Nonlinear Model Predictive Control of a Power-Split Hybrid Electric Vehicle With Consideration of Battery Aging

The electrification of transportation has drawn a great deal of attention with the development of various types of new energy vehicles, such as electric vehicles (EVs), hybrid electric vehicles (HEVs), and plug-in hybrid electric vehicles (PHEVs), in recent years. These new energy vehicles with an electric propulsion system are developed and engineered to have a better fuel economy and lowered emission. Batteries are essential for EVs, HEVs, and PHEVs. Different from internal combustion engines or electric motors, which have more research and technological accumulation, high voltage lithium ion battery packs are still relative new to the automotive industry and require further study. For example, the high voltage lithium ion battery packs are still facing issues, including the inaccurate estimation of state of charge (SOC), the inaccurate prediction of voltage response that results in the decrease of usable power, and the degradation of total energy along with its usage. Through the research on battery materials, packaging and control, it is hoped that possible downsizing of the battery pack will lower down the price of HEV/PHEV without compromising the dynamic performance and the prolonged battery life will make the HEV/PHEV more comparable with conventional vehicles in terms of depreciation rate and long-term performance.

Powertrain control in HEVs is achieved by an energy management system (also referred as supervisory control or power management system). The major task of the energy management system is to determine the percentages of power that come from engine and motors at every instant of time. A common goal of the system is to have a better fuel economy performance compared to a conventional vehicle powertrain with only a single power source. Researchers have explored various methods to find increasingly better solutions for the supervisory control of the HEVs. The adaptive equivalent consumption management strategy (A-ECMS) method [1,2] is used to have the suboptimal control strategy calculated while the control system is running online. The A-ECMS method is an improved approach based on the equivalent consumption management strategy (ECMS) [3–5] where an equivalent factor between fuel consumption and SOC change is considered. The A-ECMS method has improved the ECMS using an adaptive converting equivalent factor [1].

Some researchers, such as Wang and Lukic [6] and Wang et al. [7], have used the dynamic programming (DP) method to calculate the global optimized solution for a known driving cycle. In Wang's research [6], the offline generated optimized control strategy has been used to populate an online lookup table for real-time control. While using DP, the knowledge of an entire driving cycle is required, which is hard to realize in real application; other adjustment and improvement toward this approach such as the stochastic DP has been explored. In the research of Johannesson et al. [8], different levels of future driving cycle information have been provided to the controller to see how the controller performs and adjusts to different information. Lin et al. [9] have modeled the future driving cycles and driver's behavior as Markov Chain and solved the infinite horizon stochastic DP problem for the control of HEVs.

Similarly, the model predictive control (MPC) method [10–14] has also been applied to the power management system of HEVs with a limited knowledge of driving cycle in the next few minutes. With the same constraints of future driving cycle information, the stochastic model predictive control has also been explored for HEV energy management with prediction of future driving cycle and driver's power request [15]. In recent years, other factors have been looked into to realize the application of MPC in vehicle controls [16,17]. Sun et al. [17] studied different methods for vehicle speed prediction of a hybrid electric vehicle with model predictive control method. The performance of the predictors as well as the resulted fuel economy by applying the model predictive control is compared and evaluated. Besides MPC and DP, other methods such as improved particle swarm optimization and genetic algorithm [18–20] have also been considered.

To simulate and control HEV/PHEV batteries, a number of researchers have studied battery modeling for electric vehicle simulation [21–24], battery state estimation [25], battery degradation due to charging/discharging [26], and HEV energy management with a consideration of battery life [27]. In paper [27], the Pontryagin's minimum principle (PMP) method and an energy throughput-based battery aging model have been employed for the energy management of a parallel HEV. Serrao et al. [28] have also applied PMP to minimize fuel consumption and battery aging for a midsize parallel hybrid sedan with a certain driving cycle. In Serrao's research [28], the battery aging has been modeled as a severity factor map of aging with inputs of battery SOC and C-rate. Ebbesen et al. [29] augmented the ECMS method with a battery aging effect using a throughput-based capacity fade model for battery aging. In the paper of Moura et al. [30], an electrochemical battery model has been built to study the aging mechanism of the battery pack used in a PHEV. The stochastic DP method has been used for the minimization of fuel consumption and battery aging.

The purpose of the presented research is to explore HEV energy management using nonlinear model predictive control (NMPC) method with the consideration of battery aging. To fulfill such goal, an electrochemical lithium ion battery model is built and integrated with the Prius 04 power split HEV model from autonomie [31] software. The original equivalent circuit model of nickel metal hydride battery pack is replaced by an electrochemical model for the battery pack of A123 26650 LiFePO4-Graphite Li-ion cell inherited from the work of Prada et al. [32,33]. The built model has battery aging features of capacity loss and resistance increase. The studied battery cells produced by A123 system have been widely used in commercial and passenger vehicles.

The NMPC method is investigated to improve the battery aging performance due to its capability of optimizing the energy management of HEV powertrain with very short horizon of preview. The NMPC method is more feasible for real-time control compared with DP and PMP, which require the entire driving cycle's information. The predictive model estimates the fuel consumption, the battery SOC, and the battery aging index over the prediction horizon. A customized cost function is designed to combine the fuel economy and battery aging both as the objectives of the optimization problem. The simulation results of the proposed NMPC are compared with the NMPC without the consideration of battery aging.

The major contributions of the paper are follows: (1) The research work develops a predicative HEV energy management strategy with the consideration of both fuel economy and quantified lithium ion battery aging, which can be applied for real-time HEV control. Different from DP and PMP, which optimize control over a predefined driving cycle, the NMPC method allows the optimization of HEV energy management within a finite horizon without the need of knowing entire driving cycle. This feature makes NMPC-based HEV energy management feasible for real-world driving scenarios. (2) Battery aging rate is quantified based on an electrochemical lithium ion battery model, which is able to provide quantified aging characteristics for the increase of internal resistance and decrease of battery capacity. The battery aging factor is quantified by changing factors of the battery, including SOC, charging rate, and the internal temperature of the cell. These factors are influenced by the energy management system power-split decisions. (3) A comprehensive commercial HEV vehicle model is modified and integrated with battery aging model for the validation of NMPC energy management strategy using US06 driving cycle.

The rest of the paper is organized as below:

1. (1)

   The vehicle model and the battery model are introduced first. The methods that the engine and the planetary gear being modeled are discussed. The method to solve the equations for the electrochemical Li-ion model is also presented.

2. (2)

   The problem formulation for the NMPC in our application of a power-split hybrid electric vehicle is elaborated including the predictive model for fuel consumption, SOC change, and battery aging index. Certain simplification is needed for the battery model to reduce the computational load.

3. (3)

   The simulation results of the integrated HEV model using proposed NMPC algorithm are compared with those of NMPC without the consideration of battery aging. The comparison among different control algorithms is performed. The NMPC methods with different combinations of weighting factors in the cost function are also compared and analyzed.

The vehicle model is modified from an HEV model in autonomie software, a commercial software from Argonne National Laboratory to facilitate the research on fuel economy, and power performance of different HEVs and EVs [31]. This specific HEV model used in our study is built based on the Toyota Prius MY04, the first generation of Prius power-split hybrid electric vehicles. The planetary gear of the Prius 04 is shown in Fig. 1. The sun gear of the planetary gear is connected to a small motor (MG1) while the ring gear is connected to a larger driving motor (MG2). The internal combustion engine is connected rigidly to the carrier of the planetary gear. The peak powers of MG1, MG2, and the internal combustion engine are listed in Table 1.

Different from a single-shaft parallel HEV, in which the engine torque and driving motor torque are coupled electrically, the power-split HEV couples the power from engine and motor mechanically using one set of planetary gear. The purpose of such setup is to have the engine running at optimal operational speed and torque despite the range of vehicle speed and power request. The planetary gear serves as a continuously variable transmission in a certain way. The driving motor, with peak power of 50 kW, is rigidly attached to the ring gear of the planetary, which can accelerate (propulsion) or decelerate (regeneration) the vehicle. The other motor mounted at the sun gear of the planetary is used for the speed control of the engine.

In Eqs. (1)–(3), the ring gear shaft, sun gear shaft, and carrier shaft have the inertias of $Ir$⁠, $Is$⁠, and $Ic$⁠, respectively. $Tr$⁠, $Ts$⁠, and $Tc$ are the torque exerted on the ring gear, sun gear, and carrier. $F$ is the internal force between each component. $R$ is the radius of the ring gear and $S$ is the radius of sun gear. And $ωs$⁠, $ωr$⁠, and $ωc$ are the rotational speed of sun gear, ring gear, and carrier.

where $Ic$ is the carrier inertia, which includes the inertia of the engine crankshaft; and $Is$ is the inertia of sun gear shaft, which includes the inertia of MG1 shaft. $I′r$ includes the inertia of the ring gear and the equivalent inertia of final drive and equivalent vehicle rotational inertia calculated from the vehicle mass. $T′r$ includes the torque of the MG2 and the equivalent torque calculated from the resistance on the wheel, including the rolling resistance and wind drag. The details on the definition of $I′r$ and $T′r$ can be found in the papers of Moura [35] and Liu [34].

The model of the 57 kW internal combustion engine from Autonomie is map-based with maximum wide open throttle (WOT) torque, hot fuel rate map, and various emission maps. Figures 2 and 3 are the WOT torque map and hot fuel rate map, respectively.

From Fig. 3, we can tell that the higher engine speed and the torque are, the higher fuel consumption is.

From the above-mentioned maps, we can see that usually the operating conditions with higher engine speed and torque will consume more fuel instantaneously. However, such operational condition will often yield higher fuel efficiency by looking at the break-specific fuel consumption map. With the same power request for the engine, the fuel consumption under different operational regions will yield different fuel efficiency. And thus, to facilitate the hybrid controller, another map as shown in Fig. 4 is generated and used in the control system. For the hybrid controller, if certain power request is made toward the engine, the corresponding optimal rotational speed will be found through this map and the engine speed controller will stabilize the engine speed at the target speed through a closed-loop controller of sun gear motor.

The map of Fig. 4 was utilized in the original rule-based controller of the supervisory energy management control for the Prius HEV in the autonomie software. It is also inherited and employed in cooperation with the NMPC controller studied in this paper. With the application of the optimal engine operation speed map, the optimized variables can be downsized to one variable at each instant moment of time instead of two variables. In NMPC algorithm, the engine power is determined with the consideration of battery power limits. The engine speed is then determined by engine power request.

Certainly, the speed request of this map cannot always be met due to other constraints, such as the speed of the vehicle, the speed limits of the MG1, the torque limits of MG2, and the power limits of the battery pack at each time instant. In the closed-loop controller of the engine speed, if the engine speed request from the map cannot be reached due to any of these constraints, it will be regulated and kept at the closest point to the desired speed. In the predictive model of NMPC, such factors are also considered to have a more accurate prediction on future engine speed.

The goal of the supervisory hybrid controller is to have the engine running within the most efficient operational regions with given driving cycles.

In this paper, an electrochemical model for the lithium iron phosphate cell—A123 26650 developed by Prada et al. [32,33] is modeled and integrated with the Prius 04 HEV model.

The averaged single particle battery model can be described as shown in Fig. 5. The assumption is that the current is averaged across the cell from the left to the right side (from negative current collector to the positive current collector). And because of that, kinetic overpotential at the solid phase and electrolyte surface is directly determined by the current applied. And there is only one large solid phase particle in positive and negative electrodes, respectively.

As shown in Fig. 5, the lithium ions will be transported into or out of the single solid phase as the battery is being charged or discharged. Because of the diffusion phenomenon of the lithium ions, uneven distribution of ions inside the solid phase is considered.

In Eq. (6), $cs$ is the concentration of lithium ions in solid phase (mol/cm²), $Ds$ is the solid phase diffusion coefficient for lithium ion, $r$ is the radial coordinate. And for spherical Li-ion particles, $m$ = 2 [36].

where $Rs$ is the radius of particle size at each electrode as shown in Fig. 5.

Equation (7) indicates that the center of the electrode particle of the lithium ion concentration stays the same along the radial coordinate direction (zero flux). Equation (8) indicates that at the surface of the electrode solid phase particle, the derivative of lithium concentration along the radial should be proportional to the electrode current density as in the Butler–Volmer equation.

where $Ea_k$ is the activation energy in the Arrhenius law accounting for the temperature dependency; $Tint$ is the internal temperature of the cell; $Tref$ is the reference temperature for the Arrhenius law; $Sn$ is the electroactive surface of the negative electrode; $δSEI$ is the thickness of the SEI layer; the $β$ is the SEI ionic conductivity; $κSEI$ is the charge transfer coefficient for the side reaction; $ϕs,n$ is the ohmic drop due the SEI layers; $Us$ is the electric potential of the solvent reduction; and $is0$ is the exchange current density of the side reaction.

The overall purpose of the model predictive control is to calculate a path of future control $u$ to optimize the performance of plant output over the prediction horizon [40]. From Eq. (12), we can see that the battery aging process is highly nonlinear and the calculation of exchange current density of the side reaction is very complicated. And it can be inferred from the equations that the rate of battery aging influenced majorly by the concentration of lithium ions at the particle surface of negative electrode, the cell temperature, the applied charge current, and the current state of health (SOH). The concentration of lithium ions at the particle surface of negative electrode is mostly determined by the battery SOC and overpotential due to the lithium ion diffusion inside the negative particle calculated by Eqs. (9)–(11).

While the battery thickness of the SEI layer $δSEI$⁠, which represents the SOH, and diffusion overpotential in negative electrode particle are hard to quantify, the internal temperature of the cell, the battery SOC, and C rate are observable and controller variables during the HEV operation. As a result, the battery aging rate can be considered as a function with inputs of battery SOC, temperature, and the applied current as they are the major factors that determine the battery aging.

In Figs. 6 and 7, the current density of the side reaction is simulated at different SOC, temperature, and charge rate. The current density is always negative indicating the battery aging side reaction cannot be reversed and is happening continuously even when the battery is being stored. And the smaller this current density value is, the faster the battery is aging. From the figures, we can see that the battery aging is significantly influenced by SOC that the higher the battery SOC is, the faster battery is aging. High battery charge rate as well as high temperature also has a negative influence toward aging.

It should be emphasized that the lookup table does not necessarily give the accurate speed of aging for the battery cell because of the change of SEI layer sickness, the change of diffusion overpotential inside the negative particle, and other factors. However, the lookup tables are still chosen to present the aging of battery in the NMPC because: (1) The calculation speed is significantly slowed down if the electrochemical model is directly used in the predictive model; (2) The calculation of the electrochemical model needs the values of extra variable inside the battery model (such as the lithium ion concentration on particle surface), which will be hard to implement for a real vehicle controller; and (3) The lookup table presents the same trend of aging performance under different operational conditions.

For the simulation of an entire vehicle to validate the control strategy, the first principle electrochemical model is still used to have the more accurate battery performance simulated.

where $SOC$ denotes the battery pack state of charge at the $kth$ time instant; $ωMG1k$ is the sun gear speed; and $ωMG2k$ is the ring gear speed both at the $kth$ time instant.

where $Peng$ stands for the power request for engine.

where $Vbattk$ and $Ibattk$ are the instant measured values of battery terminal voltage and applied current.

where $ffuel$ is a nonlinear fuel consumption function as shown in Fig. 3, $SOCcost$ being the matrix to calculate the cost generated by SOC state, and $X1$⁠, $X2$⁠, and $X3$ being the system state over the prediction horizon calculated using control $U$⁠, $α$⁠, $β$⁠, $λ$⁠, and $γ$ are the weights of different components. $is$ represents the rate of battery aging side reaction as shown in Figs. 6 and 7. The temperature used in the model is the cell temperature. There are several physical limits considered in the cost function calculation, including the engine and motor maximum torque for certain speed and the battery power limits, which are interpreted using the penalty values. The penalties used in the above equation are referring to the large value added to avoid vehicle to operate in unwanted areas such as battery power requested being higher than its limits, or SOC out of the preferred operational range for the battery packs, which are treated as soft boundaries in this study.

For HEVs, the SOC operating range is relatively narrower, which can be achieved by a quadratic term of SOC error in the cost function. However, the SOC range of PHEVs is wider compared to HEVs. The current trend of hybrid electric vehicles is mostly multimode PHEVs with blended charge-depleting and charge-sustaining modes. This study does not constrain the SOC to a narrow band. However, the NMPC algorithm does include one constraint to limit SOC between a lower limit and an upper limit.

To evaluate the performance of the proposed NMPC with the consideration of battery aging, a comprehensive vehicle model integrated with an electrochemical lithium ion battery model is used to validate the control performance. The US06 (high-speed aggressive driving) is used as a testing driving cycle for two control methods: the NMPC method with the consideration of battery aging and the NMPC without the consideration of battery aging, to evaluate the controller performance in both fuel economy and battery aging.

The difference between the NMPC with and without battery aging lies in the definition of the cost function. In the cost function of former one as shown in Eq. (23), the battery aging index is considered. In the cost function of the latter one, the last term in Eq. (23) is neglected. When the battery aging feature is considered in the NMPC as a part of the cost function, the battery aging will be slowed down compared with the NMPC without the consideration of battery aging. As shown in Figs. 8 and 9, battery aging rate is slower while using the NMPC with battery aging consideration. And the battery aging rate is faster when using the NMPC without the battery aging concerned.

The NMPC with and without the consideration of battery aging will yield different vehicle performance in terms of battery SOC and fuel economy as shown in Figs. 10 and 11. The NMPC with battery aging consideration will tend to charge the battery less frequently except in the case of regenerative braking and keep the SOC at a relatively lower level.

Figures 12 and 13 show the difference in the change of SEI thickness and battery capacity loss when applying different weighing factor values in the cost function of NMPC. In the simulation results presented in Figs. 12 and 13, the parameter to be changed is the weighting factor $γ$ in the cost function of Eq. (23). The $γ$ value for para set 1, para set 2, or para set 3 is 0.006, 0.008, or 0.01, respectively.

As the battery aging process is controlled to be slowed down, the fuel economy and the region of the battery to be operated may also be shifted to meet the revised control goal. Table 2 shows the comparison of fuel economy and battery aging performance between two controllers. The MPC controller without the consideration of battery aging has the worst performance in terms of battery aging. As seen in other research projects on the battery aging conscious optimization control, the improvement of battery aging performance will tradeoff certain fuel economy performance. The equivalent fuel consumption is calculated as the battery being charged back to the original value with the vehicle being stopped. The target performance can be achieved by tuning the weighting factors in the proposed cost function. The aging rate of the battery varies together with the battery temperature, which adds complexity to the tuning of the weighting factors for the NMPC. However, the battery temperature will be more dynamic at the start of the vehicle, but kept within a reasonable range by an heating, ventilating, and air conditioning system later on once the vehicle has been running for a while. The weighting factors can be tuned for vehicles running within the normal battery operating temperature range to simplify the problem.

From Table 2, we can see that, by tuning the weighting factor $γ$ only, the performance of the NMPC with the consideration of battery aging could be improved. It is believed that for specific driving cycles, there should be an optimized selection of weighting factor to have the combined best fuel economy and battery aging performance. As shown in Table 2, when the value of $γ$ is increased from 0.006 (para set 1) to 0.01 (para set 3), the fuel economy drops but the battery aging performance improves. It is noticed that both the battery aging and fuel economy have been improved when the para set 1 weighting factor is selected comparing with the NMPC without the consideration of battery aging. The current selection may not be the most optimal selection but it demonstrates the potential for the improvement of fuel economy and battery aging by selecting appropriate weighting factors. The degradation rate in terms of SOH is also listed in the table to describe different rate of battery aging. Note that the degradation over the driving cycle is currently faster at the beginning of life for the battery cell but will stabilize and slow down as the electrochemical reaction reaches a balanced point.

In this paper, the NMPC-based HEV energy management strategy with the consideration of battery aging is presented and tested using an integrated hybrid electric vehicle model. The presented NMPC has taken the battery aging into consideration in the design of supervisory control of a power-split hybrid electric vehicle. Different from existing research work on the MPC or NMPC control for HEVs, this paper has incorporated an electrochemical battery model in a detailed vehicle model for the simulation validation to yield a more reliable battery performance especially on battery aging. For the NMPC model, the battery aging model is simplified to speed up the optimization process. It is shown that certain level of battery model simplification will still work with the NMPC algorithm and the study of physics-based model will help in the design of according controller.

The NMPC model is structured with the prediction of battery aging under different SOC levels, battery temperatures, and charging currents. The cost function for the NMPC incorporates the battery aging as one key factor in the optimization process. The proposed NMPC can improve the battery aging performance of the studied vehicle with similar or better fuel economy compared with the NMPC without the consideration of battery aging. By having different combination of weighting factors for the proposed NMPC with the consideration of battery aging, different control performances of the proposed algorithm can be achieved.

$cs$ =

the concentration of lithium ions in solid phase

$Ds$ =

the solid phase diffusion coefficient for lithium ion

$Ea_k$ =

activation energy in the Arrhenius law

$F$ =

internal force between sun, carrier, and ring gear

$Ic$ =

inertia of the carrier shaft

$Ir$ =

inertia of the ring gear shaft

$is$ =

rate of battery aging side reaction

$Is$ =

inertia of the sun gear shaft

$I′r$ =

inertia of the ring gear and the equivalent inertia of final drive and equivalent vehicle rotational inertia calculated from the vehicle mass

$is0$ =

exchange current density of the side reaction

$Ibattk$ =

battery current

$Pbattk$ =

the total power of battery

$Pbrake(k)$ =

deceleration power request

$Peng$ =

power request for engine

$Preq(k)$ =

driver's acceleration power request

$r$ =

the radius of particle

$R$ =

radius of the ring gear

$Rbatt(k)$ =

battery internal resistance

$S$ =

radius of sun gear

$Sn$ =

electroactive surface of the negative electrode

$SOC$ =

battery state of charge

$Tc$ =

torque exerted on the carrier

$Tr$ =

torque exerted on the ring gear

$Ts$ =

torque exerted on the sun gear

$T′r$ =

torque of the MG2 and the equivalent torque calculated from the resistance on the wheel including the rolling resistance and wind drag

$Tint$ =

internal temperature of the cell

$Tref$ =

reference temperature for the Arrhenius law

$u(k)$ =

control variables

$Us$ =

electric potential of the solvent reduction

$v(k)$ =

disturbances

$Vbattk$ =

battery terminal voltage

$Voc(k)$ =

battery open-circuit voltage

$xk$ =

system state

$α, β, λ, γ$ =

weighting factors of the cost function

$β$ =

SEI ionic conductivity

$δSEI$ =

thickness of the SEI layer

$ωc$ =

rotational speed of carrier

$ωr$ =

rotational speed of ring gear

$ωs$ =

rotational speed of sun gear

$κSEI$ =

charge transfer coefficient for the side reaction

$ϕs,n$ =

ohmic drop due the SEI layers