*4.3. Control Strategy*

In the BEV case, the aim of control strategy is to find an optimal gear ratio *h\** in each sampling instant, which minimises the battery discharging power and maximises the battery recharging/ regenerative power:

$$h^\* = \underset{h}{\text{argmin}} \left\{ \begin{array}{ll} \eta^k\_{httt} P\_{htt}(P\_{d\prime\prime} \alpha\_{W\prime} h), & \text{for } P\_{httt} < 0 \text{ (charging)}\\ \eta^k\_{httt} P\_{httt}(P\_{d\prime\prime} \alpha\_{W\prime} h), & \text{for } P\_{httt} \ge 0 \text{ (discharging)} \end{array} \right. \tag{6}$$

where η*batt* is the battery efficiency and *Pd* is the transmission input power demand calculated from *vv* and τ*<sup>w</sup>* as shown in Figure 12. The optimal gear ratio *h*\* is calculated offline and mapped as *h*\* (*Pd*, ω*w*). The SoC dependence of *Pbatt* has a minor effect on *h*\* and is neglected in Equation (6), and further on.

Similarly, in the CONV case, the control strategy finds an optimal gear ratio *h*\* that minimises the fuel mass flow *m˙ <sup>f</sup>*:

$$h^\* = \underset{h}{\text{argmin}} \dot{m}\_f(P\_{d\prime\prime} \omega\_{w\prime} h) \,. \tag{7}$$

The off-line obtained optimal solutions are mapped as *h*\* (*Pd*, ω*w*).

**Figure 12.** Block diagram map-based RB+ECMS control strategy.

The HEV and PHEV control strategy determines the transmission gear ratio *h* and the engine torque τ*<sup>e</sup>* based on combining a rule-based (RB) controller and an equivalent consumption minimisation strategy (ECMS) [17,24]. The original RB+ECMS strategy is modified here to allow for computationally-efficient map-based realization, which is shown in Figure 12. A proportional-like battery SoC controller commands the battery recharging power—*Pbatt*, which is added to the transmission input power demand *Pd* to obtain the engine power demand *P*\* *<sup>e</sup>*. The demanded engine power *P*\* *<sup>e</sup>* is compared with engine on and off thresholds *Pon* and *Po*ff < *Pon*, respectively, in order to determine the engine on/off state *ENst*. The engine is exceptionally kept switched on in the case *P*\* *e* < *Po*ff when the speed-dependent M/G machine power limit is not high enough to satisfy the driver power demand *Pd* in the fully electric driving mode. If the engine is switched on, the signal *P*\* *e* is fed to the ECMS to find the optimal values of *h* and τ*e*. Otherwise, the electric driving mode is activated and the gear ratio is determined according to Equation (6).

In the original RB+ECMS strategy [17], the equivalent fuel consumption *m˙ eq*(*Pe*, *Pd*, ω*w*, *h*, *SoC*) is minimised instantaneously and on-line with respect to both control variables *h* and τ*e*. In the simplified map-based RB+ECMS version considered here, the equivalent fuel consumption is minimised in two stages. In the first stage, the ECMS is applied to discrete operating points along the constant power curve *P*\* *e* (denoted in Figure 11 by blue circles) to determine the optimal gear ratio:

$$h^\* = \arg\min\_h \dot{m}\_{t\eta}(P\_{a^\*}^\* \ P\_{d \cdot \prime} \omega\_{w \cdot \prime} h) \,, \tag{8}$$

where *m˙ eq* is the equivalent fuel consumption rate containing the actual fuel consumption rate *m˙ <sup>f</sup>* and a battery power-equivalent fuel rate (see [17,24] for details). The off-line obtained optimal solutions are stored in a three-dimensional (3D) map *h*\* (*P*\* *<sup>e</sup>*, *Pd,* ω*w*) representing the RB+1D-ECMS control map. In the second stage, the ECMS is applied along the engine torque axis (see green arrows in Figure 11):

$$
\pi\_{e,2D}^\* = \underset{\pi\_{\mathfrak{t}}}{\text{argmin}} \; \dot{m}\_{a\mathfrak{q}}(P\_{d\mathfrak{q}}, \omega\_{\mathfrak{t}}, \pi\_{\mathfrak{t}}) \; . \tag{9}
$$

The off-line optimisation results are stored in a 2D map τ*\* e,*2*D*(*Pd,* ω*e*) representing the 2D-ECMS control map. Finally, the engine torque obtained by the RB+1D-ECMS (as τ*\* e,RB* = *P*\* *<sup>e</sup>*/(*h\* io*ω*w*)) and the one obtained by the 2D-ECMS are combined/blended on-line using the SoC control error (*eSoC*)-dependent weighting factor *W*(*eSoC*) [17,24]:

$$
\pi\_{\varepsilon}^\* = \pi\_{\varepsilon, \text{RB}}^\* \mathcal{W}(\varepsilon\_{\text{SoC}}) + \pi\_{\varepsilon, \text{2D}}^\* (1 - \mathcal{W}(\varepsilon\_{\text{SoC}})) \,, \tag{10}
$$

where *W*(.) is an SoC control error-dependent weighting function illustrated in Figure 12. For small SoC control errors, the 2D-ECMS is dominant, while for large SoC control errors, the RB+1D-ECMS is preferred to satisfy the engine power demand *P\* <sup>e</sup>* (and, thus, the battery power demand *P\* batt*).

Finally, gear shift delay logic is implemented to prevent frequent gear switching [17]. The goal is to prevent gear shift occurrence in *<sup>k</sup>*th time step, i.e., rather use the gear ratio from the previous (*k*−1)th step, *hk*−1, if the time elapsed since the last gear shift *tsh* is lower than an arbitrarily set shift delay threshold *tth* and if *hk*−<sup>1</sup> gives feasible set **<sup>u</sup>***<sup>k</sup>* of engine and M/G machine operating points in the *<sup>k</sup>*th step (denoted by Π):

$$h\_k^\* = \begin{cases} h\_{k-1'}^\* & \text{for } t\_{sh} < t\_{th} \text{ and } \mathbf{u}\_k \big( P\_{e'}^\* P\_{d,k'} \boldsymbol{\omega}\_{w,k'} h\_{k-1}^\* \big) \in \Pi\\ h^\*(P\_{e,k'}^\* P\_{d,k'} \boldsymbol{\omega}\_{w,k}) & \text{otherwise} \end{cases} \tag{11}$$

The same gear shifting delay logic is applied in the CONV and BEV cases. The shift delay threshold is set here to *tth* = 2 s.

The above described simplified, map-based control strategy has been found to result in a negligible model response deviation when compared to the use of original strategy, which was proven to be close to the dynamic programming-based global optimum [17]. On the other hand, the execution time is reduced by around 200 times. The achieved execution time, expressed as the amount of microseconds needed to simulate one second of real time (for a workstation having 16 GB RAM and Intel® Xeon® Processor E5-1620 v3 @ 3.50GHz) falls in the range from 50 to 87 μs/s depending on vehicle type. This results in approximate yearly 10-bus fleet simulation time ranging from approximately 4.5 h to 7.5 h, which is deemed acceptable for such a large-scale fleet simulation. Note that the execution time could further be reduced by using parallel computing.

The PHEV can operate in two characteristic modes [20]: (i) charge depleting (CD) followed by charge sustaining (CS), where the former involves the engine only when absolutely needed and the latter correspond to hybrid operation at the target SoC of 30%; (ii) blended mode where engine is regularly used all over the driving cycle for additional energy savings. For the sake of simplicity, the CD/CS mode is considered in this paper.

#### *4.4. Simulation Results*

The results related to relative fuel and/or electricity consumptions for different city bus types are given in Table 3 for the full recording period. The relative difference between the simulated (Sim) and recorded (Rec) fuel consumptions for the CONV bus is equal to only 1.4%. Therefore, the CONV simulation model used as a basis for e-bus modelling can be considered accurate. Note that although the real and simulated buses are different (MAN Lion City and Volvo 7900), the validation is considered fair, as the two buses are rather comparable in terms of size, mass, engine power, number of passengers and other similar factors.

The simulated electricity consumptions of PHEV- and BEV-type buses are close to recorded ones documented in the ZeEUS project report [25] for Volvo 7900 bus series (Table 3). In the PHEV case, the simulated fuel consumption is by 30% higher than the ZeEUS recorded one, but this discrepancy is compensated for by 26% higher recorded electricity consumption when compared to the simulated one. In the BEV case, the relative difference in electricity consumption equals 6%. The simulated HEV fuel consumption is reduced by 50% when compared to CONV simulation results, while the manufacturer states the fuel consumption reduction from 39% to 45% reported by operators [19]. The observed, relatively modest discrepancies in fuel/electricity consumption may be related to difference in considered bus weights (passenger weight is fixed to 1250 kg), road slope and traffic congestion conditions, as well as regenerative braking capacity (set to the maximum amount of 100% in simulation).


**Table 3.** Recorded and simulated relative fuel and electricity consumptions for different bus types and full recording period.

\* Recorded PHEV and BEV fuel and electricity consumptions are taken from the ZeEUS project report [25]. \*\* Estimated based on information on fuel consumption reduction from 39% to 45% for Volvo 7900 Hybrid vs. Volvo 7900 according to [19].

Figure 10 shows the PHEV-case simulation results for the recorded driving cycle shown in Figure 4 repeated 15 times and the initial battery SoC equal to 90%. When the CD mode is active, the engine is used only when needed and the cumulative fuel consumption is often constant (a stepwise-like response, Figure 10a). After entering the CS mode, the engine is more active to sustain the battery SoC (Figure 10b). The control strategy deploys the operating points of engine (when switched on) and M/G machine in the high efficiency areas of corresponding maps (Figure 10c–d), thus minimising the energy consumption.
