Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle

Plug-in hybrid electric vehicles (PHEV) are proven to be viable transition technology towards fully electric vehicles (EV), as they overcome main deficiencies of EVs such as high prices and short range, while allowing recharging from power grid. PHEVs typically operate in charge depleting (CD) and charge sustaining (CS) regimes, where pure electric (CD) driving is active until the battery is discharged to a predefined lower-limit level, when hybrid driving (CS) is activated in order to sustain the battery state-ofcharge (SoC). In the case of knowing a trip length in advance, it is possible to discharge the battery more gradually under blended regime (BLND) and thus further reduce fuel consumption [1, 2, 3] (typically from 2% to 5% when compared to the CD/CS regime [1]). The optimal SoC trajectory (expressed with respect to travelled distance) tends to have a nearly-linear minimum-length shape for the zero road grade case [1, 2, 3], while it can significantly deviate from the linear trend in the presence of variable road grade [1, 3], low emission zones [4], and different driving patterns on different route segments [2].


Introduction
Plug-in hybrid electric vehicles (PHEV) are proven to be viable transition technology towards fully electric vehicles (EV), as they overcome main deficiencies of EVs such as high prices and short range, while allowing recharging from power grid. PHEVs typically operate in charge depleting (CD) and charge sustaining (CS) regimes, where pure electric (CD) driving is active until the battery is discharged to a predefined lower-limit level, when hybrid driving (CS) is activated in order to sustain the battery state-ofcharge (SoC). In the case of knowing a trip length in advance, it is possible to discharge the battery more gradually under blended regime (BLND) and thus further reduce fuel consumption [1,2,3] (typically from 2% to 5% when compared to the CD/CS regime [1]). The optimal SoC trajectory (expressed with respect to travelled distance) tends to have a nearly-linear minimum-length shape for the zero road grade case [1,2,3], while it can significantly deviate from the linear trend in the presence of variable road grade [1,3], low emission zones [4], and different driving patterns on different route segments [2]. This paper deals with an analysis of optimal SoC trajectory profiles obtained by using dynamic programming (DP) optimisation of PHEV control variables in the blended regime. A convex analysis of the relevant powertrain functions is conducted to explain the observed optimal SoC trajectory patterns, in order to further gain insights into the optimal powertrain operation for different operating conditions. The main contributions of the paper include: (i) proposing a method of generating optimal SoC trajectories of different length and conducting correlation analyses of obtained results, (ii) clarifying the cause and conditions under which the optimal SoC trajectory has a minimum-length linear pattern, and (iii) analytical proof of optimal SoC trajectory pattern for the simplified scenarios of battery-only discharging system. Fig. 1a illustrates the parallel PHEV configuration of a city bus powertrain considered herein for the purpose of analysis. The powertrain consists of internal combustion engine (ICE), electric machine (M/G), lithium-ion battery and automated manual transmission with 12 gears [5]. When being switched off, the engine can be disconnected from the powertrain by using a clutch, thus enabling electric-only driving. The PHEV powertrain is modelled in the backward-looking manner, where the engine i.e. M/G machine rotational speed ωe = ωMG is calculated from the vehicle velocity vv:

Modelling of PHEV Powertrain
while the total engine and M/G machine torque is obtained from the demanded torque at wheels τw while accounting from the drivetrain losses: In Eqs. (1) and (2) h and io denote the transmission gear ratio and the final drive ratio, respectively, ωw is the wheel speed, rw is the effective tire radius, ηtr is the transmission efficiency, while P0 denotes the idlemode power losses (see Figs. 1b and 1c). The demanded power can then be defined as The engine specific fuel consumption and M/G machine efficiency are modelled by means of 2D maps, while the corresponding maximum torque characteristics are modelled by 1D maps (Fig. 2). The specific fuel consumption map (Aek), expressed in g/kWh unit, can readily be transformed to the fuel consumption rate map (ṁf; expressed in g/s unit) by using the following expression: ̇= ( , ) 3.6 · 10 6 . The battery is modelled as a charge storage by an equivalent electrical circuit (Fig. 3a), where the open circuit voltage Uoc and internal resistance R are set to be dependent on the battery state-of-charge (SoC) (Fig. 3b). Finally, the battery model is represented by the following state equation [6] ̇= where Qmax is the battery charge capacity, while Pbatt is the battery power which is determined by M/G machine power PMG as where ηMG denotes M/G machine efficiency (see Fig. 2b), and k is equal to 1 for the case of battery charging (Pbatt<0) and −1 for the case of battery discharging (Pbatt>0).

Optimisation of PHEV Control Variables
This section deals with DP optimisation of PHEV control variables for the blended regime, which is aimed at finding optimal SoC trajectories for different driving cycles and conditions. More details on the optimisation approach can be found in [1,7] and references given therein.

Optimal Problem Formulation
The optimisation problem is to find the PHEV control variables in each discrete time step in order to minimise the cumulative fuel consumption, while satisfying the state-and control variables-related constraints, as well as a requirement on the value of state variable in the final time step. By introducing the substitutions for the state variable x, control vector u and external input vector v: the following discrete-time cost function including cumulative fuel consumption is set: where k denotes the discrete time step, N is the total number of discrete time steps, and ∆T is the discretisation time step. Apart from the fuel consumption within discrete time step, ṁf ∆T, additional terms aimed to penalise violation of different constraints are included into Eq. (9). The function H − (.) represents the inverted Heaviside function which is equal to 1 when its argument is negative, while otherwise it equals 0. The factor Kg is weighting factor which is set to a relatively large value (here Kg = 10 12 ) in order to avoid constraints violation. The state equation given by Eq. (5) can be discretized in time to assume difference equation form: The initial state variable at k = 0 and final state variable at k = N are defined as An additional term Jf penalising the deviation of the final SoC from the target value SoCf is included in the cost function (8), so that the final optimisation problem reads where Kf denotes a weighting factor (here Kf = 10 6 ).
The above-formulated optimisation problem is solved by using a dynamic programming (DP), which provides globally optimal results for given discretisation resolution of the state and control variables (set as a trade-off between computational efficiency and the optimisation accuracy).

Optimisation Results
DP optimisations of PHEV control variables are conducted for the blended regime and different repeating driving cycles defined in Fig. 4, where DUB1 driving cycle is considered both for zero and varying road grade. The driving cycles are repeated three times to provide discharging the battery to its minimum allowable SoC level which is set here to 30%. The optimised SoC trajectories given in dependence of distance travelled (blue colour plots in Fig. 5) are close to linear profile, which represents the minimum length between two SoC boundary points. This is confirmed by the values of correlation index K (also given in Fig. 5 and obtained by using Matlab function corrcoef(.)), which are close to 1. The linear trend is slightly deteriorated in the case when the road grade is introduced (Fig. 5a), which is also reflected in somewhat reduced correlation index. In this case low frequency oscillations appear in the SoC trajectory, which are caused by the battery recharging during regenerative braking on negative slopes.
The observed, approximately linear optimal pattern of SoC trajectory can serve as a basis for synthesis of SoC reference trajectory applied within a powertrain control strategy for the blended regime and the case when a trip distance is known in advance [1,6].

Generation and analysis of SoC trajectories of different length
Based on the results presented in Fig. 5 it can be hypothesised that the optimality is closely related to the SoC trajectory length, i.e. that the shortest-length trajectory is optimal. In order to test this hypothesis, the optimal SoC trajectories of different length are generated by introducing the following additional SoC soft constraint to the cost function included in Eq. (12) which penalises the deviation of SoC from several (Nj) prescribed values SoCconstr,j in the j th discrete time steps (the weighting factor KSoC is set to 5·10 5 ).
Apart from the total fuel consumption Vf, the total electric energy losses EEL,loss consisting of battery losses Ebatt,loss and M/G machine losses EM/G,loss are also considered in this analysis: The battery losses are dissipated as a heat on its internal resistance R and have quadratic dependence with respect to battery current Ibatt (i.e. ∫ibatt 2 Rdt), while the M/G machine losses depend on the efficiency ηMG (see Fig. 2b).
The normalised SoC trajectory length is calculated as where ∆SoCk and ∆sk represent the difference of SoC and travelled distance between two consecutive time steps (i.e. ∆SoCk = SoCk −SoCk−1, ∆sk = sk −sk−1), respectively, while sf denotes the total travelled distance.   . In order to understand these causes better, the total fuel consumptions Vf are shown versus the mean engine specific fuel consumptions Aek,mean in Fig. 6c, and versus the total electrical losses EEL,loss in Fig. 6d. The results shown in Fig. 6c reveal that the cause (i) may be discarded since the larger total fuel consumption often corresponds to lower mean specific fuel consumption. On the other hand, Fig.  6d confirms that the increased fuel consumption is predominantly caused by the increased total electrical losses (the correlation index is very close to 1). BLND-case SoC trajectory has minimum length and results in minimum fuel consumption, while max LSoC,norm-case SoC trajectory (obtained for Nj = 2 in Eq. (14)) has the maximum length and results in maximum fuel consumption (Fig. 6b).
Based on the presented analysis it can be concluded that the observed optimal pattern related to SoC trajectory length minimization (i.e. linear-like trend) is closely related to minimisation of the electrical losses. More detailed analyses are presented in the following section.

Analysis of Optimal SoC Trajectory Patterns
This section is aimed to further explain the observed DP-based optimal SoC trajectory patterns, starting by an analysis of the optimal operation of a battery-only system and following by an analysis of the whole powertrain including the engine, M/G machine and battery.

Simplified Case of Minimizing Solely Battery Energy Losses
First, the problem of discharging battery from the initial SoC value SoCi (here SoCi = 0.9) to some Similarly, the derivative of SoC with respect to travelled distance is expressed as The battery power losses equal Pbatt,loss = Ibatt 2 R(SoC), which when combining with Eqs. (17) and (18)  where Qmax is omitted since it is constant and does not have impact on the optimisation problem solution. Next, the following substitution can be introduced leading to Since the quadratic function (.) 2 is convex, the following expression based on Jensen's inequality can be established By assuming constant vehicle velocity vv, constant internal resistance R, and constant length of each route segment ∆s, the expression (24) leads to ∆SoCr/∆sr = (SoCf -SoCi)/sf, where sf represents the total travelled distance. In that case the optimal operation would be to discharge the battery with constant SoC depletion rate, i.e. the SoC trajectory would follow the linear trend and have the minimum length.
The same battery discharging problem is further analysed numerically by using DP algorithm to study the impact of varying battery parameters on SoC trajectory shape. Fig. 7 shows SoC trajectories obtained by constant SoC depletion rate (SoClin) and by DP optimisations for: (i) the constant battery parameters (the mean values from Fig. 3b are used), and (ii) the SoC-dependent battery parameters (Fig. 3b). In the case of constant battery parameters, the optimal operation is related to a constant SoC depletion rate ( Fig. 7b; slight deviation from the constant value in the case of DP occurs due to discretisation effects and the requirement on the final SoC value). It can be observed from Fig. 7a that the impact of variable battery parameters on the optimal SoC trajectory shape is negligible. The slight deviation from the constant SoC depletion rate in the case of variable battery parameters (Fig. 7b) is caused by the SoC dependence of internal battery resistance R; namely, slightly lower absolute values of dSoC/dt are observed until SoC falls around 75%, because the resistance R has somewhat larger values for SoC > 75% than in the range from 30% to 65% (see Fig. 3b).

More Realistic Case of Minimizing Fuel Consumption
The analysis is extended here to the overall powertrain, which includes the engine, M/G machine, transmission, and battery (see Fig. 1a). In order to study the optimal SoC trajectory with respect to fuel consumption minimization while discharging the battery (i.e. from 90% to 30%), the fuel consumption rate ṁf is expressed in dependence on SoC depletion rate ̇ for different values of the battery SoC, power demand Pd, and the engine speed ωe: The optimal solution for ̇ which minimises the fuel consumption can be found analytically if the function g in Eq. (25) is convex, under assumption of constant values of Pd, SoC, and ωe (i.e. constant vehicle velocity). It can be shown that the optimality is achieved if ̇ is kept constant during whole driving cycle and set to the value which would discharge the battery to the predefined minimum value (the same reasoning as in the case of deriving optimal SoC depletion in Eq. (24)). The analysis is given here in the time domain, and it is equivalent to the travelled distance domain considered in previous sections because of the constant vehicle velocity assumption considered here. The corresponding second derivatives are positive over the whole range thus confirming the convexity of the analysed functions (Fig. 8b). This convexity analysis is also conducted for a wide set of Pd and ωe values, and the results are shown in Fig. 9 (the function is categorised as non-convex if its second derivative is not strictly positive). According to the results from Fig. 9, the function g in Eq. (25) is convex for a majority of Pd and ωe values.   From the standpoint of lower engine specific fuel consumption and regardless of type of engine fuel consumption characteristic (original or modified), Scenario OP2 is preferable over Scenario OP1, and Scenario OP3 is preferable over Scenario OP2 (see Figs. 11a and 11b). However, from the standpoint of overall powertrain fuel consumption, Scenario OP1 related to linear SoC trajectory should be optimal if the function ṁf vs. SoĊ is convex (as it is the case with the original characteristic in Fig. 10a), while it should be suboptimal in the case of non-convex function (modified characteristic in Fig. 10a). This is confirmed by the results presented in Fig. 12, where the comparative fuel consumption time profiles are shown for different scenarios. This finding can be explained by the fact that it is advantageous to place the engine operating point to somewhat larger specific engine fuel consumption (OP1 vs. OP2 and OP3) in the case of original engine characteristic (see Fig. 11a), and thus avoid relatively large total electrical losses whose increase is progressive with the M/G and battery power (Fig. 11c). In this case the optimal SoC trajectory is that of minimum length. However, in the case of modified engine characteristic, where the difference in the specific fuel consumption between OP2 vs. OP1 and OP3 vs. OP2 is more significant than in the original case (Fig. 11a), it is advantageous to move the engine operating point in reduced specific fuel consumption region (OP3 and OP2; see Fig. 11a and also Fig. 11b) despite the increased electrical losses in Phase 2 (Fig.  11c).
The above analysis contributes to understanding of the tendency of optimal SoC trajectories to be of minimum length (as observed in Fig. 5), taking into account that the function g in Eq. (25) is originally convex in a great majority of (typical) operating region (Fig. 9). Certain deviations of the SoC trajectories in Fig. 5 from the minimum length may be explained by the fact that the assumption on operating parameters (i.e. constant Pd, SoC, and ωe) is not satisfied for realistic driving cycles.

Conclusion
The paper has dealt with analysis of the optimal battery state-of-charge (SoC) trajectory for the blended operating regime of a parallel plug-in hybrid electric vehicle (PHEV). The analysis is based on optimal control results obtained by using dynamic programming optimisations for various driving cycles and based on a backward-looking powertrain model. It has been found that the optimal SoC trajectories (when expressed with respect to distance travelled) tend to have nearly-linear shape for different driving cycles, which corresponds to the minimum SoC trajectory length. The minimum SoC trajectory length has been proven to be optimal both analytically and numerically for a simplified battery-only system based on battery power loss minimisation. The analysis has been extended to the whole powertrain including the engine, electric machine and battery, where the main aim was to minimise the total fuel consumption. It has been shown that the linear SoC trajectory is also optimal for the whole powertrain in the actual case of convex shape of fuel mass flow vs. SoC depletion rate characteristic.
The linear SoC trajectory is optimal because of its feature to minimise the total electrical losses and because of flexibility in setting the engine operating points due to a relatively flat engine specific fuel consumption vs. engine power characteristic in a wide range. It has also been demonstrated that when modifying the engine specific fuel consumption characteristic to some extent, the optimal SoC trajectory can have significantly different pattern than the minimum-length linear one.