An Optimal Approach to Energy Management Control of a FuelCell Vehicle
1. Introduction
The next section presents the developed tanktowheel model and the results of the validation phase. Next, the vehicle speed profile optimization problem is formalized and solved. Finally, the cost function and constraints of the MPC are designed, the simulation results are discussed, and conclusions are drawn.
2. Problem Formulation
This paper presents an EMS for an FC vehicle equipped with a secondary PS. The SEM scenario is analyzed, and constraints on average vehicle speed and SC voltage are introduced. To enhance the effectiveness of the EMS design, it is assumed that all powertrain components remain unchanged.
The first step in the optimization process is to calculate the speed profile that the controller must track. This profile takes into account several factors, including the maximum allowed speed to prevent rollover in curves, the road gradient, and the maximum allowed armature current. To improve the robustness of the optimization process and ensure the reliability of the results, only the vehicle dynamics and the steadystate conditions of the electric motor are considered. The cost function used in the optimization is the integral of the armature current squared.
A nonlinear MPC algorithm with a binary optimization variable is used to control the plant. Fuel consumption optimization is achieved through the use of an appropriate cost function that selects the optimal PS to feed the powertrain. The arguments of the cost function are the FC current and the SC voltage. Additionally, the SC voltage is constrained to adhere to the SEM rules and ensure a valid run attempt.
3. TanktoWheel Model
The FC uses the hydrogen stored in the tank to produce electrical energy. The generated power can be used to supply the EM or to recharge the SC. Torque is applied to the vehicle’s rear drive wheel by a suitable mechanical transmission. In addition, a freewheel is inserted between the electric motor’s shaft and the mechanical transmission to decouple the system when the vehicle is freerolling.
3.1. Equation Model
$${V}^{fc}$$
$${\dot{V}}^{sc}=\frac{d{V}^{sc}}{dt}=\frac{{I}^{sc}}{{C}_{sc}}$$
$${\dot{I}}^{a}=\frac{d{I}^{a}}{dt}=\frac{{V}^{a}}{{L}_{a}}\frac{{R}_{a}}{{L}_{a}}{I}^{a}\frac{{k}_{e}}{{L}_{a}}{\omega}^{m}$$
$${\dot{\omega}}^{m}=\frac{d{\omega}^{m}}{dt}=\frac{{k}_{t}}{{J}_{m}}{I}^{a}\frac{{T}^{m}}{{J}_{m}}$$
$$\Delta \omega ={\omega}^{m}{\omega}^{p}$$
$${T}^{d}$$
$${i}_{t}=\frac{{n}_{ag}}{{n}_{p}}=\frac{{\omega}^{p}}{{\omega}^{ag}}$$
$$\dot{v}=\frac{dv}{dt}=\frac{{F}^{d}{F}^{aero}{F}^{climb}{F}^{roll}}{{m}_{eq}}$$
$${F}^{aero}=\frac{1}{2}{\rho}_{air}S{c}_{x}{v}^{2}$$
$${F}^{climb}=mgsin\alpha $$
$${F}^{roll}=\mu mgcos\alpha $$
$$\mu ={\mu}_{0}tanh\left(\frac{v}{{v}_{th}}\right)$$
3.2. Model Validation
$${V}^{fc}{E}_{oc}=\left[\begin{array}{cc}A& {R}_{ohm}\end{array}\right]\left[\begin{array}{c}{N}_{c}log\left(\frac{{I}^{fc}}{{i}_{0}}\right)\\ {I}^{fc}\end{array}\right]$$
4. Reference Speed Profile Optimization
In this study, the scenario is assumed to be known. In particular, the information about the race track that hosts the SEM is available and can be studied in advance. For this reason, the first step in the proposed EMS is to compute a speed profile that minimizes the energy used to complete a run attempt. Specifically, the Circuit Paul Armagnac in Nogaro, France, where SEM 2022 took place, was analyzed.
$$\begin{array}{cc}\hfill \dot{v}=\frac{{\eta}_{t}{i}_{t}{k}_{t}}{{r}_{r}{m}_{eq}}{I}^{a}\frac{{\rho}_{air}S{c}_{x}}{2{m}_{eq}}{v}^{2}& \frac{mg}{{m}_{eq}}({\mu}_{0}cos\alpha +sin\alpha )\hfill \end{array}$$
$$\begin{array}{cc}\hfill \underset{{I}^{a}}{min}& \phantom{\rule{1.em}{0ex}}{\int}_{{t}_{0}}^{{t}_{end}}{\left({I}^{a}\right)}^{2}\left(t\right)dt\hfill \end{array}$$
$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \phantom{\rule{1.em}{0ex}}{s}_{i}^{end}={s}_{i+1}^{0}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\in \{1,\dots ,{N}_{phase}1\}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}{v}_{i}^{end}={v}_{i+1}^{0}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\in \{1,\dots ,{N}_{phase}1\}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}{v}_{i}\left(t\right)\le {v}_{i}^{max}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\in \{1,\dots ,{N}_{phase}\}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}0\le {I}^{a}\left(t\right)\le {I}_{max}^{a}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}s\left({t}_{0}\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}v\left({t}_{0}\right)=0\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}s\left({t}_{end}\right)={l}_{track}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}\frac{{l}_{track}}{{t}_{end}{t}_{0}}\ge 25\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{h}\hfill \end{array}$$
5. MPC Design for Energy Management
The plant states are the vehicle speed v and the traveled distance s, while the DC–DC converter duty cycle ${d}^{a}$ and the switching variable $\Omega $ are the inputs. The former determines the duty cycle of the brushed motor driver and, based on the powertrain PS voltage, the armature voltage ${V}^{a}$, while the latter defines the power supply of the driveline (i.e., FC or SC).
The cost function (26) optimizes the vehicle’s fuel consumption while taking into account the fuel cell current ${I}^{fc}$ and the SC voltage ${V}^{sc}$ by acting on the binary control variable $\Omega $. According to the working principle of a proton exchange membrane fuel cell, the supplied current is linearly proportional to its consumption. On the other hand, by exploiting the SC’s recharge architecture, the hydrogen used in this task can be approximated by a linear function of the SC voltage ${V}^{sc}$. The scaling factor $\gamma $ is introduced to equalize the two quantities (i.e., ${I}^{fc}$ and ${V}^{sc}$). In addition, the SEM rules require that the SC voltage at the end of the run must be equal to that measured at the starting line.
$$\begin{array}{cc}\hfill \underset{\Omega}{min}& \sum _{k=1}^{{H}_{p}}\Omega {\left({I}_{k}^{fc}\right)}^{2}+(\Omega 1){\left(\gamma {V}_{k}^{sc}\right)}^{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\Omega \in \{0,1\}\hfill \end{array}$$
$$\begin{array}{cc}& \phantom{\rule{1.em}{0ex}}{V}_{k}^{sc,low}=\left\{\begin{array}{cc}{V}_{k}^{s{c}^{*}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{V}_{k}^{s{c}^{*}}\ge {\overline{V}}_{sc,min}\hfill \\ {\overline{V}}_{sc,min}\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}$$
$$\left\{\begin{array}{cc}{I}_{k}^{fc}={I}_{k}^{a}{d}_{k}^{a,fc}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{SC}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{recharged}\hfill \\ {I}_{k}^{fc}={I}_{k}^{a}{d}_{k}^{a,fc}+(\beta {V}_{k}^{sc}+\delta )\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$$
$$\left\{\begin{array}{cc}{I}_{k}^{sc}={I}_{k}^{a}{d}_{k}^{a,sc}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{SC}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{recharged}\hfill \\ {I}_{k}^{sc}={I}_{k}^{a}{d}_{k}^{a,sc}+{I}_{rch}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
$${V}_{k+1}^{sc}=\frac{{I}_{k}^{sc}}{{C}_{sc}}{T}_{s}+{V}_{k}^{sc}$$
$${s}_{k+1}={v}_{k}{T}_{s}+{s}_{k}$$
$$\begin{array}{c}\hfill {T}_{k}^{{d}^{*}}=\left[\frac{{v}_{k+1}^{ref}{v}_{k}}{{T}_{s}}+\frac{{\rho}_{air}S{c}_{x}}{2{m}_{eq}}{v}_{k}^{2}\right.\\ \hfill \left.+\frac{mg}{{m}_{eq}}\right]\left({\mu}_{0}cos{\alpha}_{k}sin{\alpha}_{k}\right)\end{array}$$
$${V}_{k}^{a}=\left(\frac{{I}_{k+1}^{a}{I}_{k}^{a}}{{T}_{s}}+\frac{{R}_{a}}{{L}_{a}}{I}_{k}^{a}+\frac{{k}_{e}}{{L}_{a}}\right){L}_{a}$$
$${d}_{k}^{a}=\frac{{V}_{k}^{a}}{{V}_{k}^{ps}}=\frac{{I}_{k}^{ps}}{{I}_{k}^{a}}{\eta}_{a}$$
To predict the system state at the next instant, it is assumed that the vehicle properly tracks the reference speed profile. As a result, the proposed Algorithm 1 can be used iteratively until the end of the run attempt.
Algorithm 1 Armature state prediction algorithm. 
Algorithm 2 EMS algorithm. 
6. Testing and Simulations
Extensive simulations were performed to evaluate the proposed EMS. Two different scenarios were used to evaluate the performance of the designed solution. The first, referred to as the simplified scenario, consisted of only one lap of the Circuit Paul Armagnac in Nogaro, France, with a length of 1571 m, and was used to analyze the influence of different factors on the performance. The second, called the full scenario, consisted of ten laps (15,710 m) and aimed to simulate an effective race test.
The MPC utilizes a prediction horizon of 2 and operates with a sampling time of 0.5 s. The latter selection ensures that the EMS operates at a slower pace compared to the other powertrain controllers, such as the DC–DC driver and FC cooling fan. The optimal solution is computed using a bruteforce approach that evaluates all possible combinations of the control sequence to determine the optimal result.
The scaling factor $\gamma $ is critical for the performance of the controller. A preliminary value of this parameter is estimated by calculating the relationship between the SC voltage and the FC current. This value is then adjusted in each simulation setup using a heuristic approach. The scaling factor is subjected to a gain between 13.3 and 3.6 with respect to the initial estimation.
The controller’s performance was evaluated using two different simulations with the same setup; in the first, only the FC was used to supply the powertrain, while in the second the designed MPC was used to perform EMS.
In the simplified scenario, the influence of different factors on the controller’s performance was evaluated, specifically, the impact of the prediction horizon on fuel saving and the potential delayed response of the FC. The results indicate that while an increase in the prediction horizon leads to higher computational time (approximately five times more), it does not provide significant performance improvements. On the other hand, there is an overall reduction in fuel consumption when a delay in the FC behavior is present. These findings can be attributed to the assumption of a constant FC voltage in the prediction horizon.
7. Conclusions
This paper proposes a lowcomputationalcost EMS for FC hydrogen vehicles with a secondary power source. The proposed solution can effectively handle possible constraints on the secondary PS state at the end of the test, such as the SC voltage or the battery’s state of charge. The results obtained in the simulation phase highlight that the controller is able to reduce fuel consumption, handle constraints, and overcome the presence of disturbances.
To simplify the investigation, a wellknown scenario was used in the study; this condition can be traced back to a driving cycle. Nevertheless, the architecture can be improved by the introduction of an online speed profile optimizer and a road profile estimator.
In future research, onbench and ontrack tests can be used to further validate the results. In addition, the control performance can be improved by exploring techniques such as MPC gain scheduling to increase the accuracy of online optimization or by developing a new DC motor driver capable of handling continuous commutation between the primary and secondary PSs.
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