2.2.1. Transient Fuel Consumption Model

The transient fuel consumption model consists of a steady fuel consumption and a transient correction factor. The steady fuel consumption is based on an engine universal characteristic map and calculated by two-dimensional interpolation, which can be shown as

$$B\_{\rm 6} = T\_{\rm 6} n\_{\rm 6} b e \left( T\_{\rm 6}, n\_{\rm 6} \right) / \left( 9549 \cdot \rho\_{\rm fu} \right), \tag{1}$$

where *B*s is the steady fuel consumption, *T*e is the engine torque, *n*e is the engine speed, *be* is the specific fuel consumption on the basis of the look-up table, and ρfu is the fuel density.

The BPNN was selected to identify a transient fuel consumption correction factor due to its excellent learning ability. The engine torque *T*e, engine speed *n*e, and their calculated differentials and product were employed as the input layer of the BPNN, considering the engine transient process, including speed and torque variation. The output layer is the correction factor employed to correct the steady fuel consumption model. The BPNN was first trained by a cluster of data collected in the actual engine operation. Figure 3 shows the structure of the transient fuel consumption model based on BPNN.

**Figure 3.** Structure of the transient fuel consumption model based on the back propagation neural network (BPNN).

The transient fuel consumption can be expressed as

$$B\_{\rm t} = B\_{\rm s}(n\_{\rm e}, T\_{\rm e}) \times R\_{\rm bp}(n\_{\rm e}, T\_{\rm e}, dn\_{\rm e}, dT\_{\rm e}), \tag{2}$$

where *B*<sup>t</sup> is the transient fuel consumption; *R*bp is the correction factor based on a trained BPNN calculation; and *dn*<sup>e</sup> and *dT*<sup>e</sup> are the differential of engine speed and torque, respectively.

This BPNN employed the trainlm function as the train function, log-sigmoid as the hidden layer transfer function, and the purelin function as the output layer transfer function. This trained model finally achieved a good simulation performance compared to the experimental data. More details on the model can be found in our previous research [61].

### 2.2.2. Output Torque Model

The steady output torque can be calculated by interpolation of an engine torque-speed-throttle characteristic map as

$$T\_{\mathbf{e},\mathbf{s}} = f\_{\text{Tr},\mathbf{e}}(n\_{\mathbf{e}}, \mathbf{y}\_{\mathbf{e}}),\tag{3}$$

where *T*e,s is the engine steady torque, *f*Tn,e is the interpolation function of the torque-speed-throttle characteristic map, and γ<sup>e</sup> is the engine throttle.

Considering the transient operation process, a delay of the throttle voltage, injection pressure, and air inflow would lead to a delay of the output torque. For describing the transient process, the dynamic output torque can be expressed as

$$T\_{\mathbf{e}} = T\_{\mathbf{e},\mathbf{s}} \cdot \frac{1}{1 + \tau\_{\mathbf{e}}\mathbf{s}'} \tag{4}$$

where *T*<sup>e</sup> is the engine dynamic output torque, τ<sup>e</sup> is the engine time constant, and *s* is the Laplace operator.

### *2.3. Driveline Components' Models*
