Model predictive control based on single-phase shift modulation for triple active bridge DC-DC converter

Abstract

The triple-active bridge (TAB) converter is widely used in various applications due to its high efficiency and power density. However, the high-frequency (HF) transformer coupling between the ports presents challenges for controller design. This article presents a model predictive control (MPC) approach based on single-phase shift modulation for the TAB converter. The developed MPC offers improved transient performance, control flexibility, and precision, ensuring compliance with DC voltage regulations and achieving optimal solutions for port decoupling. The MPC utilizes a cost function to provide robust voltage regulation, and an algorithm based on Karush-Kuhn-Tucker (KKT) conditions is developed to derive closed-form solutions for optimal control parameters. To validate the performance of the TAB converter with the proposed MPC control, Typhoon 602 hardware-in-loop (HIL) experimental case study is conducted. Additionally, a comparison with previous works is carried out to confirm the effectiveness of the proposed method. The results of the HIL experimental setup and the comparative analysis demonstrate that the developed method is effective, providing faster dynamic characteristics and port power decoupling operation capability.

Introduction

Power electronics have become increasingly popular in the development of power systems due to their unique ability to convert electrical energy and transform parameters. Among various power electronics components, DC-DC converters are highly efficient electrical devices that combine both power electronics and high-frequency electrical energy conversion technology, providing power regulation and effective energy management¹. This type of converter is widely used in applications such as electrified vehicles, fuel cells, and more electric aircraft, making it one of the most popular topologies in the field of power electronics.

The dual active bridge (DAB) converter is a widely used DC-DC topology that consists of two H-bridges and an HF transformer. It allows for bidirectional power transfer through phase shift control. This topology offers several advantages including low device stresses, fixed-frequency operation, and the use of transformer leakage inductance as the energy transfer element. Moreover, it has a significantly higher power density than other low-frequency transformer solutions, making it an excellent option for onboard electric transportation systems. In^1,2, the DAB transfers power between the 270 VDC and 28 VDC busses of an electric aircraft’s distribution system. In³, an innovative interleaved DAB structure with a coupled inductor was introduced to improve the maximum transmission power.

Previous studies have introduced a multiport power conversion concept using triple active bridge (TAB) converters to improve power density and simplify the integration of converter groups into onboard electrical power distribution systems^4,5,6. Nevertheless, the design of a TAB converter faces a significant challenge related to the power coupling between ports. This is because the converter utilizes a high-frequency isolation transformer with a common magnetic core to transmit power and isolate the ports. Consequently, the power of different ports is coupled together through the magnetic circuit of the same transformer. As a result, a power disturbance in one port can cause voltage and current fluctuations in the other ports.

The coupling problem refers to the dynamic performance issue of the converter’s ports. Existing research is based on a traditional proportional-integral (PI) controller that controls the power flow among the ports. However, for multi-input multi-output (MIMO) systems, the linear control range limitations and oscillation characteristics of the PI-based voltage and current controllers, due to unquantified coupling relationships, have reduced the accuracy of steady-state control and dynamic performance. In^7,8, a TAB DC-DC converter is presented, which utilizes the same principle as the DAB converter^9,10. This principle uses a determined phase shift based on the PI controller between the transformer voltages. Furthermore, the three-port topology proposed in^11,12 combines phase-shift and PWM modulation to extend the operating range, leading to a more complex control system.

Researchers have investigated quantified modeling and control methods for specific MIMO converters, such as TAB and QAB. The study¹³ proposed a modeling technique that utilizes a decoupling matrix-based PI controller for a TAB converter to enhance its dynamic performance. Furthermore, the authors in¹⁴ presented a voltage balancing technique with reduced control variables for the TAB converter, which was achieved through hardware modification. In contrast, reference¹³ explored and validated the MAB converter’s generalized input impedance model, comparing it with TAB and QAB converters.

Model predictive control (MPC) has recently been recognized as a potential solution to enhance dynamic response time compared to classical techniques employed in power converter control. MPC is a model-based control technique that minimizes a cost function specifying the system’s behavior. It aims to reduce the difference between the predicted output and the reference¹⁴. This technique offers several advantages, including the ability to incorporate a system’s parameters/states into the prediction model, the achievement of fast dynamic response while avoiding the complexity of nested loops, the capability of multivariable control, and the inclusion of nonlinear effects and constraints in the control law. Furthermore, unlike classical controllers such as the PI controller, the effectiveness of MPC schemes is not dependent on properly tuning their parameters k_i and k_p, as MPC schemes have no parameters to adjust.

The MPC has been developed for the DAB converter, aiming to achieve a fast, dynamic response and high precision. In¹⁵, a lookup table-based MPC was presented for the DAB converter. However, it exhibited low control accuracy. Consequently, the finite control set-model predictive control (FCS-MPC) was introduced in¹⁶ to improve control accuracy. Furthermore, a non-linear MPC scheme was developed in¹⁷ to address sudden changes in the load, input voltage, and reference, which resulted in a low transient response with overshoot. However, the MPC schemes in references^18,19 exhibited a significant drawback of slow dynamic response and a considerable overshoot. Additionally, MPC may still suffer from a computational burden due to its requirement to solve optimization problems when predicting a long time horizon.

A power decoupling-based configurable model predictive control (PDC-MPC) strategy was developed in²⁰ to deal with problems related to the control strategy of the converter. The strategy aims to achieve good transient performance, high control flexibility, and high precision to comply with DC voltage regulations. The PDC-MPC operates in two phases: prediction of the DC current through a binary search and decoupling of the desired power of each isolated virtual branch to its phase shift angles under the single-phase shift (SPS) modulation strategy. Due to the need for additional decoupling links when designing the controller, the control’s computational complexity increases, thus affecting the system’s dynamic performance. However, the proposed method’s design and implementation are too comprehensive to utilize. Furthermore, the binary search algorithm-based MPC still needs to improve when parameters are uncertain because inaccurate prediction using models with inaccurate parameters leads to performance degradation or even instability.

This paper proposes an MPC to improve the dynamic characteristics of a TAB converter operating beyond its rated values while eliminating the effect of coupling among the ports and ensuring a fast and accurate response. To achieve this, a novel algorithm based on the Karush–Kuhn–Tucker (KKT) conditions is utilized to derive closed-form solutions for the global optimal control parameters, resulting in simplified calculations and succinct closed-form solutions for the optimization problem. The paper’s contribution can be summarized as follows:

• The MPC developed here can extend the control range and improve control precision during dynamic operations when compared to an optimal PI controller. This is achieved by quantifying the prediction model and accounting for the nonlinear voltage characteristics of the system. As a result, the MPC can eliminate steady-state voltage ripples and reduce oscillation during transition.

• The adaptive step in the MPC algorithm has been designed to enable a fast transition in the system. This feature allows the system to adjust to changing conditions and respond promptly and effectively.

• The model’s robustness to parameter uncertainties is another noteworthy feature of the developed MPC.

• The developed MPC has a low computational burden as it uses the Karush-Kuhn-Tucker (KKT) conditions, simplifying the calculation and providing concise closed-form solutions for the optimization problem without the need for complex optimization methods.

The paper is structured as follows: "Configuration of TAB converter topology" section provides a comprehensive outline of the topology of the TAB and SPS modulation. "Proposed Model Predictive Control (MPC)" section presents the proposed adaptive model predictive control. "Parameter sensitivity analysis" section discusses the robust term of the model parameters analysis. The operating principle of the proposed control strategy is described in "Operating of the proposed control strategy" section. In "Simulation verification" section, the hardware-in-loop (HIL) experimental and simulation results are discussed to validate the effectiveness of the proposed control strategy. Finally, "Experimental verification" section shows the conclusions of this work.

Configuration of TAB converter topology

Circuit description

The topology of the TAB converter is presented in Fig. 1^21,22. The converter comprises three H-bridges that generate a square-wave voltage, which is then coupled to a three-winding transformer. As depicted in the figure, the fuel cell source is connected to port 1, while ports 2 and 3 are connected to the load and battery energy storage system (BESS). V_FC, V’_L, and V’_Bat illustrate the voltage levels at ports 1, 2, and 3, respectively. Thus, u_T1, u’_T2, and u’_T3 denote the AC voltages of the three ports. i_L1- i’_L3 and v_L1- v’_L3 indicate the currents and voltages of the leakage inductance L₁- L^’₃, which are employed for the energy transfer elements.

Figure 1.

Isolated three-port DC-DC converter topology^6,7,8,21.

In this scenario, a single-phase shift control method has been utilized to determine the direction and amount of power flowing through the other ports. The voltage signals, shown in Fig. 2a, are shifted by φ₁₂ and φ₁₃ with respect to u_T1, which is used as the reference, and both signals have the same duty cycle. T is the switching period, while φ₁₂ and φ₁₃ ∈ [-0.5, 0.5].

Figure 2.

(a) Simplified high-frequency voltages and currents. (b) Y-model. (c) Δ-equivalent circuit for the TAB converter²².

All variables and parameters of ports 2 and 3, which are referred to as the transformer primary side port 1, as follows: V_L= V’_L/n₂, u_T2= u’_T2/n₂, v_L2= v’_L2/n₂, i_L2= i’_L2n₂ and L₂ = L^’₂/n₂² for port 2. Further, V_Bat= V’_Bat/n₃, u_T3=u’_T3/n₃, v_L3=v’_L3/n₃, i_L3=i’_L3n₃, and L₃ = L^’₃/n₃² for port 3, where the turn’s ratios V_FC:V_L:V_Bat between the ports are: 1:n₂:n₃ for port two and port three, respectively.

Figure 2c presents a Δ-equivalent circuit for a TAB converter, which simplifies power flow analysis between the ports. By transforming from Y-mode to Δ-mode, the inductances’ relationships can be expressed as follows²²:

1

Power flow management

Based on the Δ-equivalent circuit in Fig. 2c and the instantaneous voltage in Fig. 2a and Y-model in Fig. 2b, the instantaneous current of the Δ-equivalent circuit is presented as:

2

The direction and amount of power transfer can be controlled by adjusting the phase shift between three-port converters^23,24. The expressions for powers P₁, P₂, and P₃ can be obtained using the following equations:

3

where I_x and V_x represent the AC current and voltage of the respective port. The transmission power between ports, denoted as P₁₂, P₂₃, and P₁₃, refers to the power that flows through the corresponding branches L₁₂, L₂₃, and L₁₃ in the Δ-equivalent circuit. To calculate P₁₂, P₂₃, and P₁₃, use the formula (2):

4

The power flow at each port results from the combined power flow through two inductor branches. To summarize, we can calculate the active power of the three ports as follows²¹:

5

Modeing of the TAB converter

The initial stage in controller design involves deriving the small signal of the converter, with the DAB converter viewed as a current source exhibiting controlled average current throughout a single switching cycle. This method enables us to obtain the model of the three-port converter, with the average currents of the three ports i₁, i₂, and i₃ denoted by I₁, I₂, and I₃, respectively. These average values depend on the two-phase shifts and can be represented using Eq. (5)²¹.

6

Figure 3 depicts a typical model equivalent circuit. The average switching periods of i_FC, i_L, and i_Bat are denoted as I_FC, I_L, and I_Bat, respectively. Additionally, any variation in the DC source voltage is neglected, leading to I_FC = I₁. As a result, the stabilizing capacitor C₁ can be omitted.

Figure 3.

Average model of the TAB converter²².

Applying KCL with 𝑉₂ and 𝑉₃ as state variables within the average equivalent circuit illustrated in Fig. 3, the state-space equation for the TAB converter can be expressed as follows:

7

Model predictive control requires a discretized mathematical model of the control system. To achieve this, the forward Euler method is applied to discretize Eq. (7).

8

The variables V_x(k) and I_x(k) (where x = 2, 3) represent the average voltage and average output current at port x during the k^th switching cycle, respectively. V_x(k + 1) represents the average voltage at port x during the k + 1th switching cycle. By substituting Eq. (6) into (8), the state-space equation of the discretized TAB converter can be obtained.

9

The mathematical model of the TAB converter, as described by Eq. (9), confirms that it is a nonlinear multi-input multi-output (MIMO) system. The voltages and currents at ports 2 and 3 are coupled through the phase-shift duty cycle φ₁₂ and φ₁₃. Neglecting this coupling during the controller design process can lead to interference between the ports, potentially compromising their dynamic control performance. Designing the controller for the TAB converter requires additional decoupling links, which increase the computational complexity and affect the system’s dynamic performance. To overcome this issue, the paper proposes model predictive control, which optimizes the dynamic performance of the port and achieves decoupling control of the TAB converter port.

Developed Model Predictive Control (MPC)

Prediction model

By ensuring that the voltage ratio of the three-winding transformer aligns with the rated voltage ratio of the three DC ports, V₂ and V₃ can be expressed as functions of V₁ and the transformer turns ratios. Specifically, V’₂ = V₁n₂/n1, and V’₃ = V₁n₃/n₁. This simplifies the discrete state-space Eq. (9) of the TAB converter using V₁ as a reference voltage, ensuring that the proper relationships between V₁, V₂, and V₃ are maintained through their respective turn ratios. Consequently, the discrete state-space Eq. (9) of the TAB converter is rewritten, resulting in the following prediction model:

10

According to the proposed topology, the load is connected to port two, while port three is linked to a battery energy storage system. Port two is set to voltage control mode, while port three requires both current and voltage control modes to manage the battery’s SOC effectively. The prediction model for the output current can be articulated as follows:

11

Cost function

The MPC process can be explained as follows: At each switching cycle (k), the control variable (Δ(k + 1)) is calculated based on the current state variables (ε(k)) and control quantity (q(k)). This calculation aims to minimize the cost function over a finite time horizon (µ_m) of switching cycles, which helps to achieve optimal control in the next switching cycle. To reduce the computational complexity of the controller and maximize the switching frequency, the predictive optimization time domain horizon (µ_m) is set to 1 in this study. Therefore, to achieve optimal control, a multi-objective optimization cost function is established based on the control objectives. The cost function used in this work is as follows:

12

To solve the MPC problem while meeting the phase shift duty ratio constraints and minimizing the cost function, we can express it in the following way:

13

where V₂^ref and V₃^ref represent the voltage references of ports two and three, while I_Bat^ref is the current reference for port three. The parameter ξ represents the control mode associated with the energy storage at port three: ξ = 1 signifies that the energy storage port operates in voltage control mode, whereas ξ = 0 indicates current control mode. The cost function is designed to consider three control objectives: the voltage at port two and both the voltage and current at port three. Furthermore, the optimal phase shift duty ratios φ₁₂ and φ₁₃ are calculated during each switching cycle (k) to minimize deviation in all three control objectives. This approach effectively decouples the dynamic processes at each port.

Controller design

To address the problem outlined in Eq. (13), the KKT conditions will be employed. These conditions are beneficial for handling complex issues involving constraints. The primary objective is to minimize the cost function J(k), which quantifies the converter’s performance. By applying the KKT conditions, the optimal phase shift duty ratios φ₁₂ and φ₁₃ can be determined to minimize the cost function while satisfying the specified constraints^{25,26,27,28,29, 21}.

To apply the KKT conditions, we must formulate the Lagrangian (L). This Lagrangian includes the original cost function and the constraints along with their associated Lagrange multipliers:

14

where λ₁, λ₂, ε₁, and ε₂ are the Lagrange multipliers. At the optimum, Tucker’s conditions must be satisfied:

Stationarity

The partial derivatives of the Lagrangian with respect to φ₁₂ and φ₁₃ must be zero.

15

Primal feasibility condition states

that the constraints must be satisfied.

16

Dual feasibility

The Lagrange multipliers must be non-negative.

17

Complementary slackness

The product of each Lagrange multiplier and its corresponding constraint must be zero.

18

Thus, the overall conditions can be restated as follows²¹:

19

The process of MPC strategy for converters operates in the following manner:

1. Initialize Variables: Start with initializing the variables and parameters, including V₂^ref, V₃^ref, I_Bat^ref and ξ.

2. Set Control Mode: Based on the value of ξ, determine whether the control mode for the energy storage port is voltage control (ξ = 1) or current control (ξ = 0).

3. Calculate Phase Shift Ratios: For each switching cycle k, calculate the phase shift duty ratios φ₁₂ and φ₁₃ using the equations provided, ensuring that − 0.5 ≤ φ₁₂, φ₁₃ ≤ 0.5.

4. Update Control Quantities: Update the control quantities V₂, V₃, and I_Bat for the next switching cycle (k + 1) using the formulas (10) and (11).

5. Compute Cost Function: Calculate the cost function J(k) to evaluate the deviation of the control quantities from their reference values.

6. Optimization: Apply the Tucker’s conditions to minimize J(k), subject to the constraints on φ₁₂ and φ₁₃.

7. Iterate: Repeat the process for the next switching cycle until the system reaches the desired control objectives.

8. Check Convergence: If the deviation of the control quantities from their reference values is within acceptable limits, the process is complete.

9. Adjustment: If not, adjust the control parameters and return to step 3.

By solving Eq. (19), the optimal values of φ₁₂ and φ₁₃ can be determined while satisfying the constraints to minimize the cost function J(k). This minimization is essential to achieve dynamic decoupling control of the multi-port system, as it reduces the deviation of control quantities from their reference values. Subsequently, the expressions for φ₁₂ and φ₁₃ under MPC are derived as follows:

20

Normalize these current expressions by using the base current I_base to obtain the per-unit currents for Eq. (20):

21

Thus, from Eq. (7) the current at ports two and three at the next time step I₂(k + 1) and I₃(k + 1) are computed as;

22

Based on Eqs. (20), (21), and (22), the subsequent outcomes can be derived:

23

Thus,

24

Based on Eq. (25), it is evident that φ₁₂ and φ₁₃ demonstrate a dual relationship, which further simplifies the equation set to be solved as follows:

25

From the numerical solution of Eq. (25), we can determine the relationship between phase-shift duty cycles φ₁₂ and φ₁₃ and the corresponding currents ΔI₂ and ΔI₃.

Parameter sensitivity analysis

MPC is a control strategy that relies on the mathematical model of the control object. However, some parameters in the control model may require actual measurements to be determined accurately. The precision of the mathematical model is affected by the measured error, which, in turn, affects the control accuracy of model prediction. Therefore, analyzing the parameter sensitivity of the control is crucial and is usually done in the design process of the model predictive controller. If the sensitivity is too high, additional compensation control links may be required to reduce parameter dependence.

While solving the MPC problem mentioned in this chapter, three critical TAB converter model parameters, C₂, C₃, and L_s, are utilized. Afterward, an analysis is conducted to determine the impact of errors in these three parameters on control accuracy.

In order to facilitate quantitative analysis, the degrees of mismatch for the three model parameters are defined as follows:

26

where Q_Ls, Q_C2, and Q_C3 are the mismatch degrees of inductance L_s and DC capacitors C₂ and C₃ at ports two and three. Each is defined as the ratio of the component’s nominal value used in control (δ_Ls, δ_C2, δ_C3) to its actual value (L_s, C₂, C₃).

The control parameters δ_Ls, δ_C2, and δ_C3 are used to calculate the currents ΔI₂ and ΔI₃ for ports two and three in Eq. (23), which are defined as δΔI₂ and δΔI₃, respectively. The calculation process is as follows:

27

The phase-shift duty ratios φ₁₂ and φ₁₃, calculated based on the currents δΔI₂ and δΔI₃, satisfy the following equation:

28

After modulating with φ₁₂ and φ₁₃, the actual current of the TAB converter follows this relationship:

29

Further, by substituting Eq. (28) into (29), we can get:

30

Meanwhile, substitute Eq. (27) into (30), which gives us the following result:

31

Since items Q_Ls = δL_s/L_s, Eq. (31) can be rewritten as:

32

Considering the steady state, I₂ = I_L, I₃ = I_Bat, Eq. (32) can be further expressed as:

33

Since the energy storage device connected to port three operates in two different modes: voltage control and current control, it is necessary to analyze separately the influence of parameter errors on the voltage and current control parameters when analyzing parameter sensitivity.

Parameter sensitivity in voltage control mode

Assuming the converter operates at rated conditions and the power transmitted through ports two and three is at the rated power P_rated, the following can be established:

34

Substituting the above equation into (33) yields the following result:

35

The formula above shows that if ports two and three are in the voltage control mode, the steady-state error of the port voltage will only be affected by the mismatch degree of the inductance parameters Q_LS. On the other hand, the mismatch degree of the capacitance parameters Q_C2 and Q_C3 will not have an impact on the steady-state error of the voltage. Thus, an error in voltage can be defined as follows:

36

An evaluation of the voltage error, as delineated in Eq. (36), utilized the circuit parameters of the TAB converter specified in Table 1. A graphical representation derived from Eq. (35) illustrates the voltage error relative to variations in the inductance parameter mismatch, depicted in Fig. 4. The analysis reveals that within an inductance mismatch range of 0.5 to 1.5, the steady-state voltage error consistently remains at 0.32% for ports two and three. These results suggest that the MPC implemented exhibits minimal dependency on model parameters, thereby facilitating high control accuracy without additional compensatory components.

Table 1.

Circuit parameters of the TAB converter model.

Description	Value
Port 1 fuel cell voltage, VFC	300 V
Port 2 load side DC voltage, VL	150 V
Port 3 battery voltage, VBat	90 V
Transformer turns ratio, 1:n2:n3	1:0.5:0.3
Leakage inductance, L1	1µH
Leakage inductance, L2	48.897µH
Leakage inductance, L3	16.275µH
Capacitor, C1	120µF
Capacitor, C2	300 µF
Capacitor, C3	250 µF
Switching frequency, fs	20 kHz
Load Resistance, RL	48Ω
Nominal power, Pn	5 kW

Figure 4.

The correlation curve between voltage error and transformer inductance mismatch.

Parameter sensitivity in current control mode

Based on Eq. (33), it can be concluded that the steady-state error of the current in the current control mode is only affected by the mismatch degree of the inductance parameter Q_LS. It is not affected by the mismatch degrees of the capacitance parameters Q_C2 and Q_C3. By substituting ξ = 0 into Eq. (33), we can derive the following equation:

37

Further, the current error can be obtained as follows:

38

According to the formula provided, a curve of the steady-state error of the current can be drawn with respect to the mismatch of the inductance parameters, as shown in Fig. 5. This curve demonstrates that the steady-state error in port current control is greatly influenced by the mismatch in inductance parameters when operating in the current control mode. This impact is particularly noticeable when the value of Q_LS is less than 1. In such cases, the error increases sharply as Q_LS decreases. Therefore, applying additional compensatory control stages when operating in the current control mode is crucial to reduce the steady-state error and achieve the desired control objectives.

Figure 5.

The correlation curve between current error and transformer inductance mismatch.

Operating of the proposed control strategy

Figure 6 illustrates the proposed MPC block diagram for the TAB converter. This approach is based on optimizing solutions and parameter sensitivity analyses from the previous MPC problem. The control strategy includes a voltage control loop, a current control loop, and a final phase shift duty cycle acquisition.

Figure 6.

Block diagram of the proposed MPC scheme for TAB dc-dc converters.

Initially, the voltage control loop is designed using Eq. (22). The voltage error is calculated by comparing the reference voltage and feedback voltage values and then amplified through C₂/T_sh and C₃/T_sh. The load current is then added in a feedforward manner to quickly reflect load changes, improving the control system’s dynamic response time. Finally, the results are normalized to obtain the normalized currents ΔI₂ and ΔI₃.

Next, the normalized current ΔI₃ is obtained directly through reference current normalization in the current control loop. However, the model’s parameter dependency is too high, and the steady-state error is too large to meet the control requirements. To reduce the parameter dependency, an additional compensating integral component is added to the current control loop to compensate for the steady-state error in the current control. The integral parameter k_i must be obtained through tuning.

Lastly, the voltage/current control loop is used to obtain the normalized currents ΔI₂ and ΔI₃. These are then used to solve Eq. (25) to acquire the final phase shift duty cycle, which becomes the control variable for the next switching cycle.

Simulation verification

To validate the improved dynamic response characteristics, mitigate port coupling effects, and simulate the parameter resistance capability of the TAB using the MPC control strategy, a MATLAB/Simulink control model was developed. A comparison was conducted between this approach, PI control, and the unit matrix decoupling control method (UMD) outlined in⁷, using the same simulation model parameters. The specific parameter configurations are provided in Table 1.

Soft switching analysis

ZVS can be achieved in the switching devices of HB1 by ensuring that the current through the leakage inductor L₁ is negative at the switching instant t₀ when switches S₁₁ and S₁₄ are activated, as illustrated in Fig. 2a. This means that before S₁₁ and S₁₄ turn on, the current i_L1 flows through the antiparallel diodes of S₁₁ and S₁₄ if i_L1(t₀) < 0. Consequently, switches S₁₁ and S₁₄can be turned on under ZVS conditions¹². The same ZVS conditions apply to HB2 and HB3.

The simulation waveforms of the currents and voltages for the three bridges under PI control, unit matrix decoupling control, and proposed MPC control are depicted in Fig. 7a, b, and c, respectively. All control methods demonstrate that all three bridges are soft-switched. As explained in⁶, the ZVS conditions for each bridge require negative currents at u_T1, u_T2, and u_T3 rising edges and positive currents at falling edges.

Figure 7.

Simulation results of the TAB converter, illustrating soft-switching under different control strategies: (a) PI control, (b) unit matrix control, and (c) proposed MPC control strategy.

Performance during the initial stage of the transition

Figure 8 illustrates the comparison of output voltage waveforms from the TAB converter system using both the PI control and unit matrix decoupling (UMD) control strategy. In the initial stage, the PI-controlled system had a settling time of 0.9s, while the UMD control strategy achieved faster transition performance, reaching a steady state in just 0.045s. Despite this, the PI controller exhibited a more significant steady-state voltage ripple of 8 V.

Figure 8.

Simulation voltage and phase shift ratio waveform: initial and steady state of ports two and three with the PI control and unit matrix decoupling control.

Figure 9 presents the output voltage waveforms obtained using the proposed MPC control strategy. The initiation transition is faster compared to both the PI control strategy and the unit matrix decoupling control strategy, as illustrated in Table 2. The steady-state voltage exhibited a reduced peak-to-peak ripple, decreasing from 8 to 1.2 V.

Figure 9.

Simulation voltage and phase shift ratio waveform: initial and steady state of ports two and three with the proposed MPC control strategy.

Table 2.

Comparison between three control strategies.

	PI [6]	UMD [7]	Proposed MPC
Initiation	0.9 s	0.045 s	0.03 s
Steady-state ripple (p-p)	8 V	5.3 V	1.2 V
Steady-state error	0.7 V	0.9 V	None
Load Change 30-75%	0.83s	0.25s	0.05s

Load disturbances

A specific design was implemented to evaluate the response characteristics of the TAB converter using the proposed MPC for load power step changes. In the simulation process, the control parameters for ports two and three voltage loops, under PI control, were uniformly set to k_p= 0.0004 and k_i= 0.055. In the case of unit matrix decoupling control, the parameters were k_p= 0.42 and k_i= 5. Figure 10 depicts the simulation waveform resulting from the sudden change in load power at port two.

Figure 10.

Simulation analysis of voltage, current, and phase shift angle waveforms in response to step changes in load power at port two under various control strategies: (a) PI control, (b) UMD control, and (c) proposed MPC control.

Under PI control, as depicted in Fig. 10a, a sudden step increase in load power occurs at t = 0.1s. This results in a fluctuation of 2 V in the output voltage at port two and a fluctuation of 0.2 A in the current at port three. The output voltage at port two reaches the target of 150 V after 6ms. Similarly, at t = 0.2s, when the load at port is reduced, the output voltage at port two experiences a maximum fluctuation of 1.8 V, while the current at port three fluctuates by 0.3 A. The output voltage at port two reaches the control target after 8ms.

With the unit matrix decoupling control depicted in Fig. 10b, a sudden step increase in load power at t = 0.1s resulted in a voltage fluctuation of 1.3 V at port two and a current fluctuation of 0.15 A at port three. The output voltage reached the control target of 150 V after 4ms. At t = 0.2s, the load port was reduced, causing the port two output voltage to fluctuate by a maximum of 1.5 V and the port three current to fluctuate by 0.25 A. After 5ms, the output voltage reached the control target. These results highlight the significant coupling characteristics of PI control while demonstrating that unit matrix decoupling control can effectively mitigate power coupling between the two ports.

In the proposed MPC control strategy, at t = 0.1s, as depicted in Fig. 10c, the voltage at port two was stabilized within 0.01ms. It exhibited a fluctuation amplitude of 0.25 V (± 0.125 V), while the current at port three remained stable. Subsequently, at t = 0.2s, during the load drop at port two, the proposed MPC control significantly enhanced the dynamic response speed of the converter, with a dynamic response time of about 0.05ms and minimal load voltage fluctuations. Additionally, MPC ensured power decoupling between ports two and three, preventing power transients at the load port from causing a disturbance at the energy storage port. This decoupling effect was superior to that of the diagonal matrix decoupling control.

Changes in current of battery energy storage port

The simulation setup was developed to evaluate the response characteristics of the MPC control of a TAB converter to sudden changes in the current of battery energy storage devices at port three.

At t = 0.1s, there was an increase in the current control target at port three, resulting in a sudden change in current at the energy storage port. Under PI control, depicted in Fig. 11a, the output voltage drop of 1.8 V at port three stabilizes after 6ms, with the current fluctuating at 1.5 A. Meanwhile, with unit matrix decoupling control, as illustrated in Fig. 11b, a voltage drop of 1 V is observed at port three, stabilizing after 4.5ms, with a current fluctuation of 0.7 A. In contrast, the MPC control scenario presented in Fig. 11c showcased superior port dynamic control performance, as indicated by the stable load port voltage. These results highlight the effectiveness of the proposed MPC strategy in this paper.

Figure 11.

Simulation analysis of voltage, current, and phase shift angle waveforms in response to step changes of the current at port three under various control strategies: (a) PI control, (b) UMD control, and (c) proposed MPC control.

At t = 0.2s, a sudden drop in current occurred at port three, causing the output voltage at port three to fluctuate by a maximum of 1.9 V. After 9ms, the voltage stabilized, while the current experienced a fluctuation of 0.9 A for PI control. With the unit matrix decoupling control, a maximum voltage fluctuation of 0.7 V at port three was observed, stabilizing after 7.5ms, with a current fluctuation of 0.5 A, as illustrated in Fig. 11b. In contrast, the load port voltage remained stable under the MPC, indicating improved port dynamic control performance offered by the proposed MPC strategy.

Furthermore, these simulation experiments demonstrate that the controller, based on the proposed MPC control theory, not only achieves power decoupling between ports but also significantly enhances the dynamic control performance of TAB ports.

Disturbances in fuel cell input at port one

The fuel cell input side is susceptible to disturbances, which can affect the accuracy of compensation control. To evaluate the effectiveness of the proposed MPC control, the design sets control targets for ports one, two, and three at 300 V, 150 V, and 90 V, respectively. Port two maintains a constant power output of 2 kW, with the input voltage fluctuation set at 50 V over a cycle of 0.05s. The simulation result is illustrated in Fig. 12.

Figure 12.

Simulation waveform of the converter’s input and output voltage in response to step variations in the fuel cell input voltage under various control strategies: (a) PI control, (b) UMD control, and (c) proposed MPC control.

Based on the findings from Fig. 12a and b, it is evident that when utilizing PI and unit matrix decoupling control, variations in the input voltage at port one lead to fluctuations in both the output voltage and current at port two, as well as in the output current at port three. This reveals the significant coupling characteristics between the ports and the slow dynamic performance of the PI and matrix decoupling control in response to fluctuations in the input voltage step.

In contrast, the proposed MPC control strategy stabilized the voltage and current at port two, with no fluctuations observed. Similarly, the current at port three also showed no fluctuations. The MPC demonstrated a fast dynamic response and essentially zero overshoots, with a very short control current adjustment time. These results indicate the effectiveness of the proposed MPC strategy.

Simulation Verification of mismatch conditions on fuel cell side supporting capacitance

The performance of MPC is affected by the circuit parameters of the converter. This study analyzes the sensitivity of the parameters, as detailed in "Parameter sensitivity analysis" section, to validate the proposed MPC control and examine its accuracy. The control objective is to maintain a constant output voltage of 150 V and 90 V for ports two and three, respectively, while keeping the fuel cell voltage input at a constant 300 V. Figure 13 illustrates the variations in system input voltage and output currents at port three for different values of the supporting capacitor C₁ (100µF, 200µF, 300µF, and 150µF).

Figure 13.

Simulation waveform of the converter’s input and output voltage under conditions of mismatched fuel cell support capacitance.

The simulation experiment shows that the output characteristics of the TAB converter based on MPC remain substantially unaffected within the range of capacitance variation. Furthermore, the voltage on the fuel cell side maintains a stable input, validating the analysis that fuel cell-supporting capacitance variations have no impact on the proposed MPC control.

Experimental verification

In order to evaluate the effectiveness of the proposed MPC strategy, as depicted in Fig. 1, we have established a semi-physical experimental platform using the Typhoon real-time simulation device HIL 602³⁰. The simulated three-port converter uses HIL 602, and the control component is developed with a control board that includes a TMS320F28335 DSP and other interfacing circuits. This control board is directly connected to the HIL 602 for real-time simulation. A visual representation of the platform setup can be seen in Fig. 14, and the main circuit parameters of the experiment align with the simulation. Additionally, Fig. 15presents a synoptic scheme of the proposed HIL setup. To assess the effectiveness and feasibility of the proposed MPC strategy, we conduct a comparative analysis of its control performance against PI control, the diagonal matrix decoupling control proposed in⁷, as well as a power decoupling-based configurable model predictive control (PDC-MPC)outlined in²⁰.

Figure 14.

HIL real time experimental platform.

Figure 15.

Synoptic scheme of the proposed HIL setup.

Soft switching analysis

The steady-state waveforms of voltage and current at the transformer side for PI control, UMD control, PDC-MPC control, and the proposed MPC strategy are illustrated in Fig. 16(a)–(d). It is evident that all control methods exhibit the operation of the three bridges in a ZVS manner.

Figure 16.

Experimental waveforms of leakage inductance current and AC voltage for ports 1, 2, and 3, illustrating soft-switching with the control strategy: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

Load disturbances

In the initial test case shown in Fig. 17a, it was observed that under the PI control, an increase in i_L from 10 A to 25 A caused coupling effects on the voltage and current at ports 2 and 3, as well as on the overall dynamic performance. This resulted in fluctuations of 25 V in V_L and 1.3 A in i_Bat. Additionally, a reduction in load at port 2 led to a maximum fluctuation of 30 V in V_L and 1.8 A in i_Bat.

Figure 17.

Experimental waveforms of voltage and current in response to step changes in load power at port two under various control strategies: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

Similarly, under the UMD control illustrated in Fig. 17b, a sudden increase in load power caused a 10 V fluctuation at port 2 and a 1 A current fluctuation at port 3. Upon subsequent load reduction at the port, V_L fluctuated by a maximum of 15 V, while i_Bat fluctuated by 1.2 A.

Under the PDC-MPC control shown in Fig. 17c, a sudden load power increase resulted in an 8 V fluctuation at port 2 and a 0.8 A current fluctuation at port 3. Upon reducing the load at the port, V_L experienced a maximum fluctuation of 10 V, with i_Bat fluctuating by 1 A.

In contrast, the proposed MPC control strategy, depicted in Fig. 17d, effectively stabilized V_L with a fluctuation amplitude of 4 V (± 2 V) when i_L increased from 10 A to 25 A, while i_Bat remained stable. During the load reduction at port 2, the proposed MPC control significantly improved the dynamic response speed, resulting in minimal load voltage fluctuations and a rapid response time. Moreover, MPC facilitated power decoupling between ports 2 and 3, effectively preventing load port power transients from affecting the energy storage port. This decoupling performance exceeded that of the PDC-MPC, UMD, and PI control methods.

Changes in current of BES port

In the second scenario, an abrupt increase in i_Bat from 20 A to 35 A at port three occurred. Under PI control, as depicted in Fig. 18a, the V_Bat decreased by 20 V, with the current fluctuating at 5 A, and a fluctuation occurred in port two. Similarly, when i_Bat decreased from 35 A to 20 A, the output voltage at port three fluctuated by a maximum of 15 V, while the current experienced a fluctuation of 6 A. In contrast, under UMD control shown in Fig. 18b, an increase in i_Bat led to an 8 V voltage drop at port three, with a current fluctuation of 3 A. Likewise, a drop in i_Bat resulted in a maximum voltage fluctuation of 11 V at port three, with a current fluctuation of 3 A.

Figure 18.

Experimental waveforms of voltage and current in response to step changes in the current of battery energy storage at port three under various control strategies: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

Under the PDC-MPC control, as illustrated in Fig. 18c, the increase in i_Bat caused a 5 V voltage drop at port 3, with a current fluctuation of 1.5 A. Similarly, a decrease in i_Bat resulted in a maximum voltage fluctuation of 9 V at port 3, with a current fluctuation of 2 A.

Furthermore, the MPC control scenario in Fig. 18d demonstrated superior port dynamic control performance, indicated by the stable load port voltage with minimal fluctuation in port three voltage. Similarly, when i_Bat decreased, the load port voltage remained stable under the MPC, showing improved port dynamic control performance offered by the proposed MPC strategy. These results highlight the effectiveness of the proposed MPC strategy in decoupling between ports two and three.

Disturbances in fuel cell input at port one

The experiment confirmed the effectiveness of the MPC control approach in managing disturbances on the fuel cell input side. The input voltage fluctuation was maintained at 50 V over a cycle and assessed using the exciting method. The results are illustrated in Fig. 19(a- d).

Figure 19.

Experimental waveforms of the converter’s input and output voltage in response to step variations in the fuel cell input voltage under various control strategies: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

The results depicted in Fig. 19a and b reveal that variations in the input voltage at port 1 led to fluctuations in both the output voltage and current at port 2, as well as in the output current at port 3 when using PI and UMD control strategies.

On the other hand, the PDC-MPC control, illustrated in Fig. 19c, input voltage variations caused minimal fluctuations in the output voltage and current at ports 2 and 3. While this control approach may induce an overshoot in the control output, it does not compromise control speed.

In contrast, the proposed MPC control strategy shown in Fig. 19d, effectively maintained stable voltage and current without any observable fluctuations. Consequently, the MPC exhibited a rapid dynamic response, virtually zero overshoots, and a very short control current adjustment time.

Dynamic response analysis under load current saturation at port two

The proposed MPC strategy’s effectiveness was evaluated under conditions where the load current at port two operates under saturation. In this context, saturation occurs when the load current exceeds the maximum allowable limit, activating the control system’s constraints. The system’s dynamic response was examined after a sudden change in load power from 4.5 kW to 9 kW. Figure 20 illustrates the resulting waveform from a 50% increase in load, followed by a subsequent return to the original value.

Figure 20.

Experimental waveforms of voltage and current in response to step changes when the load current exceeds the maximum allowable limit under various control strategies: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

In Fig. 20a, a sudden increase in load current from 30 A to the maximum allowable limit of 60 A resulted in coupling effects and fluctuations in the output voltage and current at port two, as well as in the output current at port three. With PI control, the output voltage at port two fluctuated by 40 V, while the current at port three fluctuated by 12 A. The output voltage at port two slowly returned to the target value of 150 V, showing a slow dynamic response under saturation conditions. Furthermore, reducing the load at port two caused a maximum voltage fluctuation of 60 V at port two, with the current at port three fluctuating by 15 A. The output voltage at port two eventually stabilized at the control target of 150 V, but only after a considerable delay.

In contrast, under UMD control, as illustrated in Fig. 20b, the voltage fluctuation at port two decreased to 30 V, while the current fluctuation at port three remained at 12 A. Following a load reduction at port two, the output voltage fluctuated by a maximum of 40 V, with the current at port three fluctuating by 16 A. These results indicate that while PI control demonstrates significant coupling effects, UMD control effectively minimizes power coupling between the two ports and reduces the time needed to reach the control target.

Under PDC-MPC control, as illustrated in Fig. 20c, the voltage fluctuation at port two was further reduced to 25 V, and the current fluctuation at port three was minimized to 6 A. Upon reducing the load at port two, the output voltage fluctuated by a maximum of 30 V, with the current at port three fluctuating by 5 A. Consequently, when operating under saturation conditions with PDC-MPC control, the TAB converter demonstrated reduced voltage fluctuations and achieved rapid current stabilization.

The proposed MPC control strategy, as illustrated in Fig. 20d, resulted in a voltage fluctuation of only 9 V at port two while maintaining a stable current at port three without any fluctuations. This outcome indicates that the proposed MPC effectively enabled power decoupling between ports two and three, preventing load port power transients from affecting the energy storage port. Additionally, during load reduction, the output voltage fluctuated by at most 12 V, with negligible fluctuations in the current at port three. The proposed MPC control not only ensured superior dynamic performance compared to PDC-MPC control but also provided enhanced power decoupling, effectively isolating the effects of load port transients from the energy storage port. This decoupling effect surpassed that achieved by the PDC-MPC control strategy.

Capability of the controller to resist variations in model parameters

The performance of MPC is significantly affected by the circuit parameters of the converter. This study investigates the sensitivity of these parameters, as detailed in "Parameter sensitivity analysis" section, to validate the proposed MPC control and evaluate its precision. The control objective is to maintain a consistent output voltage of 150 V and 90 V for ports two and three, respectively, while keeping the fuel cell voltage input constant at 300 V. The variations in inductance parameters L₁, L₂, and L₃ are 2µH, 48.897µH, and 20.65µH, respectively, while the capacitance parameters C₁, C₂, and C₃ are 200µF, 400µF, and 350µF, respectively. These scenarios were tested under a sudden increase in load power at port two.

Figure 21(a-d) depicts the steady-state responses of i_L, V_L, V_Bat, and i_Bat with PI control, UMD control, PDC-MPC control, and the proposed MPC approach. These responses were obtained when the values of inductance and capacitance deviated from their nominal values, and a step increase in load power occurred at port two.

Figure 21.

Experimental waveforms of the converter’s output voltage and current during a sudden increase in load power at port two under conditions of mismatched inductance and capacitance parameters, using various control strategies: (a) PI control, (b) UMD control, (c) PDC-MPC control, and (d) proposed MPC control.

Under PI and UMD control, illustrated in Fig. 21a-b, it is evident that i_L, V_L, V_Bat, and i_Bat were adversely affected by these parameter mismatches. This led to overshoots in the control output, which, in turn, impacted the response time and stability of the system.

Similarly, under PDC-MPC control, as illustrated in Fig. 21c, i_L, V_L, V_Bat, and i_Bat are sensitive to parameter mismatches. This sensitivity can lead to overshoots in the control output and compromised response time. Additionally, the system’s stability is impacted, as the binary search algorithm-based MPC requires enhancement under uncertain parameter conditions. Inaccurate predictions resulting from model inaccuracies can cause performance degradation or even instability.

In contrast, the proposed MPC control strategy, illustrated in Fig. 21d, showed a voltage fluctuation at port two while maintaining a stable current at port three without any fluctuations. Thus, while load disturbances and parameter modifications in capacitance and inductance may cause overshoots in the control output, the response speed and stability of the proposed MPC control remain unaffected.

Figure 22 displays the current waveforms of the transformer during the transition and under the parameter mismatches discussed above. Specifically, Fig. 22(a, b, e, and f) illustrate the step changes in the transformer current due to corresponding power changes at port two and parameter mismatches under PI control, UMD control, PDC-MPC control, and the proposed MPC approach. A zoomed-in view is provided in Fig. 22(c, d, g, and h), showcasing the steady-state current waveforms. Remarkably, the proposed MPC strategy demonstrates the ability to promptly and automatically adapt the phase shift to the actual operating conditions.

Figure 22.

Transformer current waveforms during the transient response of port two and under parameter mismatches using PI control, UMD control, PDC-MPC control, and the proposed MPC approach: (a), (b), (e), and (f) transformer currents during the transient response and parameter mismatches; (c), (d), (g), and (h) a zoomed-in view.

Control complexity

To measure the complexity of control, Typhoon HIL 602 evaluates the execution time. The computation time for the proposed MPC is illustrated in Fig. 23. This evaluation includes the time taken for code implementation, from receiving the DC current (ΔI₂(k) and ΔI₃(k)) to generating output phase shifts (φ₁₂ and φ₁₃) using four control methodologies. The PI controller is relatively straightforward to implement, especially when compared to the UMD controller. While the PDC-MPC controller takes 7.16µs, the proposed MPC only requires 4.8µs. This 49% decrease in average execution time significantly enhances dynamic performance, as evidenced in subsequent results. Since a 20 kHz switching frequency is utilized, 50µs is available in one sampling period. Therefore, there is sufficient headroom for implementing A/D sampling, digital filters, protections, etc.

Figure 23.

Measurement of the execution time of four control approaches.

Efficiency comparison

The power loss distribution for the triple active bridge was evaluated using HIL experimental results, as illustrated in Fig. 24. The proposed MPC strategy effectively decreased switch conduction losses across all ports, achieving reductions of 10.3 W, 9.1 W, and 16.3 W at ports 1, 2, and 3, respectively. Port 3, known for its high-current and low-voltage setup, exhibited the most notable reduction in conduction losses. Furthermore, the application of MPC led to a 5 W decrease in transformer core losses and a 7.2 W reduction in transformer copper losses. By operating the semiconductor switches under ZVS conditions, all other losses were minimized, resulting in a substantial decrease in turn-off losses to 2.4 W.

Figure 24.

Power loss distribution in TAB converter with PI, UMD, PD-MPC, and MPC schemes.

Figure 25 presents efficiency curves derived from experimental data using the proposed MPC method in comparison with the UMD and PI methods. With the implementation of the proposed MPC method, an efficiency of over 92% is achieved for power transmission ranging from 2 to 8 kW. Notably, at the nominal power transmission point of P_n= 5 kW, a maximum efficiency of 97.2% is attained, representing a 1.7% increase over UMD control and a 2.2% increase over the PI method. Therefore, the proposed MPC method effectively enhances the efficiency of the TAB converter and minimizes power transmission loss.

Figure 25.

Efficiency curves with different control strategies.

Comparison with existing control methods

A comparison of the proposed MPC with three existing MPC control methods is detailed in Table 3. The comparison encompasses control method, modulation, complexity, decoupling performance, parameter variations, switching frequency, dynamic performance, and maximum efficiency. It is clear that the proposed MPC method offers significant advantages over the other methods in terms of complexity, decoupling performance, parameter variations, dynamic performance, and efficiency. The notably higher maximum efficiency associated with the proposed MPC represents a significant advantage of the proposed design. This indicates that the proposed MPC exhibits lower switching losses, leading to improved efficiency compared to existing MPC methods. Additionally, the dynamic performance of the proposed control method surpasses that of the other methods. Based on this comparison, the proposed MPC controller is more favorable for use in TAB converters for general and industrial applications.

Table 3.

Comparison of three control methods with proposed control method.

Description	20	31	32	Proposed
Control Approach	MPC	MPC	MPC	MPC
Modulation	SPS	SPS	SPS	SPS
Complexity	Very complicated due to using a binary search and feedback from the DC voltages.	complicated	Moderate	Simple
Decoupling Performance	High	Moderate	Moderate	High
Parameter Variations	Moderate	The effect of parameters variations is not studied	High	High
Switching Frequency	50 kHz	10 kHz	2 kHz	20 kHz
Dynamic Performance	Moderate	Low	Low	Excellent
Maximum Efficiency	Not reported	Not reported	Not reported	97.2%
Controller Computation Burden	High	Low	Moderate	Low

Conclusions

In this study, a MPC approach based on single-phase shift modulation is introduced for efficiently controlling TAB DC-DC converters. The MPC algorithm, which utilizes Karush-Kuhn-Tucker (KKT) conditions, delivers notable enhancements in transient performance, control precision, and robustness against parameter uncertainties. By implementing MPC, dynamic decoupling control is achieved, mitigating the adverse effects of port coupling and ensuring compliance with DC voltage regulations. Experimental validations using a Typhoon HIL 602 hardware-in-the-loop setup demonstrate the proposed MPC method’s superior performance compared to traditional PI and unit matrix decoupling controls. The MPC-controlled system exhibits faster dynamic response times, reduced steady-state voltage ripples, and effective power decoupling among ports. The results also highlight the method’s high efficiency, achieving over 92% efficiency for power transmission within the 2 to 8 kW range, with a maximum efficiency of 97.2% at the nominal power transmission point.

Author contributions

Data curation, A.H.A.A. and J.C; formal analysis, M.X. and S.K; investigation, S.K. and G.A; methodology, A.H.A.A. and J.C; supervision, A.H.A.A., J.C., M.X., and G.A; writing—review and editing, S.K., G.A., M.X., and J.C.

Data availability

All data generated or analyzed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.