Robust Tracking Control of a Three-Phase Bidirectional Charger for Electric Vehicle

&is paper presents a robust control strategy for an electric vehicle’s three-phase off-board bidirectional AC-DC battery charger. &e conventional constant current (CC) and constant voltage (CV) charging mode are considered to provide a fast-charging performance for the batteries.&e bidirectional charger also allows using of the vehicle as an energy storage system for the grid i.e., charging during the peak-off times and delivering the energy back to the grid during peak times of electrical consumption. In discharging mode, the bidirectional charger maintains constant active power flow to the grid with a given reference. For both cases, user of a robust state feedback controller with integral action is made in the DQ-synchronous frame. &e set of stabilizing gains of this controller are determined by a linear matrix inequality (LMI)-based optimization so that the convergence time to steady stead is minimized in the occurrence of the parametric uncertainties of the L-filter.&e efficacy of the proposed controller is verified through simulation and experimental results on 102.4V Lithium iron phosphate (LiFePO4) batteries.


Introduction
With a tremendous demand for renewable energy throughout the world, sustainable transportation methods draw a lot of attention in comparison to conventional transportation [1]. An enormous number of electric vehicles (EV) and plug-in hybrid electric vehicles (PHEV) are currently being utilized due to their eco-friendly behavior. For that reason, researchers, governments, and automakers worldwide continue to pursue efforts through policy and design to increase the EV market share. Many control techniques have been proposed to answer to the demands of rapid charging and energy-efficient battery charger.
In [1,2] conventional proportional integral (PI) control has been studied for single-phase bidirectional chargers. ese methods were proposed using a constant-current (CC) and a constant-voltage charging stage that produce faster-charging capability than that of only fixed voltage charging methods. e topology of these methods is singlephase based, so the amount of charging current of these chargers is less than those of the three-phase topology. Moreover, another main drawback of the PI controller of [1,2] is to gain tuning efforts for both inner-loop and outerloop controllers.
A bidirectional three-phase charger has been proposed using model predictive control [3]. is scheme provides a bidirectional power transfer with instantaneous mode charging capability and fast dynamic response. However, due to its only one charging stage (CV) from the grid to a vehicle (G2V), the batteries need more time to be fully charged. is method also requires a high computational power which results in high sampling frequency. Moreover, without integration of an integral control or a disturbance observer, this method may result in output offset-state error.
Deadbeat control has been presented in [4] for the DC-DC part of the bidirectional charger to regulate the charging current to the battery while PI control is also used for the AC-DC converter side to maintain the DC-link voltage constant. It is known that deadbeat control produces fast transient performance in which settling time reaches the steady state in just a few sampling periods.
is control method is sensitive to system uncertainty and measurement noise, particularly for high sampling frequency. e topology of [1,2], and [4] consist of a DC-DC converter for the CC/CV charging stage and an AC-DC converter for power factor and DC-link voltage control. In [5,6], DC-DC converters have been used for single-phase chargers and [7,8] for the three-phase charger to improve charging efficiency. ree operations of bidirectional chargers such as grid-to-vehicle (G2V), vehicle-to-grid (V2G), and vehicle-to-home (V2H) were considered in [5,8] to provide a full charging capability of bidirectional charger for an electric vehicle. Yet, classical controllers such as PI and PR were adopted which results in multiloop gain tuning.
Taking battery lifespan into account, AI-based manufacturing and management have been reviewed in [9].
is review provides a systematic survey of AI-based manufacturing and management solutions for enhancing battery health performance with a focus on recent challenges and opportunities. Data science-based full-lifespan management strategies have also been discussed in [10] to furnish useful reference points to support the design of data sciencebased battery management solutions during its lifespan, while a brand-new hologram to make full use of battery during full-lifespan will be formulated.
In the case of advanced control methods, battery temperature is considered a key part of the battery thermal management for battery operation safety and behavior. In [11], a constrained generalized predictive control was proposed based on a newly developed coupled thermoelectric model.
is method can be easily implemented in other battery charging applications to control the charge current and guarantee charging efficiency with a long lifespan. A leader-follower-based approach has been discussed in [12] to enable optimal charging control for the Li-ion battery pack.
is method is capable of reducing the computational burden and enhancing the robustness to minimize the negative impact of the cells' model bias.
In this paper, a robust tracking control of a three-phase bidirectional charger is presented for electric vehicle applications without using a DC-DC converter as an interface between a three-phase AC-DC converter and batteries. e LMI-based robust tracking control is a well-known method and has been proposed for three-phase inverters [13,14] and three-phase chargers [15,16]. is proposed bidirectional charger is capable of charging Tesla Model S batteries which range between 352 V and 402 V. e battery is charged with a constant current until the voltage reaches the recommended maximum voltage, then the voltage is maintained constant until the current consumed by the battery falls to a residual value. During the discharging operation mode, the energy stored in the batteries can be delivered back to the power grid. e vehicle-to-grid (V2G) technology is crucial from the viewpoint of the European Union and the bidirectional charger allows to use of the Full Electric Vehicle as an energy storage system for the electric grid, charging them in the peak-off times and delivering the energy back to the grid in peak times of electrical consume. For both cases, the use of a robust state feedback controller with integral action is made in the DQ-synchronous frame to provide stability and eliminate the steady-state error. Unlike a conventional MPC, this proposed controller is capable of removing an offset error to provide a good reference tracking output. is method also provides a systematic controller design by reducing the effort of gain tuning compared to the conventional PI controller and guaranteeing stabilized performance under parameter uncertainty. e set of stabilizing gains of this controller are determined by a linear matrix inequality (LMI)-based optimization so that the convergence time to steady stead is minimized in the occurrence of the parametric uncertainties of the L-filter. e consideration of an uncertainty model in this proposed method provides a wider range of good performance under the uncertain value of the L-filter than a deadbeat control. In the case of charging control, an outer-loop PI controller is employed to maintain the dc-current and dc-voltage for CC and CV control, respectively. e conventional phase-locked loop (PLL) is considered in this paper to obtain grid voltage phase angle.

System Description
A three-phase bidirectional charger circuit is shown in Figure 1. e dynamic of the line current is expressed in the abc-axis as follows: where Here, e dynamic in abc-axis (1) can be transformed into the dq-axis as follows [17]: 2L ω is the angular frequency of the AC voltage source. e inductor current i dq (t) and the control input u(t) in the dqframe satisfy the relationship as follows: where T(t) ≔ cos(ωt) cos(ωt + 2π/3) cos(ωt + 2π/3) −sin(ωt) −sin(ωt + 2π/3) −sin(ωt + 2π/3) , On the other hand, the output voltage v o (t) in the state equation (2) is governed by the following dynamic: where where i con. is the converter current and i bat. is an output current to the battery. e control input variable u(t) ≔ u d (t) u q (t) T must be constrained as follows [18]: e dynamic (2) can be transformed in the following discrete time with sampling period h [19] as follows: where

Model Uncertainties and Offset-free Control
In this section, the uncertainties model of the system and offset-free control are discussed. Suppose that the value of L and R in each phase are equal but vary in certain ranges below as follows: Here, we denote the matrices (A d , B d ) corresponding to the four possible combinations of the immoderate value of 1/ L and 1/R as (A i , B i ) (i � 1,2,3,4) and suppose that the matrices (A d , B d ) belong to the polytopic uncertain set Ψ below as follows: Journal of Advanced Transportation e uncertainties of the system can be any kind of variation but should lie within the range (9). e system uncertain range can be determined as follows: where R 0 and L 0 are the nominal value of the filter resistance and inductance, respectively, and µ (>1) can be considered as a tuning parameter.
To compensate for the offset error despite the system's uncertainty model, the control law based on [10] is employed for (8)as follows: where K and L are state feedback and integrator gains, respectively. Because of the integrator in (12), the steadystate error between the reference state x ref and the gridcurrent x will be compensated provided that the closed-loop system is stable.
generated by the outer-loop controller in the case of charging mode and can be computed with a given power reference in the case of discharging stage.

Robust Optimal Gain
Here, let us determine the gains of (12) so that the closedloop stability is provided to the system in the occurrence of parametric uncertainties. A systematic design method is proposed to obtain stabilizing state feedback gain K and integral gain L using LMI. From relations (8) and (12), we get the following: where output matrix C � 1 0 0 1 . Relation (13) can be simplified as follows: e control input u(k) can be given as follows: Suppose that D (k) � 0 to determine stabilizing gain F, then the closed-loop system can be computed as follows: e closed-loop system (16) is stable [20] if there exists a positive-definite matrix W such that It can be seen that the condition (17) holds for some By employing Schur complement [20] to (18), we get where H ≔ FS 0 , S ≔ W − 1 and S 0 ≔ S −1 0 . It should be noted that the matrices A u and B u contain the uncertain matrices A d and B d ; thus, (19) should hold for all (A d , B d ) ∈∈Ψ. To ensure that (19) is met for all (A d , B d ) ∈∈Ψ, we should verify that the condition is satisfied at all corners of the set Ψ i.e., Summarizing the above discussion, a closed-loop system (16) is asymptotically stable if there exist symmetric positive definite matrices S, S 0 and a matrix H such that (20) holds, and the stabilizing gain is given as follows: Suppose that W 0 < αW or en, (18) implies that for k > 0, It can be expected that a small α would give a fast convergence of z to the origin. erefore, to obtain optimal gain F such that the convergence time is minimized, the following optimization problem should be solved. (20)and (22) .
is optimization scheme is a generalized eigenvalue problem that can be solved efficiently by MATLAB Toolbox. e convergence of this control method can be found in [20].
It can be noted that the controller with optimal gain F determined by solving problem (23) satisfies the condition (17) and guarantees the overall closed-loop stability for any of the variations on the filter's inductance L and resistance R as long as it stays within the uncertain range (9).

Outer-Loop Charging Control
To perform the battery charging process, most of the battery manufacturers recommend two charging stages: constant current (CC) followed by constant voltage (CV). e battery is charged with a constant current until the voltage reaches the recommended maximum voltage, then the voltage is maintained constant until the current consumed by the battery falls to a residual value. e control of these two charging states is discussed in this section.

Constant Voltage (CV) Charging Mode.
Here, the outerloop PI control for constant voltage charging is discussed. Suppose that the dynamics of the inner-loop control in the previous section are considerably fast so that we could assume.
for some current reference I r (t). Let I r be the output of the outer-loop control, i.e., where v ref o is the constant voltage reference and v o (t) is the output voltage. From (5) and (25), we get e K i and K p gain can be determined by considering the characteristic polynomial of (27) and can be given as follows: for some appropriate value of ζ and ω 2 r ; we get e control diagram of the constant voltage control of the three-phase bidirectional charger is shown in Figure 2.
e battery voltage is fed-back to the out-loop controller, which produces a reference current i r � i ref q . A conventional phase-locked loop (PLL) is used in the controller to obtain the phase-angle of the grid voltage.

Constant Current (CC) Charging
Mode. In the constant current charging stage, the battery pack is charged with a fixed current until the voltage reaches the recommended maximum voltage, then switch to the constant voltage charging stage. For the control of this constant current charging mode, an outer-loop PI is utilized to generate a reference signal i r � i ref q for inner-loop robust control with the same concept as CV charging discussed in the previous section. e control structure of the proposed CC charging control is validated as shown in Figure 3. e control structure of both CC and CV charging modes are almost identical, however, the main difference is the use of outer-loop feedback; i bat and v o for CC and CV charging modes, respectively.

Discharging Control
Unlike CC and CV, this control scheme does not require the outer-loop controller; however, an uncomplicated computation of the reference state x ref is needed. e discharging controller allows the battery charger delivers constant power back to the grid with a given reference P ref .
Now, let us consider how to compute the reference state x ref for the proposed robust controller. e instantaneous active and reactive power can be represented in the αβ-frame [21] as follows:  where i αβ , e αβ , P o , and Q o are grid-current in αβ-frame, gridvoltage in αβ-frame, grid active power and grid reactive power, respectively. en, the relation (31) can be transformed to dq-frame as follows: i dq and e dq are grid-current and voltage in dq-frame, respectively. From (32), the grid-current can be computed as follows: In order to maintain a unity power factor, reactive power should be eliminated, and the reference can be computed as follows: where M � e 2 d + e 2 q . e reference state x ref allows the charger to deliver constant power back to the grid with a given reference P ref .
It can be noted that a negative power reference P ref results in reverse current flowing, in another word current flow from batteries to the grid. We can charge the batteries with constant power by the positive P ref . Moreover, the active and reactive power can be regulated directly by adjusting P ref and Q ref in (34), then the reference state can be obtained. e control structure of the proposed discharging control is validated as shown in Figure 4.

Simulation Results
is section presents the results of the simulation on a threephase AC/DC bidirectional charger to verify the efficacy of the proposed method. e simulation is implemented using the PSIM simulation tool and MATLAB LMI toolbox to obtain robust gain for the inner-loop controller. e parameters of the system are shown in Table 1. e control algorithm is conducted using a DLL block from Microsoft Visual Studio and the sampling rate is set to 10 kHz. An equivalent circuit in Figure 5 is used for the simulation studies.
e implementation of the proposed control strategy can be summarized as follows: Step 1: Derive the discrete-time model based on (8) using the nominal value of inductance L and resistance R.
Step 3: Compute the state feedback gain K and integrator gain L by solving the optimization problem (23).
Step 4: Implement the control (12) to the charger.

Journal of Advanced Transportation
Step 5: If the closed-loop system shows serious overshoot or becomes unstable, then adjust the uncertainty range, i.e., raise the value of µ and repeat the procedure from Step 3.
Step 6: After the closed-loop system becomes stable, then apply the outer-loop control for CC or CV.
In this paper, a synchronous reference frame phaselocked loop (PLL) is used to estimate the frequency and phase angle of the grid voltage [22]. e control structure of the conventional PLL is shown in Figure 6.   Here, the simulation performances of the proposed bidirectional charger are discussed. e transient response of the output current to the battery in CC mode is shown in Figure 7.
is is the first charging stage of the batteries followed by constant voltage (CV) charging mode. We can see that the proposed robust control provides a fast-transient performance and smooth output current. e steady-state performance of the three-phase grid current is validated in Figure 8. Figure 9 shows the transient performance of the output voltage in constant voltage CV charging mode. e bidirectional charger switches to this stage when the battery's voltage reaches a certain point after constant current CC charging mode. e outer-loop voltage control generates the   reference to inner-loop control (12) as shown in Figure 2. It can be seen that the output voltage to batteries is considerably fine.
For the simulation study in Figure 10(a), the discharging and charging of batteries with constant active power is validated. e battery is discharged with 200 W power at t � 0.11 s then switch to charging at t � 0.29 s with 200 W. It can be seen that the settling time to steady-state for both points is considerably fast and with smooth output power. Figure 10(b) shows phase-A grid voltage and current. We can see that the voltage and current are in 180°phase difference in discharging stage and change to in-phase during the charging period. It can be noted that all the aforementioned simulation results were carried out using the same uncertainty range (µ �1.35) because within this range we can obtain fast performance and no overshoot.
From Figure 11, a further study of using different uncertainty ranges is shown. e different selection of µ � (1.1, 1.35, 1.8, 2.5, and 4.0) in the equations (11a) and (11b) was made. From the figure, we can conclude that the use of µ � 1.1 results in overshoot performance while sluggish performances are obtained with µ � 2.5 and 4.0.
us, a choice of µ between 1.35 and 1.8 would be considerable.

Experimental Results
For the experiment, a three-phase AC/DC converter with Lfilter is used to charge 102.4 V Lithium iron phosphate (LiFePO 4 ) batteries.
e proposed robust control was implemented on the TMS320F28377D digital signal processor with a sampling rate of 10 kHz. e battery packs consist of 64 cells, and the 32 pairs are connected in series. e actual battery pack is shown in Figure 12. A slide transformer is used to drop the grid voltage to a proper level to charge 102.4 V battery packs and the experimental setup is shown in Figure 13. A simplified experimental setup circuit is provided in Figure 14. e parameters of the experimental prototype are shown in Table 2. Figure 15 shows experimental performances of battery charging current (a) and the phase-A grid-current (b) in steady stead with the reference current 5 A. It can be seen that the bidirectional charger can provide a smooth constant charging current to the batteries. Note that all the experimental data below is obtained by CAN communication from the control board.
In Figure 16, the steady state three-phase grid-current in charging mode is validated. We can see that a considerable balanced sinusoidal grid-current is obtained using this proposed control. Figure 17 shows a full process of charging using CC and followed by CV. e battery is charged with constant 5 A until t � 7400 s then switched to constant voltage mode. From the figure, we can observe that the battery voltage is increasing to maintain a constant current. However, during the constant voltage charging stage, the battery current is dropping as time goes on.
For the experimental results in Figure 18    We can see that the settling time to steady-state for both points is considerably fast. Figure 18(b) shows a three-phase grid-current in discharging mode followed by charging mode. In both discharge and charging cases, an almost unity power can be obtained.

Conclusions
is paper describes a robust control strategy for a threephase off-board bidirectional charger for an electric vehicle without using a DC-DC converter as an interface between a three-phase AC-DC converter and batteries. e conventional constant current (CC) and constant voltage (CV) charging mode are considered to provide a fast-charging performance for the batteries. e bidirectional charger also allows using of the electric vehicle as an energy storage system for the power grid. e proposed control consists of inner-loop robust control and outer-loop conventional PI control. For the inner-loop robust control, a state feedback controller with integral action is employed in dq-synchronous frame. e set of stabilizing gains of this controller are determined by an LMI-based optimization so that the convergence time to steady stead is minimized in the occurrence of the parametric uncertainties of the L-filter. It can be noted that the uncertainty range of the inductance and resistance can be considered a design parameter. us, its choice should be made depending on the resulting performances. From the simulation and experimental results, we can see that the proposed control for the bidirectional charger has considerable performance in both charging and discharging modes. e shortcoming of this proposed charger is that it is not capable of charging low voltage batteries due to the lack of a DC-DC converter as an interlink between the batteries and an AC-DC converter. e additional implementation of disturbance and state observers will be included in the future work of this research.

Abbreviations
CAN: Controller area network CC: Constant current CV: Constant voltage EV: Electric vehicle G2V: Grid to vehicle LMI: Linear matrix inequality PHEV: Plug-in hybrid electric vehicle PI: Proportional integral PLL: Phase-locked loop PR: Proportional resonant V2G: Vehicle-to-grid V2H: Vehicle-to-home L: Inductive filter R: Filter resistance C: Capacitive filter v o (t): Output voltage S abc : Switching vector in abc-frame i abc (t): Grid current vector in abc-frame i αβ (t): Grid current vector in αβ-frame i dq (t): Grid current vector in dq-frame u(t): Continuous time control input E m : Grid voltage magnitude ω: Angular frequency i con. : Converter current i bat. : Battery current x(k): Discrete-time state u(k): Discrete-time control input L min : Lower bounce filter inductance L max : Upper bounce filter inductance R min : Lower bounce filter resistance R max : Upper bounce filter resistance Ψ: Uncertainty set μ: Uncertainty range w(k): Integral state L: Integral gain K: State feedback gain x ref : Reference state W: Weighting matrix x(k): Augmented state K i : Outer loop integral gain K p : Outer loop proportional gain P o : Output active power

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.