## Abstract

This article proposes a hierarchical energy management strategy for power-split hybrid electric vehicles (HEVs) in presence of driving cycle uncertainty. The proposed hierarchical controller exploits long-term and short-term decision making via a high-level pseudospectral optimal controller and a low-level robust tube-based model predictive controller. This way, the proposed controller aims at robust charge balance constraint satisfaction and improvement in energy efficiency of the HEVs in presence of uncertainty in the future driving cycle. This article further focuses on the human-driven HEV energy management and exploits a data-driven future velocity prediction method that uses the data obtained from a drive simulator. Simulation results show an improvement in fuel economy for the proposed controller that is real time applicable and robust to the driving cycle’s uncertainty.

## Introduction

The necessity of improving fuel efficiency and reduction of greenhouse gas emissions has led to an increased attention to hybrid electric vehicles (HEVs). The additional electrical component in HEVs makes it possible to downsize the engine, capture energy during braking instead of wasting it, and preventing the engine from working at low efficiency load and speeds [1]. To make use of their potential, HEVs require an energy management strategy that determines an optimal power split between the two powertrain components while ensuring that the driver’s power demand is met and all the powertrain constraints have been satisfied. This includes charge balance constraint satisfaction for conventional HEVs that requires the final battery’s state-of-charge (SOC) to be within a specific set around its initial value.

The energy management problem of HEVs seeks an optimal solution that minimizes the overall fuel consumption of the vehicle over its trip. Solving such an optimal control problem over the entire trip in real time is not possible due to the limited computational power available on-board the current vehicles and the unknown future driver’s power demand over the whole trip. Early rule-based power management strategies [2] that are based on heuristics can result in highly suboptimal solutions and also lead to the charge balance constraint violation. Dynamic programing (DP) [3] and stochastic DP (SDP) [4,5] can provide the global or expected global optimal solution of the problem over the whole driving cycle by assuming a priori known driving cycle or a probability distribution of driving cycles, respectively. However, the unrealistic assumption of a known future driving cycle used in deterministic DP and computational complexity of both DP and SDP make them inappropriate for real-world applications. Equivalent consumption minimization strategy (ECMS) [6] and its variants (such as adaptive ECMS [7]) minimize an instantaneous cost and hence can be implemented online. However, optimality of their solution highly depends on the chosen equivalence factor [8,9] whose determination is generally nontrivial. Model predictive control (MPC)-based HEV energy management [10] offers a good compromise between optimality and computational complexity, but it also requires the knowledge of future driver’s torque demand or velocity profile over its horizon, which is generally unknown. Also, conventional MPC-based methods do not guarantee charge balance constraint satisfaction because they do not solve over the whole driving cycle’s profile.

Self-sustainability of the electrical path expressed as the charge balance constraint for conventional HEVs is a major concern for online energy management strategies. Charge balance constraint satisfaction becomes even more challenging in real-world applications in presence of driving cycle uncertainty. SDP-based methods [4,5] are computationally expensive, and robust MPC-based methods [11], although addressing the uncertainty in the prediction horizon, can be suboptimal. Despite the presence of multiple energy management methods for HEVs, current literature still lacks systematic approaches that can ensure constraints satisfaction, specifically charge balance constraint satisfaction in presence of uncertainty in the future torque demand and velocity.

Generally, hierarchical control methods, with high-level long-term and low-level short-term planners, have shown to perform well for the HEV energy management problem [1215]. However, previous research on hierarchical HEV energy management methods does not always solve the energy management problem over the whole trip [14], can be computationally expensive (DP-based methods) [12], or assume exact knowledge of the future velocity profile [1315].

In our previous work [16], we have developed an energy management strategy based on robust backward reachability analysis and MPC with tightened constraints for robust charge balance constraint satisfaction. However, Ref. [16] did not consider a hierarchical control framework. Our previous work [17] also includes development of a hierarchical energy management strategy that tries to ensure charge balance constraint satisfaction in the presence of a bounded uncertainty in the driving cycle.

In this article, we extend our previous work [17] to human-driven HEVs energy management problem by developing a human-driven vehicle’s velocity prediction algorithm and validating our proposed controller with data obtained from a drive simulator. Our proposed hierarchical control architecture, as shown in Fig. 1, exploits the advantages of long-term and short-term decision making to achieve a real-time applicable strategy that improves the fuel efficiency of HEVs while focusing on robust charge balance constraint satisfaction. It includes a high-level pseudospectral optimal controller (PSOC) that generates a near optimal solution in a tractable fashion by approximately solving the energy management problem over the entire driving cycle.

At the high level, road’s average velocity or speed limits are used as a preview of the whole driving cycle while a data-driven algorithm is used at the low-level to predict the human-driven vehicle’s velocity over each horizon. Since PSOC is not exactly real-time applicable and does not handle system’s uncertainty, a robust tube-based MPC controller is utilized in the low level. The robust tube-based MPC ensures constraint satisfaction over its time horizon and maximizes fuel efficiency while considering a bounded prediction error. Although the robust tube-based MPC can address the bounded uncertainty over the time horizon in a tractable fashion, it can provide suboptimal solutions. To address this, the high level’s approximate solution is used by the low-level controller to improve its suboptimality and facilitate in charge balance constraint satisfaction. This way a combination of the long-term and short-term planning can help us achieve a charge sustained and high energy-efficient operation of the HEVs. It is worth noting that, since the high-level controller does not function in real time, our proposed implementation strategy includes solving the high-level problem at the beginning of the driving cycle and periodically update the high-level control solution. In between the updates, the low-level controller follows the most recent solution obtained from the high-level controller.

The major contributions of this work include the development of a hierarchical control architecture for HEV energy management that (i) effectively exploits long-term and short-term decision making to improve solution’s suboptimality, (ii) applies to the human-driven HEVs by utilizing a data-driven human driver’s velocity prediction algorithm for velocity predictions over each horizon in the low-level controller, and (iii) aims at robust constraints satisfaction, specifically charge balance constraint, in presence of the future velocity and torque demand’s uncertainty while avoiding highly conservative solutions.

## Problem Description

### System Dynamics.

In the energy management problem of HEVs, battery’s SOC is the main system’s state whose dynamics is given by [4]
$dSOCdt=−Voc−Voc2−4PbattRbatt2CbattRbatt$
(1)
where Voc, Rbatt, Cbatt, and Pbatt are the open-circuit voltage, internal resistance, capacity, and power of the battery, respectively. Neglecting the motor and generator power losses, the battery power is sum of the motor power Pmot and generator power Pgen, so that Pbatt = Pmot + Pgen. Following Refs. [18,19], the inertial losses of the engine, generator, and motor are ignored in the powertrain’s inertial dynamics equations that converts them to a set of static equality constraints. This is an acceptable assumption because of much faster rotational dynamics of the powertrain components compared to the battery dynamics. Other kinematic constraints are as follows:
$NSωgen+NRωmot=(NS+NR)ωeng;ωmot=gfrwv$
(2)
where NS and NR are the radius of the sun and ring gear, and ωeng, ωgen, and ωmot are the engine, generator, and motor speeds, respectively. gf is the final transmission gear ratio, and rw is the wheel radius of the vehicle. Finally, v is the velocity of the vehicle, which follows (ignoring friction brake torque)
$mdvdt=Tdriverrw−12ρAvCdv2−Crmgcos(θ)+mgsin(θ)$
(3)
where m is the mass, Av is the frontal area, Cr is the rolling resistance, Cd is the drag coefficient, ρ is the air density, θ is the road grade, and g is the acceleration due to gravity.

The control-oriented model can be thus described as $x˙=f(x,u,w)$, where $x∈X⊆R$ is the battery SOC and $u=[Teng,ωeng]⊤∈U⊆R2$ is the control vector that includes the engine’s torque and speed. Uncertainty vector $w=[Tdriver,v]⊤∈W⊆R2$ includes the driver’s torque demand and the vehicle velocity, since they are generally unknown in most real-world applications.

### Optimal Control Problem.

In general, the energy management problem of HEVs seeks an admissible control policy that solves the following problem over the whole driving cycle [t0, tf],
$min∫t0tfm˙f(u(t))dt$
(4a)
$s.t.x˙(t)=f(x(t),u(t),w(t))$
(4b)
$SOCmin≤x(t)≤SOCmaxωengmin≤u2(t)≤ωengmax;Tengmin≤u1(t)≤Tengmax$
(4c)
$x(tf)∈x(t0)+L$
4c

Constraint (4d) is commonly known as charge balance constraint in conventional HEVs, which requires the final SOC to be within a specific bound, defined by the allowable error set $L$, of the initial SOC. Here, the rate of fuel consumption is approximated with a cubic polynomial, $m˙f(u(t))=∑i=03∑j=03ci,jTengiωengj$, which is convex in u(·) for positive Teng and ωeng [16] and its coefficients have been obtained by fitting the engine map data.

## Methodology

In this article, we are focusing on the energy management problem of a human-driven HEV that is traveling through multiple road segments with various road speed limits. Although the speed limits of the road segments the driver is traveling through is known, the way the driver will drive in the road is unknown to the energy management system. This control framework is also applicable for vehicles where an advanced driver assistance system provides suggested commands to a human-driven vehicle to follow as done in our recent work [20].

To ensure charge balance constraint satisfaction (4d), we use high-level PSOC controller, which can provide an overall energy management solution in a tractable fashion using the velocity profile (driving cycle) of the whole trip. We use the speed limits over different segments of the road as a preview of the driving cycle for the high-level controller. Since the actual HEV velocity (and the torque demand) can be different from this preview of the driving cycle, at low level, the driver’s velocity over each horizon is predicted by a data-driven algorithm. Robust tube-based MPC controller then utilizes this velocity prediction and provides the optimal control solution while accounting for a bounded uncertainty around the predicted velocity profile. By using the high level’s solution over the whole trip as the waypoints for the low-level controller, its suboptimality is improved, and it also helps to satisfy the charge balance constraint. Hence, the proposed hierarchical control structure addresses the major issues related to the HEVs energy management.

### High-Level Pseudospectral Optimal Controller.

PSOC is a direct numerical optimal control method that transcribes the original continuous optimal control problem into a nonlinear progarming (NLP) problem by approximating the state and control variables using interpolating polynomials at a set of collocation points. The resulted NLP problem is then solved using the well-known NLP solvers. Generally, based on the discretization scheme used, there are three different groups for pseudospectral optimal control method, namely, Legendre PSOC, Gauss PSOC, and Radau PSOC. It has been shown that the PSOC method can provide near optimal solutions for noninear optimal control problems in different areas, specifically in aerospace industries [21]. However, its application to HEVs energy management is very limited [15,17,22] and has not been investigated thoroughly.

At high level, the PSOC method solves the energy management problem (problem in Eq. (4)) over a preview of the driving cycle to obtain the optimal SOC trajectory xps(τ), τ ∈ [t0, tf], and the corresponding optimal control ups(τ), τ ∈ [t0, tf]. PSOC handles continuous nonlinear system dynamics (4b), state and control constraints (4c), and more importantly, the charge balance constraint (4d). To implement the PSOC method in this article, a matlab-based software package gpops-II [23], which is based on the Radau PSOC method, is utilized to obtain the approximate optimal solution of the problem in Eq. (4).

### Low-Level Tube-Based Model Predictive Control.

Optimal control problems with nonlinear system dynamics lead to nonconvex problems [24], which are computationally expensive. Hence, nonlinear system dynamics is linearized about the high level’s solution and a nominal uncertainty vector (w0) and then it is discretized to obtain
$x(k+1)=A(k)x(k)+B(k)u(k)+Bw(k)d(k)+F(k)$
(5a)
$x(k)∈X;u(k)∈U;d(k)=w(k)−w0(k)∈D$
(5b)
$z(k+1)=A(k)z(k)+B(k)v(k)+F(k)$
(5c)
where the state, control, and uncertainty constraint sets $X$, $U$, and $D$, respectively, are all considered to be polytopic. Equation (5c) describes the nominal system (z and v are the nominal system’s state and control, respectively), which is the system without disturbance. The nominal uncertainty vector w0 is predicted over each prediction horizon using the proposed velocity prediction algorithm.

#### Human Driver’s Velocity Prediction.

Existing velocity and torque prediction techniques for HEV energy management [25] include methods based on the Markov chain model [11], neural networks [26], and predefined functions (such as an exponentially decreasing torque model [19]). In this article, we employ a data-driven modeling technique [20] to predict the future driver’s velocity, given the speed limit of the road. Following the data obtained from a drive simulator, the velocity profile of a human-driven vehicle is modeled with a sigmoid function,
$v(n)=vc−v01+e−b(n−n0)+v0n≥1$
(6)
where vc represents the speed limit (or average speed) and v0 represents the velocity at the instant immediately before the change of the speed limit. n is the number of sampling instants elapsed since the change of the speed limit, and hence, it is reset to zero every time that it changes. The parameters n0 and b vary from driver to driver and from time to time. Hence, they are obtained through least square error minimization and are updated online to predict the future velocities. Data from the drive simulator in University of North Carolina Charlotte are used to evaluate the parameters n0 and b of the function in Eq. (6). The root-mean-square error for velocity fitting is 0.2257 m/s.

#### Tube-Based Model Predictive Control.

The controller u in Eq. (5) is considered to be of the form
$u(k)=v(k)+K(x(k)−z(k))$
(7)
which consists of the nominal system control v(k) and a local feedback control with gain K. The feedback control is employed to attenuate the effect of uncertainties and compensate for the difference between the actual (with state x) and the nominal system (with state z). The feedback gain K should be chosen such that the closed-loop system $(A(τ)+B(τ)K),∀τ$, is stable. For the linear time-varying system that we are dealing with in this article, it is determined by the method discussed in Ref. [17].
Considering the actual (5a) and the nominal (5c) systems, the error between the actual and the nominal system is e(τ) = x(τ) − z(τ), and its dynamics is given by
$e(τ+1)=(A(τ)+B(τ)K)e(τ)+Bw(τ)d(τ)$
(8)
If the error at any time τ is bounded by $e(τ)∈S(τ)$, then to satisfy the actual state and control constraints in Eq. (5b), the nominal system needs to satisfy $z(τ)∈X⊖S(τ)$ and $v(τ)∈U⊖KS(τ)$ (where $⊖$ denotes the Pontryagin set difference). The time-varying error sets $S(τ)$ can be computed by
$S(τ+1)=(A(τ)+B(τ)K)S(τ)⊕Bw(τ)D$
(9)
with $S(0)=0$, implying z(0) = x(0), i.e., the nominal and the actual state at the beginning of the finite time horizon are the same. $⊕$ represents the Minkowski sum of the sets. We obtain Eq. (9) following the definition of the error dynamics and the bounds on the disturbance $d(k)∈D$.
With the tightened system constraints and use of the pseudospectral solution to define a terminal cost for each horizon, the proposed MPC problem at any time k with a finite horizon T is given by
$minπT|k∑τ=kk+T−1m˙f(v(τ))Δt+‖(z(k+T)−xps(k+T))‖R2$
(10a)
$s.t.z(τ+1)=A(τ)z(τ)+B(τ)v(τ)+F(τ)$
(10b)
$z(τ)∈X⊖S(τ);v(τ)∈U⊖KS(τ)$
(10c)
where $πT|k=[v(k),v(k+1),…,v(k+T−1)]⊤$ is the finite horizon nominal control at any time k, R is the penalizing weight, and xps(k + T) is the pseudospectral value of SOC at time k + T. Addition of the terminal cost $‖(z(k+T)−xps(k+T))‖R2$, which uses the pseudospectral controller’s result over the whole driving cycle, helps improving the suboptimality issue of the low-level controller and it is also beneficial for charge balance constraint satisfaction.

## Simulation Results and Discussion

For simulation purposes, a Toyota Prius HEV is considered in this article. Simulations were performed on a personal computer with an Intel(R) Core(TM)i7 − 7700 CPU @ 3.60 GHz with 16.0 GB RAM. The state constraint set is considered to be $X=[0.45,0.8]$, the error set $L=[−0.03,0.03]$ for charge balance constraint (4c), and the uncertainty set $D$ is considered to be time varying with its upper and lower limits equal to +0.1w0(t) and −0.1w0(t) (10% of the nominal uncertainty vector w0 at that time), respectively.

For simulation purposes, a drive simulator is used where a human-driven vehicle is considered to travel through roads with speed limits as shown in Fig. 2. Since the energy management problem is solved in time domain, the road speed limits are mapped into the time domain and the final location is mapped into a final time as well using the speed limits information. This mapping error is minimal since we are not considering high congestion situations. The result of velocity prediction, as discussed in Human Driver’s Velocity Prediction section, is shown in Fig. 2. The root-mean-square error of the velocity prediction is 0.597 m/s.

To assess the performance of the high-level PSOC controller, we compare its result with DP. Figure 3 shows the optimal SOC trajectories obtained from pseudospectral and dynamic programming methods for a case with 0.7 initial SOC. It can be seen from Table 1 and Fig. 3 that the state constraint and the charge balance constraint in Eq. (4c) have been satisfied by both methods. PSOC solution is close to the DP solution (as in Table 1), but computational time required by the PSOC method was 38.49 s while DP solution required 9.37 h.

Our proposed control strategy utilizes the high level’s long-term result to guide the low-level tube-based MPC controller to an ensured charge balance constraint satisfaction. At each time instant, the nonlinear system dynamics is linearized over a time horizon T = 10, about the high level’s solution and a nominal driving profile predicted by the human driver’s velocity prediction algorithm. The blue trajectory in Fig. 3 shows the simulated SOC trajectory obtained from solving the problem in Eq. (10) using our proposed method with Δt = 1 s. The control commands are kept constant (using zero-order hold) over this sampling time of 1 s till the next discrete time instant. It can be seen in Fig. 3 and in Table 1 that the solution from our proposed method satisfies the charge balance constraint and results in 86.76 MPG fuel economy, which is lower in comparison to the high-level solution (90.86 MPG). However, MPC solution is fast enough to be implemented in real time (with an average computational time of 0.7 secs for each horizon), and it ensures robust constraint satisfaction in the presence of system uncertainty.

Next, we present a comparison between our proposed control strategy and an MPC-based baseline method. This baseline method is a standard MPC-based HEV energy management approach that penalizes the deviation of SOC at the end of each finite horizon from the initial value of SOC to achieve the charge balance constraint satisfaction. In this case, the nonlinear system dynamics is linearized about the solution obtained in the previous time instant and the predicted uncertainty vector w0. Figure 3 shows the resulting SOC trajectory in red. The resulting fuel efficiencies in Table 1 show that our proposed method is providing much higher fuel economy. We are able to obtain better fuel economy result since we exploit long-term planning to improve the low-level tube-based MPC controller’s suboptimality issue and also avoid the conservative solution of the baseline method by using the waypoints provided by the long-term high-level controller.

## Conclusion

A hierarchical predictive controller has been proposed in this article for the energy management of power-split human-driven HEVs. By exploiting the benefits of long-term planning, the low-level MPC-based controller’s suboptimality is improved, and charge balance constraint satisfaction is guaranteed. The unknown future velocity is predicted by a data-driven prediction algorithm over each horizon of the low-level controller. Uncertainty of the prediction is accounted for by tightening the actual system constraints in the low level’s tube-based MPC controller. In this way, our proposed control structure is capable of robustly ensuring charge balance constraint satisfaction and also generating higher fuel economies compared to the commonly-used methods in the literature. Future work involves studying a more comprehensive way of uncertainty quantification for modeling the time-varying disturbance sets.

## Conflict of Interest

There are no conflicts of interest.

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