Kitayama et al. 2011).

  • rxx : 2 dimensional array to store the design variables in the original design variable space.

  • xx : 2 dimensional array to store the scaled design variables.

  • upper : 1 dimensional array to store the upper bounds of the i-th design variable in the original design variable space.

  • lower : 1 dimensional array to store the lower bounds of the i-th design variable in the original design variable space.

  • rad : 1 dimensional array to store the radius given by (A7).

  • dmat : 2 dimensional array to store the distance between sampling points.

  • rchk (integer): parameter to examine the validity of the radius. The initial value is set to zero. If the radius is not valid, rchk will increase as rchk = rchk + 1.

  • nobj (integer): the number of objective functions.

  • objs (real): the scaling coefficient used for the objective functions. Note that the scaling coefficient of the design variables is used.

  • yy : 2 dimensional array to store the objective functions at the sampling points

  • syy : 2 dimensional array to store the scaled objective functions.

  • weight : 2 dimensional array to store the weight given by (A4).

All design variables are scaled in the subroutine “scal” by using the following equation:

$$ \begin{array}{ccc}\hfill xx\left(i,j\right)=\frac{rxx\left(i,j\right)- lower(j)}{upper(j)- lower(j)}\times sfact\hfill & \hfill i=1,2,\cdots, nsp\hfill & \hfill j=1,2,\cdots, ndv\hfill \end{array} $$
(A11)

Next, in the subroutine “radius”, the maximum distance between the sampling points are calculated and stored in dmat. d j,max in (A7) is obtained from this dmat, and consequently the radius given by (A7) is determined. Since the initial scaling coefficient (0.1) is used, the radius will be small. Then, the radius is checked in the subroutine “radcheck”. In this subroutine, the minimum radius is found.

$$ {r}_{\min }=\underset{1\le j\le nsp}{ \min}\left\{{r}_j\right\} $$
(A12)

If r min ≤ 1 then, the scaling coefficient (sfact) is updated as follows, and rchk is increased as rchk = rchk +1:

$$ \begin{array}{cc}\hfill sfact=\alpha \times sfact\hfill & \hfill \alpha >1\hfill \end{array} $$
(A13)

In A13, α is set to 1.2. This step is repeated till r min > 1, and the final scaling coefficient is determined. This repeated step is called the adaptive scaling technique, and the details can be found in Ref. [23].

Let us move on to the next step. When the final scaling coefficient is determined, the scaling coefficient for the objective functions is set as objs = sfact/2. Then, in the subroutine “objscal”, all objective functions are scaled and stored in syy by using the following equation:

$$ \begin{array}{ccc}\hfill syy\left(i,j\right)=- objs+\frac{yy\left(i,j\right)-{y}_{j, \min }}{y_{j, \max }-{y}_{j, \min }}\times objs\times 2\hfill & \hfill i=1,2,\cdots, nsp\hfill & \hfill j=1,2,\cdots, nobj\hfill \end{array} $$
(A14)

where y j,min and y j,max are the minimum and maximum value of the j-th objective function, respectively. Finally, by using the scaled objective functions, the weight vector w of the RBF network is calculated in the subroutine “learn”. Note that the scaled design variables with the final scaling coefficient are used. The response surface is then developed, and optimization by the DE is performed. Also note that the optimization is performed in the scaled design variable space. If you need the optimal solution in the original design variable space, the following equation is used:

$$ \begin{array}{cc}\hfill {x}^{opt}(j)={X}^{opt}(j)\times \frac{upper(j)- lower(j)}{sfact}+ lower(j)\hfill & \hfill j=1,2,\cdots, ndv\hfill \end{array} $$
(A15)

where x opt(j) denotes the optimal solution of the j-th design variable in the original design variable space, X opt (j) the optimal solution of the j-th design variable in the scaled design variable space, sfact the final scaling coefficient, upper(j) the upper bounds of the j-th design variable in the original design variable space, lower(j) the lower bounds of the j-th design variable in the original design variable space, respectively.

Here we would like to explain about (A14) that is a crucial point to develop a highly accurate response surface. By (A14), the range of the scaled objective functions are set between – sfact /2 and + sfact /2. The illustrative example in two dimensions is shown in Fig. 15, from which it is found that the design space is cube. In the case of n dimensions, a hypercube is constructed in the scaled design space. A highly accurate response surface is obtained by the scaling technique, and consequently a highly accurate optimal solution can be obtained.

Fig. 15
figure 15

Illustrative example of scaling technique in two dimensions

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Kitayama, S., Saikyo, M., Nishio, Y. et al. Torque control strategy incorporating charge torque and optimization for fuel consumption and emissions reduction in parallel hybrid electric vehicles. Struct Multidisc Optim 54, 177–191 (2016). https://doi.org/10.1007/s00158-015-1394-x

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