# Optimal Control of a PHEV Based on Backward-Looking Model Extended with Powertrain Transient Effects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Powertrain Models

#### 2.1. Powertrain Configuration

_{1}and s

_{2}) is placed between the input shaft, which rotates at the speed ω

_{MG}, and a counter-shaft that rotates at the speed ω

_{cs}and transmits the torque τ

_{ss}when engaged by synchronizers. The main gear reduction stage (gears m

_{1}, m

_{2}, and m

_{3}) is placed between the main shaft rotating at the speed ω

_{ms}and the counter-shaft, and it contains two dog clutches to achieve different gears. Note that if the gears s

_{2}and m

_{3}are engaged at the same time, the input and main shaft are directly connected (see Table 1 and Figure 2). Changing the main reduction gear requires the use of M/G machine to synchronize the speeds of target m-gear and main shaft. Note that the main gear reduction stage also comprises the reverse gear m

_{0}(not considered in this paper). Finally, the range gear reduction stage (gears r

_{1}and r

_{2}) comprises a planetary gear set in which the ring gear can be synchronized via torque τ

_{sr}of synchronizer r to the casing (gear r

_{1}) or the carrier connected to the output shaft rotating at the speed ω

_{os}(gear r

_{2}). The gears are changed by controlling the positions of synchronizers and dog clutches through pneumatic actuators. Synchronizer and dog clutch normalized position values are denoted as s

_{ps}and s

_{pr}for synchronizers s and r, respectively, and s

_{pm}for dog clutches. The list of 12 forward-gear indexes h

_{idx}, the corresponding gear ratios h and normalized synchronizers positions are given in Table 1.

#### 2.2. Backward-Looking Powertrain Model

_{v}(t), the required wheel torque τ

_{w}(t) is determined from the vehicle longitudinal dynamics equation as follows:

_{w}is the effective tire radius, M

_{v}is the vehicle mass, R

_{0}is the rolling resistance coefficient, g is the gravitational acceleration constant, δ

_{r}is the road grade, ρ

_{air}is the air density, A

_{f}is the vehicle frontal area, and C

_{d}is the aerodynamical drag coefficient. For the case of closed main clutch, the speeds of engine (ω

_{e}), M/G machine (ω

_{MG}), wheels (ω

_{w}), and vehicle (v

_{v}) are connected through the following equation (Figure 1):

_{o}is the final drive ratio and h is the transmission gear ratio. Similarly, in case of locked main clutch the transmission input shaft torque, i.e., the sum of M/G machine and engine torques, τ

_{MG}and τ

_{e}, respectively, is determined from the total wheel torque τ

_{w}while accounting for the transmission ratio and mechanical power loss:

_{t}equals 1 for τ

_{w}< 0 (regenerative braking), and k

_{t}= −1 holds for τ

_{w}> 0 (traction), η

_{tr}is the torque-dependent transmission efficiency, and P

_{0}is the transmission speed-dependent power loss (see [23] for details). Note that in case of pure electric driving, ω

_{e}= 0 rad/s and τ

_{e}= 0 Nm hold. The model parameters are given in Appendix A.

_{MG}(τ

_{MG}, ω

_{MG}) and the engine specific fuel consumption map A

_{ek}(τ

_{e}, ω

_{e}) are shown in Figure 3, along with the corresponding maximum torque curves. These maps are adopted from the respective maps published in the literature for similar engine and M/G machine and are scaled with respect to maximum speed and power ratios of the respective vehicles and the particular PHEV-type bus considered in this paper [20]. The map A

_{ek}(τ

_{e}, ω

_{e}) is used to calculate the fuel mass flow

_{fuel}is the diesel fuel density.

_{max}, where Q and Q

_{max}are actual and maximum battery charge, respectively. The SoC dynamics description is based on the equivalent battery circuit model illustrated in Figure 4a [6]:

_{oc}(SoC) and the internal resistance R(SoC) characteristics are given in Figure 4b [23]. The battery output power P

_{batt}is expressed as follows:

_{b}= 1 holds for regenerative braking (${\tau}_{MG}{\omega}_{MG}$< 0) and k

_{b}= −1 is valid for traction (${\tau}_{MG}{\omega}_{MG}$ ≥ 0). The BWD model given by (1)–(7) is discretized and implemented in the Matlab/Simulink environment with the sampling time T

_{d}= 1 s as a good trade-off between the computational efficiency and ability to capture the longitudinal dynamic transients [6].

#### 2.3. Forward-Looking Powertrain Model

_{e}, the lumped M/G machine and input shaft inertia I

_{MG}

_{1}, the counter-shaft inertia I

_{cs}, the main shaft inertia I

_{ms}, the output shaft inertia I

_{os}, the four wheel inertia I

_{w}

_{1}= I

_{w}

_{2}= I

_{w}

_{3}= I

_{w}

_{4}= I

_{w}, and the vehicle mass M

_{v}. The main model parameters can be found in [10]. The Amesim-embedded variable integration step solver is used in simulation [24].

_{MG}= 10 ms, whereas the turbocharged Diesel engine torque dynamics are modeled by a first-order lag term with a speed dependent time constant T

_{e}(ω

_{e}) in the case of torque increase (see Figure 7b), and the fixed time constant of 10 ms in the case of torque decrease. The engine drag torque characteristic τ

_{e,drag}(ω

_{e}) given in Figure 7a is applied when the engine is switched off.

_{mcl}is modelled by the classical Coulomb-type friction model, where the friction torque magnitude is proportional to the normalized clutch torque capacity c

_{mcl}(ranging from 0 to 1 for open and lock clutch states, respectively). For the sake of model implementation simplicity, two dog clutch models are replaced by a synchronizer model for engaging gears m

_{1}and m

_{2,}and a half-synchronizer model for engaging gear m

_{3}. During the synchronization, the synchronizer torques τ

_{ss}, τ

_{sm}, and τ

_{sr}are determined by a dynamic Coulomb-type friction model, where the torque magnitude is proportional to the normalized torque capacity (c

_{ss}, c

_{sm}and c

_{sr}) dependent on the normalized synchronizer positions (s

_{ps}, s

_{pm}and s

_{pr}), and where the stiction torque is modeled by a parallel spring-damper element [10].

_{ps}, s

_{pm}, and s

_{pr}assume values depending on the gear ratio h, as designated in Table 1. To account for the synchronizer pneumatic actuators dynamics, the synchronizer position dynamics are also modelled by the first-order lag term with the time constant of 20 ms.

_{brk}is modeled by the first-order lag term with the time constant of 10 ms to account for the pneumatic actuator dynamics. Equal braking torque distribution on all four wheels is assumed.

## 3. Control Strategy

#### 3.1. Structure of Overall Control Strategy

_{wd}for the high-level control strategy. The driver model parameters K

_{Dr}and T

_{Dr}are determined based on the damping optimum method for the target damping ratio ζ = 0.45 and the equivalent time constant T

_{eq}= 0.75 s [26]. The driver wheel torque demand τ

_{wd}is saturated with respect to maximum wheel torque characteristic determined by the engine and M/G machine maximum torque curves given in Figure 3 and the drivetrain gear ratios h and i

_{0}. Depending on the current values of wheel speed ω

_{w}and battery SoC, the high-level control strategy transforms the wheel torque demand τ

_{wd}to the low-level control strategy references, i.e., the transmission gear ratio h

_{R}and the engine torque reference τ

_{eR}, as well as the target engine on/off status flag EN

_{stR}. The low-level control strategy is fed by the driver wheel torque demand τ

_{wd}and the engine torque reference τ

_{eR}, the current wheel, engine and M/G machine speeds (ω

_{w}, ω

_{e}, and ω

_{MG}), and the current gear ratio h

_{R}, and it outputs the main clutch torque capacity reference c

_{mclR}, the synchronizer normalized position references (s

_{psR}, s

_{pmR}and s

_{prR}), the M/G machine torque reference τ

_{MGR}, the mechanical brake torque reference τ

_{brkR}, and the engine torque reference τ

^{*}

_{eR}that may differ from high-level controller-commanded reference τ

_{eR}in the case of transients (Section 3.3). The braking torque reference τ

_{brkR}is determined by the low-level controller as the excess of driver brake torque with respect to M/G machine regenerative braking torque limit.

#### 3.2. High-Level Control

_{d}including the mechanical loss described within Equation (3). The SoC controller sets the battery power demand P

^{*}

_{batt}which is added to the propulsion power demand P

_{d}to obtain the engine power demand P

^{*}

_{e}. The engine start/stop logic requests the engine to be switched on (EN

_{stR}= 1) when the engine power demand P

^{*}

_{e}is greater than the engine-on power threshold P

_{on}(P

^{*}

_{e}> P

_{on}), and to be switched off (EN

_{stR}= 0) if P

^{*}

_{e}is lower than engine-off power threshold P

_{off}(P

^{*}

_{e}≤ P

_{off}< P

_{on}). Exceptionally, the engine will be kept switched on regardless of P

^{*}

_{e}if the M/G machine itself cannot deliver the power demand P

_{d}due to its speed-dependent torque [5,23].

_{eR}and h

_{R}to minimize the equivalent fuel consumption ${\dot{m}}_{eq}$ defined by [23]:

_{battc}and η

_{battd}denote the battery charging and discharging efficiencies, respectively, while ${\overline{A}}_{ek}$ is the mean engine specific fuel consumption accounting for engine efficiency during past battery charging periods [5,23]. The subscript k in Equation (8) stands for the discrete time step of control strategy execution, where the sampling time is equated with the BWD model sampling time T

_{d}= 1 s (see Section 2). When calculating the quantities ${\dot{m}}_{f}$ and ${\dot{m}}_{batt}$ in Equation (8), the ECMS relies on the computationally efficient BWD model given by Equations (1)–(7).

_{eR}[23]. Namely, the lower and upper engine torque reference limits vary between the absolute lower limit P

_{off}/ω

_{e}and the absolute upper limit τ

_{e,}

_{max}(ω

_{e}) depending on the SoC control error e

_{SoC}= SoC

_{R}− SoC, i.e., a smooth weighting function w(e

_{SoC}). For e

_{SoC}= 0, the limits are wide open, i.e., they correspond to the absolute limits. As e

_{SoC}increases, the limits narrow and eventually converge to the operating point set by the RB controller τ

_{e}= P

^{*}

_{e}/ω

_{e}at high values of e

_{SoC}. In this way, the ECMS provides a 1D control variable search over the hyperbolic, constant power curve P

^{*}

_{e}= const. in the (ω

_{e}, τ

_{e}) plane if the SoC control error e

_{SoC}is high, in order to respect the power demand P

^{*}

_{e}and suppress the SoC control error. On the other hand, if the control error e

_{SoC}is low, the ECMS is allowed to give up from the power demand P

^{*}

_{e}and provides a 2D control variable search for reduced fuel consumption.

_{stR}= 0, τ

_{eR}= 0), the M/G machine alone propels the vehicle and the transmission gear ratio is selected to minimize the total electric energy losses [23]:

_{R,k}remains the same as the current gear h

_{k-}

_{1}and the time elapsed since the last gear shift, t

_{sh}, is shorter than the time threshold t

_{th}. The discount factor r

_{f}varies from r

_{0}set to 0.6 and the nominal value of 1. The time threshold t

_{th}is selected as a trade-off of powertrain efficiency and shift comfort (i.e., drivability). The discount factor r

_{f}given by Equation (11) is also applied to the cost function given in Equation (9) related to the pure electric operating mode.

#### 3.3. Low-Level Control

_{eR}including the engine-on status reference EN

_{stR}, the wheel torque demand τ

_{wd}, and the gear ratio target h

_{R}set by the high-level control strategy (Figure 8). This is achieved by coordinating the main clutch torque capacity reference c

_{mclR}, the synchronizers s, m and r normalized position references s

_{psR}, s

_{pmR}and s

_{prR}, respectively, the modified engine torque reference τ

_{eR}, and the M/G machine torque reference τ

_{MGR}. The low-level control operation is illustrated below for the following three characteristic powertrain transient modes: (i) engine-on switching, (ii) gear shifting while engine is switched on, and (iii) brake control. The low-level control strategy is implemented in C programing language within the Amesim model, with the sampling time set to 20 ms to capture the fast powertrain dynamics (e.g., those related to engine torque development).

#### 3.3.1. Generation of M/G Machine and Mechanical Brake Torque References

_{eR}to zero; otherwise, i.e., if the clutch is locked, τ

_{eR}= τ

_{eR}holds. The M/G machine torque reference τ

_{MGR}is determined as the difference between the wheel torque demand (τ

_{wd}) referred to transmission input shaft and the modified engine torque reference τ

_{eR}:

_{MGR}is saturated during regenerative braking to its limit curve τ

_{MG}

_{,min}(ω

_{MG}), the total (four-wheel) braking torque reference τ

_{brkR}is set to fill the gap between the M/G machine reference and limit values, i.e., it is determined as the following:

_{eR}is reset to zero during braking intervals.

#### 3.3.2. Low-Level Control for Engine-On Transient Mode

_{mclR}which is determined assuming linearly falling clutch slip speed profile and prescribing the engagement time to Δt

_{mcl}= 0.27s while considering the actuator dynamics with the time constant T

_{mcl}(see Appendix B):

_{mcl}

_{,max}is the maximum clutch torque capacity and ω

_{mcl,start}is the initial clutch slip speed. The engine torque reference τ

_{eR}is reset to zero and the M/G machine reference τ

_{MGR}is set to deliver the driver-commanded wheel torque τ

_{wd}(see Equation (12)).

_{mcl}falls below a zero-speed threshold. In this phase, the engine torque reference τ

^{*}

_{eR}is linearly increased from zero to the reference τ

_{eR}set by the high-level control strategy within the period of 0.1 s.

_{mclR}is linearly increased from the value determined by Equation (14) to the fully closed torque capacity value c

_{mclR}= 1. Once the clutch is fully locked, i.e., when c

_{mcl}= 1 is achieved, the transient mode is completed (see Figure 10a).

#### 3.3.3. Low-Level Control during Gear Shifting

_{R}. In the engine-on case, the first gearshift phase (Phase 1 in Figure 11) starts with opening the main clutch by setting the clutch torque capacity and engine torque references to zero: c

_{mclR}= 0 and τ

_{eR}= 0.

_{ps}is commanded to change and synchronize the input shaft speed ω

_{MG}with the counter shaft speed ω

_{cs}and engaging the s-gear (s

_{1}, Table 1). At the same time, the dog clutch m is commanded to be fully disengaged by setting its reference position s

_{pmR}to zero. Phase 3 corresponds to synchronization of new m-gear (m

_{2}, Table 1) with the main shaft speed ω

_{ms}. The synchronization is performed by using a proportional-integral (PI) controller of the M/G machine speed ω

_{MG}, with the reference ω

_{MGR}set to reflect the synchronization speed ω

_{ms}. The PI speed controller is tuned according to the damping optimum method [26], and its gains are scheduled to reflect the change of equivalent inertia when changing the gears. Note that Phase 3 is omitted for shifts that do not involve the change in m-gear.

_{m}

_{2}approaches the speed ω

_{ms}(Figure 10c). In this phase, the position reference s

_{pmR}of synchronizer m is finally set to the value corresponding to gear m

_{2}. Once the synchronizer position reference is reached (i.e., when s

_{pm}approaches s

_{pmR}), the target gear ratio h

_{R}is set, and Phase 4 ends. For the sake of better visibility of the overall response in Figure 10, the response of fifth phase (Phase 5) covers only the initial interval of main clutch closing corresponding to Phase 1 in Figure 10. The remaining part of response is omitted as it is presented and discussed with Figure 10 as Phases 2 and 3. Note that for shifts where engine-off transition was commanded (i.e., if transition to electric mode occurred, EN

_{stR}= 0) the main clutch would stay open, i.e., Phase 5 would be omitted. Similarly, in gear shifts occurring during pure electric operation, Phase 1 is omitted (see Figure 12).

#### 3.3.4. Low-Level Control during Braking Event

_{brkR}if the braking torque demand τ

_{wd}< 0 exceeds the M/G machine torque limit τ

_{MG}

_{,min}(ω

_{MG}) (Section 3.3.1). Low-level brake control is illustrated in Figure 12, where 12–10 downshift is commanded by the high-level control strategy during an interval of vehicle deceleration and pure electric operation (EN

_{st}= 0). Before the downshift was commanded, the M/G machine regenerative braking torque τ

_{MG}could fully meet the driver brake demand τ

_{wd}< 0 (see the initial period of response in Figure 12b), and the braking torque reference was set to zero (τ

_{brkR}= 0 Nm).

_{pmR}= 0. Once the dog clutch 1 is fully opened, the downshift transfers to Phase 2, where the new m-gear (m

_{2}, Table 1) is to be synchronized. Since the M/G machine is disconnected in this phase by the open dog clutch, the driver-demanded wheel braking torque τ

_{wd}< 0 is briefly not met. This results in increase of the vehicle velocity tracking error and consequent increase of absolute value of braking torque demand τ

_{wd}. Once the new m-gear speed ω

_{m}

_{2}is synchronized with main shaft speed ω

_{ms}by means of closed-loop M/G machine control, Phase 3 starts, in which the dog clutch assumes its positions s

_{pm}= 2 and the gear shift is completed. Although the mechanical brakes could have been used during Phases 1–3, the braking torque reference τ

_{brkR}is deliberately set to zero (τ

_{brkR}= 0 Nm) to avoid energy dissipation on mechanical brakes and maximally utilize regenerative braking after the shift is completed (Phase 4 in Figure 12). Since the regenerative M/G machine torque τ

_{MG}< 0 becomes eventually saturated, the mechanical brake torque reference τ

_{brkR}is determined according to Equation (13) to satisfy the braking torque demand τ

_{wd}< 0.

## 4. Extended Backward-Looking Powertrain Model

#### 4.1. Model Structure Overview

^{*}

_{eR}is reduced to zero during the powertrain transients that involve manipulation of main clutch or dog clutches. During those intervals, the M/G machine predominantly delivers the traction power, and at the same time it covers additional transient-related losses such as the main clutch and synchronizers slippage losses, engine starting mechanical loss, and additional electrical loss related to M/G machine-based synchronization action for dog clutches.

_{e,k}, and (ii) transient power losses P

_{dyn,loss}. To calculate the torque and power losses, it turns out that it is necessary to know the engine on/off status in the previous sampling time (EN

_{st,k}

_{-1}), the previous gear ratio (h

_{k}

_{-1}), and the previous wheel/vehicle speed (ω

_{w,k}

_{-1}). To this end, the EXT-BWD model includes a one-step memory block 1/z for each of those three variables (Figure 13), as additional dynamic blocks to the battery model SoC integrator. The calculated engine torque cut Δτ

_{e}is simply subtracted from the engine torque reference τ

_{eR}to obtain the engine torque τ

_{e}(Figure 13).

_{dyn,loss}is divided by the M/G machine speed ω

_{MG}to obtain the corresponding torque loss, which is also added to the M/G machine torque reference to cover both static and dynamic torque losses (Figure 13). This calculation process is formally justified by the following M/G machine torque equation obtained by modifying the transmission input shaft torque balance Equation (3):

#### 4.2. Engine Torque Loss

_{e,k}from Figure 13 is defined as

_{c}≤ 1 is the engine torque reduction coefficient which is represented by a map (actually a table due to discrete amplitude inputs, see Appendix C) parameterized based on the results of exhaustive FWD model simulations. To extract the values of coefficient r

_{c}, the engine torque reference commanded by the high-level control (τ

_{eR}) is related to the delivered engine torque τ

_{e}obtained from the forward model and averaged over the high-level controller sampling period T

_{d}= 1 s (τ

_{e,avg}) as illustrated in Figure 14:

_{c,k}calculated according to Equation (17) for different combinations of r

_{c}-map inputs, which describe five distinct powertrain transient modes discussed in Appendix C, are stored in the engine torque reduction coefficient map r

_{c}(EN

_{st,k}, EN

_{st,k}

_{−1}, h

_{k}, h

_{k}

_{−1}). In the special case of no engine status and shift transient, the coefficient r

_{c}is set to zero, i.e., τ

_{e}= τ

_{eR}applies as in the case of BWD model. If the engine is switched off, the coefficient r

_{c}is set to 1, i.e., τ

_{e}= 0 holds.

#### 4.3. Powertrain Transient Power Loss

_{dyn,loss}from Figure 13 is determined in each sampling instant t

_{k}= kT

_{d}of the high-level control strategy (T

_{d}= 1 s) from the following energy loss contributions: (i) main clutch and synchronizer slippage losses E

_{mcl,loss}and E

_{sync,loss}, respectively, (ii) engine-on switching energy loss E

_{e,ON,loss}, and (iii) M/G machine-based synchronization loss E

_{MG,sync}:

#### 4.3.1. Main Clutch Slippage Energy Loss

_{eR*}= 0 Nm; Figure 10) and the M/G machine speed is nearly constant (${\dot{\omega}}_{MG}=0$), the clutch slip speed dynamics can be described as:

_{e}is the engine inertia. Integrating Equation (19) while accounting for the initial condition ω

_{mcl}(k) = ω

_{mcl}

_{,start}for the particular (k

^{th}) sampling interval yields

_{mcl}is the target clutch engagement time (see Equation (14)). Based on the assumption that ${\dot{\omega}}_{MG}=0$, the initial clutch slip speed may be expressed as

_{mclR}, taking into account the actuator dynamics, the main clutch torque τ

_{mcl}during the engagement period can be expressed as

_{mcl}given by

_{mcl}, b

_{mcl}and c

_{mcl}are included in Appendix A.

#### 4.3.2. Synchronizer Slippage Energy Loss

_{MG}

_{1}is the lumped M/G machine and input shaft inertia. Note that the synchronizer s slip speed ω

_{ss,k}in k

^{th}step (i.e., after the transient) equals to zero (ω

_{ss,k}= ω

_{MG,k}– ω

_{cs,k}= 0 rad/s), and by assuming near constant counter shaft speed ω

_{cs}(${\dot{\omega}}_{cs}=0$, ω

_{cs,k =}ω

_{cs,k}

_{−1}) the slip speed at the start of the transient ω

_{ss,k}

_{−1}equals to difference of M/G machine speed prior and after the transient (ω

_{ss,k}

_{−1}= ω

_{MG, k}

_{−1}− ω

_{MG, k}, see Equations (26) and (2)). Similarly, assuming the constant output shaft speed during the r-gear synchronization (${\dot{\omega}}_{os}=0$), negligible influence of synchronizer actuator dynamics, and inactive s- and m-gears, the r-gear synchronization loss is given by

_{MG}

_{2}is the lumped M/G machine, input, counter, and main shaft inertia, h

_{r}is the h-gear speed ratio of the engaged r-gear, and ω

_{sr,k}

_{−1}is the synchronizer r slip speed before the gear shift transient, where, similarly to the case of synchronizer s, the slip speed ω

_{sr,k}

_{−1}equals to the difference of main shaft prior and after the transient (ω

_{sr,k}

_{−1}= ω

_{ms,k}

_{−1}− ω

_{ms,k}).

#### 4.3.3. M/G Machine-Based Synchronization Energy Loss

_{MG}

_{3}is the lumped M/G machine, input and counter-shaft inertia.

#### 4.3.4. Engine-On Energy Losses While Switching On

_{e,ON,loss}includes the loss related to change in engine kinetic energy E

_{e,kin,loss}and the engine drag-related loss E

_{drag,loss}:

_{e,idle}is the engine idle speed, τ

_{e,drag}is the engine drag torque (see Figure 7a), and Δt

_{idle}is the average time of reaching the idle speed, which is calculated from a rich set of FWD model responses. The integral in Equation (29) is numerically and off-line solved based on the assumption of constant engine acceleration (equal to ω

_{e,idle}/Δt

_{idle}, Section 3), and the resulting constant engine-on energy loss is applied on-line, i.e., when evaluating the model.

#### 4.3.5. Inertial Load of Powertrain Components

_{v}within the longitudinal dynamics Equation (1).

#### 4.3.6. Use of EXT-BWD Model within ECMS+RB Control Strategy

## 5. Dynamic Programming-Based Control Variable Optimization and EXT-BWD Model Validation

#### 5.1. Optimal Problem Formulation

_{k}= SoC

_{k}(see Section 2). In the case of EXT-BWD model, the state variable SoC

_{k}is supplemented by the engine on/off status and the gear ratio variables in the previous, (k−1)

^{st}step, which are designated as EN

_{st,prev,k}and h

_{prev},

_{k}, respectively (see Figure 13 and note that the third signal for which the memory block z

^{−1}is applied therein is not a state, but rather an external input ω

_{w}, which is known in advance):

_{eR,k}and the transmission gear ratio h

_{k}are set as elements of the control input vector

_{w}

_{,k}and the current wheel speed ω

_{w,k}, as well as the previous wheel speed ω

_{w,k}

_{-1}in the case of EXT-BWD model, are combined into the external input vector

_{d}= 1 s, as well as the following one-step delay state equations for the additional states EN

_{st,prev,k}and h

_{prev},

_{k}:

_{EN,st}(u) is the step-type activation function, which equals 1 if u > 0, while it is set to 0 if u ≤ 0. The overall discrete-time state-space system is described in the following vector form:

^{−}(x) is defined as H

^{−}(x) = 1 for x < 0, and H

^{−}(x) = 0 otherwise. The penalization factor K

_{g}is set to a sufficiently high value (10

^{12}, herein) to ensure that constraints are satisfied.

_{f}) is equal to the final condition SoC

_{f}, where the weighting factor is set to K

_{f}= 10

^{6}.

_{k}

_{−1}and EN

_{st,k}

_{−1}, respectively, are inherently discrete variables, and have only 2 and 12 discrete levels, respectively. The gear ratio control input ${h}_{k}$ has 12 discrete levels, while the battery SoC state SoC

_{k}and the engine torque control input ${\tau}_{e,Rk}$ are originally continuous variables and are discretized into 200 levels each.

#### 5.2. Optimization Results

_{i}= SoC

_{f}= 30%, and three heavy-duty certification driving cycles (HDUDS, WHVC and JE05) and a recorded city bus driving cycle with the road grade set to zero (denoted as DUB; [30]). The optimization results obtained for both BWD and EXT-BWD models are given in Table 2. They include the total fuel consumption V

_{f}and the related final SoC value SoC(t

_{f}) as well as the number of gear shifts N

_{g}and engine-on switching N

_{e}. In the case of realistic DUB driving cycle, the fuel consumption predicted by EXT-BWD model is 9% higher than that predicted by BWD model. This indicates that the BWD model is largely optimistic in predicting the fuel consumption because of neglected transient losses. Furthermore, the number of gear shifts N

_{g}and engine-on events N

_{e}is approximately halved in the case of EXT-BWD model when compared to BWD model. This is because the transient events are discouraged when accounting for the transient losses within the EXT-BWD model. In the case of artificial/certification driving cycles, the fuel consumption predicted by the EXT-BWD model is around 5% higher than that of the BWD model, while the level of drivability improvement is comparable to that of DUB cycle (around 50% less shifting/switching events).

#### 5.3. Validation of EXT-BWD Model

**u**

_{EXT-BWD}and

**u**

_{BWD}, are fed to the original, more accurate FWD model in an open-loop manner. The fuel consumption and SoC trajectories predicted by the EXT-BWD, BWD, and FWD models, and given in travelled distance x-axis, are shown in Figure 15 for the four driving cycles considered in Table 2. These results show that, when applying the input

**u**

_{EXT-BWD}, the EXT-BWD and FWD models predict very similar SoC and fuel consumption trajectories where the final values match within the error margin of 1.5%. On the other hand, when applying the input

**u**

_{BWD}to BWD and FWD model, the fuel consumption and SoC trajectories of the two models deviate substantially, especially for more dynamic (and realistic) DUB driving cycles. The BWD model-predicted final fuel consumption and SoC are underestimated to a large extent (at least −10% offset for SoC(t

_{f}) and 30% reduced V

_{f}). This is, again, due to neglected transient losses in the case of BWD model.

## 6. Control System Simulation Results

_{f}) in the case of control system, the DP optimizations have been conducted for a number of final SoC conditions SoC

_{f}around the target of 30%. The respective optimal fuel consumption values V

^{*}

_{f}are linearly interpolated with respect to final SoC conditions SoC

_{f}, and as such they are used to calculate the indicators’ relative differences.

_{g}for all driving cycles. At the same time, the fuel consumption is mostly improved, but only marginally. On the other hand, RB+ECMS-EXT considerably reduces the fuel consumption relative excess with respect to DP benchmark (see the percentage values in V

_{f}column). The reduction is most significant in the case of realistic DUB driving cycle where the fuel consumption excess is reduced from 8.7% to 5.2%. RB+ECMS-EXT has comparable number of gear shifts N

_{g}to that of RB+ECMS with GSD algorithm included. This is achieved through physical description of transient losses rather than using a heuristic GSD algorithm. The number of engine-on switching N

_{e}is comparable for all control strategies considered.

_{f}is corrected with respect to deviation of final SoC from its target value SoC

_{R}= 0.3:

_{f}is obtained by forming a linear regression of multiple SoC(t

_{f}) vs. V

_{f}pairs obtained by a series of simulations with different values SoC

_{R}[5]. The results in Table 4 point out that RB+ECMS-EXT outperforms RB+ECMS in terms of fuel consumption, and the extent of improvement is similar as in the case (EXT)-BWD model simulations (cf. Table 3). In the most realistic case of DUB driving cycle, the fuel consumption reduction is 2.2%, which is achieved by computationally non-demanding extension of RB+ECMS.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Powertrain and Control Strategy Parameters

_{v}= 12.635 kg, R

_{0}= 0.012, i

_{o}= 4.72, r

_{w}= 0.481 m, C

_{d}= 0.7, ρ

_{air}= 1.225 g/m

^{3}, A

_{f}= 7.52 m

^{2}, Q

_{max}= 30 Ah.

_{on}= 80 kW, P

_{off}= 30 kW, ${\overline{A}}_{ek}$= 195 g/kWh, K

_{SoC}= 736,000 W, Δ

_{SoC}= 0.02.

## Appendix B. Generating Main Clutch Normalized Torque Capacity Reference

_{mcl}and considering the boundary conditions ω

_{mcl}(Δt

_{mcl}) = 0 rad/s and ω

_{mcl}(0) = ω

_{mcl,start}, the following equality applies:

## Appendix C. Engine Torque REDUCTION Coefficient

_{c}is identified for five distinct powertrain transient modes, which include: (i) engine-on transient with no gear shifting, (ii) and (iii) engine-on and gear shift transient with and without m-gear change, respectively, and (iv) and (v) gear shift transient only with and without m-gear change, respectively. The transient mode is determined based on the gear ratio h and the engine on/off status EN

_{st}in the current k

^{th}and previous (k − 1)

^{st}steps.

_{c}for each transient mode, which is then stored in the map r

_{c}(EN

_{st,k}, EN

_{st,k}

_{−1}, h

_{k}, h

_{k}

_{−1}) depending on the transient mode input obtained for the four map inputs.

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**Figure 4.**Equivalent circuit model (

**a**) and SoC-dependent open-circuit voltage U

_{oc}and internal resistance R (

**b**) for LiFePO4 battery, reprinted with permission from [23].

**Figure 7.**Engine drag torque characteristic (

**a**) and turbocharged Diesel engine torque development time constant (

**b**).

**Figure 10.**Low-level control system response during engine-on switching mode: engine status flag and main clutch normalized torque capacity (

**a**), main clutch slip speed and engine and M/G machine speeds (

**b**); engine and M/G machine reference and actual torques (

**c**). Phases: (1) main clutch engagement, (2) engine torque buildup, (3) main clutch locking.

**Figure 11.**Low-level control system response during engine-on gear shifting mode (8–9 upshift): gear index and main clutch normalized torque capacity reference and actual responses (

**a**), main clutch slip speed and main shaft and m

_{2}gear speeds (

**b**), s-synchronizer and m-dog clutch reference and actual positions (

**c**), engine and M/G machine reference and actual torques (

**d**). Phases: (1) switching off engine and main clutch opening, (2) dog clutch opening and s-gear synchronization, (3) m-gear synchronization, (4) m-gear engagement, (5) M/G machine torque buildup.

**Figure 12.**Low-level control system response during engine-off downshift transient including use of mechanical brakes (12-10 downshift): gear index and dog clutch position reference and actual responses (

**a**), total wheel demand torque and reference and actual brake torque (

**b**), main shaft, m

_{2}gear, and M/G machine speeds (

**c**), M/G machine torque with corresponding limits (

**d**). Phases: (1) dog clutch opening, (2) m-gear synchronization, (3) m-gear engagement, (4) M/G machine torque buildup and mechanical brake activation.

**Figure 14.**Illustration of identification of engine torque reduction coefficient r

_{c}based on FWD model responses.

**Figure 15.**SoC and fuel consumption trajectories given over travelled distance and predicted by BWD, EXT-BWD, and FWD models fed by DP-optimized control variables of BWD and EXT-BWD models for DUB (

**a**), HDUDDS (

**b**), WHVC (

**c**) and JE05 (

**d**) driving cycles.

Gear h_{idx} [-] | Ratio h [-] | s_{1} | s_{2} | m_{1} | m_{2} | m_{3} | r_{1} | r_{2} | s_{ps} | s_{pm} | s_{pr} |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 14.94 | x | x | x | 1 | 1 | 1 | ||||

2 | 11.73 | x | x | x | 2 | 1 | 1 | ||||

3 | 9.04 | x | x | x | 1 | 2 | 1 | ||||

4 | 7.09 | x | x | x | 2 | 2 | 1 | ||||

5 | 5.54 | x | x | x | 1 | 3 | 1 | ||||

6 | 4.35 | x | x | x | 2 | 3 | 1 | ||||

7 | 3.44 | x | x | x | 1 | 1 | 2 | ||||

8 | 2.70 | x | x | x | 2 | 1 | 2 | ||||

9 | 2.08 | x | x | x | 1 | 2 | 2 | ||||

10 | 1.63 | x | x | x | 2 | 2 | 2 | ||||

11 | 1.27 | x | x | x | 1 | 3 | 2 | ||||

12 | 1.00 | x | x | x | 2 | 3 | 2 |

**Table 2.**DP-based control variable optimization results for different driving cycles and two types of BWD model.

Cycle | Model | V_{f} [L] | SoC(t_{f}) [%] | N_{e} [-] | N_{g} [-] |
---|---|---|---|---|---|

DUB | BWD | 1.80(+0.0%) | 29.75(+0.0%) | 64(+0.0%) | 967(+0.0%) |

EXT-BWD | 1.96 (+8.9%) | 29.76 (+0.0%) | 28 (−56.3%) | 555 (−42,6%) | |

HDUDDS | BWD | 2.02(+0.0%) | 29.74(+0.0%) | 25(+0.0%) | 168(+0.0%) |

EXT-BWD | 2.12 (+5.0%) | 29.89 (+0.5%) | 13 (−48.0%) | 92 (−45.0%) | |

WHVC | BWD | 4.22(+0.0%) | 29.85(+0.0%) | 51(+0.0%) | 365(+0.0%) |

EXT-BWD | 4.40 (+4.3%) | 29.69 (−0.5%) | 26 (−49.0%) | 190 (−47.9%) | |

JE05 | BWD | 2.54(+0.0%) | 29.80(+0.0%) | 54(+0.0%) | 466(+0.0%) |

EXT-BWD | 2.66 (+4.7%) | 29.61 (−0.6%) | 22 (−59.3%) | 224 (−51.9%) |

**Table 3.**Comparative simulation results for control strategies based on BWD and EXT-BWD models (denoted as RB+ECMS and RB+ECMS-EXT, respectively) when applied to EXT-BWD model.

Strategy | V_{f} [L] | SoC(t_{f}) [%] | N_{e} [-] | N_{g} [-] |
---|---|---|---|---|

DUB driving cycle | ||||

RB+ECMS, w/o GSD
| 2.11 (+9.0%) | 28.91 | 48 | 696 (0.0%) |

RB+ECMS, w/GSD
| 2.10 (+8.7%) | 28.93 | 48 | 383 (−45.0%) |

RB+ECMS-EXT | 2.04 (+5.2%) | 28.95 | 45 | 371 (−46.7%) |

HDUDDS driving cycle | ||||

RB+ECMS, w/o GSD
| 2.33 (+3.2%) | 32.99 | 11 | 167 (0.0%) |

RB+ECMS, w/GSD
| 2.34 (+3.4%) | 33.14 | 11 | 88 (−47.3%) |

RB+ECMS-EXT | 2.31 (+2.1%) | 33.12 | 12 | 79 (−52.7%) |

WHVC driving cycle | ||||

RB+ECMS, w/o GSD
| 4.64 (+3.8%) | 31.35 | 21 | 327 (0.0%) |

RB+ECMS, w/GSD
| 4.63 (+3.7%) | 31.35 | 22 | 187 (−42.8%) |

RB+ECMS-EXT | 4.61 (+3.1%) | 31.38 | 22 | 188 (−42.5%) |

JE05 driving cycle | ||||

RB+ECMS, w/o GSD
| 2.72 (+4.6%) | 28.15 | 24 | 480 (0.0%) |

RB+ECMS, w/GSD
| 2.72 (+4.6%) | 28.18 | 24 | 220 (−54.2%) |

RB+ECMS-EXT | 2.68 (+3.0%) | 28.24 | 24 | 236 (−50.8%) |

**Table 4.**Strategies based on BWD and EXT-BWD models (denoted as RB+ECMS and RB+ECMS-EXT, respectively) when applied to FWD model.

Strategy | V_{f} [L] | V_{f,corr} [L] | SoC(t_{f}) [%] | N_{e} [-] | N_{g} [-] |
---|---|---|---|---|---|

DUB driving cycle | |||||

RB+ECMS, w/GSD
| 2.13 | 2.28 (0.0%) | 27.78 | 61 | 499 (0.0%) |

RB+ECMS-EXT | 2.16 | 2.23 (−2.2%) | 28.45 | 59 | 452 (−9.4%) |

HDUDDS driving cycle | |||||

RB+ECMS, w/GSD
| 2.29 | 2.18 (0.0%) | 32.60 | 17 | 97 (0.0%) |

RB+ECMS-EXT | 2.30 | 2.16 (−0.9%) | 33.46 | 12 | 106 (+9.3%) |

WHVC driving cycle | |||||

RB+ECMS, w/GSD
| 4.68 | 4.56 (0.0%) | 33.07 | 22 | 288 (0.0%) |

RB+ECMS-EXT | 4.73 | 4.53 (−0.7%) | 34.79 | 29 | 266 (−7.6%) |

JE05 driving cycle | |||||

RB+ECMS, w/GSD
| 2.71 | 2.79 (0.0%) | 28.07 | 22 | 239 (0.0%) |

RB+ECMS-EXT | 2.84 | 2.77 (−0.7%) | 31.68 | 25 | 253 (−5.9%) |

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**MDPI and ACS Style**

Soldo, J.; Cvok, I.; Deur, J.
Optimal Control of a PHEV Based on Backward-Looking Model Extended with Powertrain Transient Effects. *Energies* **2022**, *15*, 8152.
https://doi.org/10.3390/en15218152

**AMA Style**

Soldo J, Cvok I, Deur J.
Optimal Control of a PHEV Based on Backward-Looking Model Extended with Powertrain Transient Effects. *Energies*. 2022; 15(21):8152.
https://doi.org/10.3390/en15218152

**Chicago/Turabian Style**

Soldo, Jure, Ivan Cvok, and Joško Deur.
2022. "Optimal Control of a PHEV Based on Backward-Looking Model Extended with Powertrain Transient Effects" *Energies* 15, no. 21: 8152.
https://doi.org/10.3390/en15218152