A Unified Controller for Multi-State Operation of the Bi-Directional Buck–Boost DC-DC Converter

Abstract

DC grid interfaces for supercapacitors (SCs) are expected to operate with a wide range of input voltages with fast dynamics. The class-C DC-DC converter is commonly used in this application because of its simplicity. However, it does not work if the output voltage (V₂) becomes smaller than the input voltage (V₁). The non-isolated bi-directional Buck–Boost DC-DC converter does not have this limitation. Its two half-bridges provide a means for controlling the power flow operating in the conventional dual-state mode, as well as multi-state, tri, and quad modes. These can be used for mitigating issues such as the Right Half Plane (RHP) zero that has a negative impact on the dynamic response of the system. Multi-state operation typically requires multi-variable control, which is not easy to realize with conventional PI-type controllers. This paper proposes a unified controller for multi-state operation. It employs a carrier-based modulation scheme with three modulation signals that allows the converter to operate in all four possible states and eight different modes of operation. A mathematical model is developed for devising a multi-variable control scheme using feedback linearization. This allows the design of control loops with simple PI controllers that can be used for all multi-state modes under a wide range of operating conditions with the same performance. The proposed scheme is verified by means of simulations.

1. Introduction

Energy storage units such as supercapacitors (SCs) can provide fast varying currents for power balancing and voltage regulation of DC grids, provided they have a suitable interface: a power electronics converter and control scheme. A two-switch class C DC-DC converter is frequently used as the interface of SCs [1]. However, its dynamic response tends to be slowed down due to the presence of a Right Half Plane (RHP) zero in the transfer function of the output voltage (V_o) by the duty cycle (D) of the Boost mode. This can be addressed with a tri-state logic that eliminates the RHP zero at the expense of an addition switch and the use of a multivariable modulation scheme [2]. Another issue with the class C converter is that it loses its power flow control ability due to the conduction of the top anti-parallel diode when the DC grid voltage falls below the voltage of the storage device [3].

The 4-switch bi-directional Buck–Boost converter, with two half-bridges and an intermediate inductor as shown in Figure 1, can be used to overcome this issue. In fact, it was used in a DC bus fault protection scheme in [3,4]. Its ability to operate in the Buck, Boost, and Buck–Boost modes can be explored in several ways, including increased efficiency [5,6,7,8]. The tri-state logic has also been applied to the four-switch converter operating in the Buck–Boost mode for enhanced dynamic performance [9]. The main issue with the tri-state logic is that it concerns a multi-variable control problem with two variables, usually the ON and OFF duty cycles (D_on and D_off) that need to be determined. The free-wheeling duty cycle (D_fw) is taken as the remainder of the switching period. In this case, if one wishes to control two state variables, the development of the small-signal model necessary for the design of linear controllers is complicated, leading to the appearance of cross-coupled transfer functions [10]. This can be addressed by keeping constant the duration of one of the states, typically D_off [9], or employing a dual-mode control scheme with parallel [11] or cascaded [12] control loops.

Linear PI-type controllers are often employed in power electronics interfaces [3,9,13]. These controllers are designed based on models linearized for a specific operating point and, as this operating point changes, their performance tends to degrade. In this scenario, for applications where a wide operating region is expected, such as the interface of SCs where the power flow direction changes and the SC voltage can also fall from rated to half-rated, non-linear control techniques have been employed [14,15,16]. However, due to their more complex nature, implementing such non-linear control approaches is still quite challenging. The use of dynamic feedback linearization employing well-known PI-type controllers has been seen as a good and effective alternative. This technique was used for the conventional dual-state control of uni-directional Buck, Boost, and Buck–Boost and bi-directional class-C converters in [17,18]. In [19], it was used for the four-switch bi-directional Buck–Boost converter operating either in the conventional dual-state Buck or Boost modes, with only one half-bridge switching at the time.

This paper presents a unified control scheme for the four-switch bi-directional Buck–Boost converter that allows it to operate in several multi-state (tri-state and quad-state) modes. In this way, the RHP zero that usually appears in the output voltage/current to duty cycle transfer functions related to the conventional dual-state Boost and Buck–Boost modes is eliminated, and the most efficient mode can be used for given operating conditions. A carrier-based Pulse-Width Modulation (PWM) scheme that allows the operation of the converter with dual-state, tri-state, and quad-state logic is presented. A model for the converter operating in multi-state with multi-variable control is developed. It is used to derive a unified control scheme based on feedback linearization for the control of the output voltage/current of the converter and the intermediate inductor current with simple PI controllers, independently. In the proposed scheme, one can have the converter operating in five different multi-state modes by converting control variables w₁ and w₂ from the control scheme, into three modulation signals u₁, u₂, and u₃ according to given rules. Simulation results are used to show that the dynamic performance of the converter with the unified control schemed is essentially the same for the converter operating in any of the multi-state modes and for a wide range of input voltage and output current.

This paper is organized as follows. In Section 2, the multi-state carrier-based Pulse-Width Modulation (PWM) scheme and modes of operation of the DC-DC converter are introduced. Section 3 presents the mathematical model of the converter and the proposed control scheme with feedback linearization. Section 4 discusses an approach for obtaining the modulation signals for the PWM (u₁, u₂, and u₃) from the control variables (w₁ and w₂) obtained in Section 3. A design example is presented in Section 5 and the performance of the proposed and conventional control schemes are verified in Section 6. The paper conclusions are stated in Section 7.

2. The Multi-State Carrier-Based Modulation Scheme

The proposed carrier-based Pulse-Width Modulation (PWM) scheme for the bi-directional Buck–Boost DC-DC converter is shown in Figure 2a. It presents three modulation signals, 0 ≤ u₁ ≤ u₂ ≤ u₃ ≤ V_m, where V_m is the peak value of the sawtooth carrier, which is usually assumed to be equal to 1. By comparing the modulation signals with the carrier using the circuit shown in Figure 3, one obtains the gating signals for switches S₁, S₂, S₃, and S₄, as shown in Figure 2b. All four possible states of operation for this converter (S₁₄, S₁₃, S₂₃, and S₂₄) are shown in Figure 2c. The subscript indicates the switches ON at a given time. From Figure 2, one can see that for V_m = 1:

$$u_2=D_1$$

(1)

$$u_3-u_1=D_3$$

(2)

where D₁ is the duty cycle of S₁ and D₃ is the duty cycle of S₃.

By setting additional constraints concerning the magnitudes of the modulation signals u₁, u₂, and u₃, one can have the converter operating in eight different modes, as shown in

Table 1. Modes of operation.

u1	u2	u3	States	Mode of Operation
0	0	u3	S23–S24	Not useful—Disregard
0	u2,3	u2,3	S13–S24	Not useful—Disregard
0	u2	1	S13–S23	1—Dual-State Buck
u1,2	u1,2	1	S14–S23	2—Dual-State Buck–Boost
u1	1	1	S14–S13	3—Dual-State Boost
0	u2	u3	S13–S23–S24	4—Tri-state Buck with free-wheeling (fw)
u1	u2	1	S14–S13–S23	5—Tri-state Buck–Boost with no fw
u1	u2,3	u2,3	S14–S13–S24	6—Tri-state Boost with fw (u2,3 ≥ u1)
u1,2	u1,2	u3	S14–S23–S24	7—Tri-state Buck–Boost with fw (u3 ≥ u1,2)
u1	u2	u3	S14–S13–S23–S24	8—Quad-state

. The modes in which the average voltage across the inductor (V_L) cannot be made zero on a switching cycle basis are not useful/practical. It should be noted that for operation in Buck modes (1 and 4), it is mandatory that v_C1 > v_C2, while for the Boost modes (3 and 6) that v_C1 < v_C2. It has been shown that operation in the Buck and Boost modes, when possible, leads to lower power converter losses and stress on the switches than in the Buck–Boost mode [20].

3. Mathematical Model of the Converter

By employing KVL and KCL, one can obtain the following differential equations for the average values of the state variables of the converter:

$$\frac{d{v}_{C1}}{dt}=\frac{V_1}{R_1{C}_1}-\frac{v_{C1}}{R_1{C}_1}-\frac{i_L}{C_1}\left(u_2\right)$$

(3)

$$\frac{d{v}_{C2}}{dt}=\frac{V_2}{R_2{C}_2}-\frac{v_{C2}}{R_2{C}_2}+\frac{i_L}{C_2}\left(u_3-u_1\right)$$

(4)

$$\frac{d{i}_L}{dt}=\frac{v_{C1}}{L}\left(u_2\right)-\frac{v_{C2}}{L}\left(u_3-u_1\right)$$

(5)

According to (3)–(5), the converter presents three state variables (v_C1, i_L, and v_C2), and three control signals (u₁, u₂, and u₃). Typically, one wishes to control either the output current of the converter (i₂) or the output voltage of the converter (v_C2), which require similar control actions. Frequently, this is achieved with an inner inductor current (i_L) control loop and an outer current/voltage loop [3,9,13]. However, with the three modulation signals and the four possible states of operation (S₁₄, S₁₃, S₂₃, and S₂₄), one can have a certain value of v_C2 or i₂, with different values of i_L, which is not the case for the conventional cascaded control scheme. This can be exploited for either reducing the losses in the converter, with a low value of i_L, or having a faster dynamic response for the v_C2, with a larger value of i_L. Therefore, in this analysis, it is assumed that there will be a reference value for the average value of the output voltage (v_C2*) and a reference value for the inductor current (i_L*). These values are not independent, and a discussion on suitable values for i_L* is presented in Section 3.4.

In order to control two of the state variables, one can employ two control variables. Therefore, by making the following change of variables:

$$w_1=u_3-u_1$$

(6)

$$w_2=u_2,$$

(7)

one converts (3)–(5) into:

$$\frac{d{v}_{C1}}{dt}=\frac{V_1}{R_1{C}_1}-\frac{v_{C1}}{R_1{C}_1}-\frac{i_L}{C_1}{w}_2$$

(8)

$$\frac{d{v}_{C2}}{dt}=\frac{V_2}{R_2{C}_2}-\frac{v_{C2}}{R_2{C}_2}+\frac{i_L}{C_2}{w}_1=-\frac{i_2}{C_2}+\frac{i_L}{C_2}{w}_1$$

(9)

$$\frac{d{i}_L}{dt}=\frac{v_{C1}}{L}{w}_2-\frac{v_{C2}}{L}{w}_1$$

(10)

Due to the cross-product between state and control variables, the equations presented above are non-linear. In such a case, one can design control loops for the average values of the output voltage (V_C2) and inductor current (I_L) based on the feedback linearization technique [17,18,19].

3.1. Output Voltage Control Loop with Feedback Linearization

For the design of the converter output voltage (v_C2) controller, one can employ the feedback linearization technique. First, an auxiliary input variable (v_x) related to v_C2 and corresponding to i_C2 is defined as:

$$\frac{d{v}_{C2}}{dt}=\frac{v_x}{C_2}$$

(11)

It provides a means for designing a linear PI-type controller (Gv_C2), as shown in Figure 4. Then, with v_x being the voltage error (v_C2* − v_C2) processed by the PI controller, (11) becomes:

$$\frac{d{v}_{C2}}{dt}=\frac{1}{C_2}\left[k_{pv} \left(v_{C2}^{*}-v_{C2}\right)+k_{iv}{\xi }_v\right]=\frac{v_{PIv}}{C_2}$$

(12)

$$\frac{d{\xi }_v}{dt}=v_{C2}^{*}-v_{C2}$$

(13)

It should be noted that v_x is the output of the output voltage loop PI (v_PIv), and it corresponds to i_C2. In steady state, I_C2 and V_PIv should be zero. Finally, the value of the control variable w₁ is obtained by substituting (9) in (12), leading to:

$$w_1=\frac{1}{i_L}\left[i_2+k_{pv} \left(v_{C2}^{*}-v_{C2}\right)+k_{iv}{\xi }_v\right]=\frac{1}{i_L}\left(i_2+v_{PIv}\right)$$

(14)

There, one can clearly see the presence of the feedforward signals i_L and i₂, which should make the dynamic response of the performance of the v_C2 control loop determined essentially by the PI-controller design specifications. Since 0 ≤ u₁ ≤ u₂ ≤ u₃ ≤ 1, according to (6), one should have 0 ≤ w₁ ≤ 1. While v_C2 and V₂ are expected to remain positive at all times, i₂ and i_L become negative as the direction of the power flow reverses, with V_C2 < V₂. When i_L crosses zero, the value of w₁ changes between ∞ and −∞, requiring a limiter to keep it within the (0–1) expected range.

From (9), one can say that in steady state, I₂ = w₁ I_L with 0 ≤ w₁ ≤ 1. i_L and i₂ should have the same sign, with the magnitude of i_L being larger or equal to the magnitude of i₂. In general, when one applies a positive step in v_C2*, the positive error will make v_PIv change from “0” to positive. If i₂ and i_L are positive, w₁ increases so as to increase v_C2. Conversely, if i₂ and i_L are negative, w₁ decreases due to the negative sign of i_L, which should make v_C2 increase, since i_C2 = w₁i_L − i₂. Therefore, the control law described in (13) should work for both forward and reverse power flows.

3.2. Inductor Current Control Loop with Feedback Linearization

The same procedure can be followed for the inductor current control loop. In this case, a second auxiliary input variable (v_y), related to i_L and corresponding to v_L, is defined as:

$$\frac{d{i}_L}{dt}=\frac{v_y}{L}$$

(15)

It provides a means for designing a linear PI-type controller (G_ciL), as shown in Figure 5. With v_y being the voltage error (i_L* − i_L) processed by the PI controller, (15) becomes:

$$\frac{d{i}_L}{dt}=\frac{1}{L}\left[k_{pi} \left(i_L^{*}-i_L\right)+k_{ii}{\xi }_i\right]=\frac{v_{PIi}}{L}$$

(16)

$$\frac{d{\xi }_i}{dt}=i_L^{*}-i_L$$

(17)

One can see from Figure 5 that v_y is the output of the inductor current PI controller, v_PIi, and it corresponds to v_L. In steady state, both V_L and V_PIi should be zero. The value of the control variable w₂ is obtained by substituting (10) in (16), leading to:

$$w_2=\frac{1}{v_{C1}}\left[v_{C2}{w}_1+k_{pi} \left(i_L^{*}-i_L\right)+k_{ii}{\xi }_i\right]=\frac{1}{v_{C1}}\left(v_{C2}{w}_1+v_{PIi}\right)$$

(18)

In the expression above, one sees the feedforward of signals v_C1, v_C2, and w₁. Neither v_C1 nor v_C2 should present any challenges to the computation of w₂, since they should be positive and not much different from V₁ and V₂. By feedforwarding w₁, the dynamic response of the system should be determined by the current PI controller. Since 0 ≤ u₂ ≤ 1 in the PWM scheme and according to (7), 0 ≤ w₂ ≤ 1. Concerning the values of w₁ and w₂ in steady state, one can see from (10) that w₂ > w₁ if v_C2 > v_C1.

3.3. Impact of the Output Current and Input Voltage on the Choice of the Reference Inductor Current

As discussed in previous sections, variable w₁ will be used for controlling v_C2 and variable w₂ will be used for controlling i_L. Both control variables should be limited between 0 and 1. This will impose constraints on the possible values of V₁, V₂, and i_L which are important to know. An equation describing this relationship can be derived from the equivalent circuit shown in Figure 6. It considers the reflection of the voltages V₁ and V₂ and resistances R₁ and R₂ to the intermediate (inductor) segment of the converter, using the principle of DC-transformer and the duty cycles of the S₁–S₂ and S₃–S₄ converter legs [21]. Based on (1), (2), (6), and (7) the duty cycles of the two legs, which correspond to the turns ratios of the DC transformers, can be computed as:

$$D_1=w_2 \mathrm{and} D_3=w_1$$

(19)

In general, by controlling v_C2, one should be able to control i₂, which is related to the inductor current by:

$$i_2=w_1{i}_L$$

(20)

Therefore, for a given desired output current (i₂) and a given value of inductor current (i_L), a suitable value for w₁ can be determined. Next, one can estimate a candidate value of w₂ for given values of V₁, V₂, and i_L for a specific value of i₂ (≤ i_L) and respective value of w₁. For the operating conditions to be feasible, 0 ≤ w₂ ≤ 1.

By using KVL in Figure 6, the following second-order equation can be used for calculating w₂:

$$i_L{R}_1{w}_2^2-V_1{w}_2+i_L{R}_2{w}_1^2+V_2{w}_1=0$$

(21)

In order to have 0 ≤ w₂ ≤ 1, with given values of V₂, i_L, R₁, and R₂ as w₁ varies between 0 and a maximum value (w_1max), there is a minimum value of V₁ (V_1min), calculated by

$$V_{1min}=i_L\left(R_1+R_2{w}_{1max}^2\right)+V_2{w}_{1max}$$

(22)

Figure 7 shows the values of w₂ as a function of i₂ with a maximum output current (i_2max) of 20 A, for V₂ = 48 V, i_L = 40 A (fixed), and R₁ = R₂ = 62.5 mΩ for various values of V₁. For this particular case, using (22) with w_1max = 0.5, one obtains V_1min = 27.125 V. From Figure 7, one can see that for values of V₁ < V_1min, one cannot have 0 ≤ i₂ ≤ i_2max with 0 ≤ w₂ ≤ 1. Conversely, for V₁ > V_1min, one can have 0 ≤ i₂ ≤ i_2max with 0 ≤ w₂ ≤ 1.

Alternatively, one can use a larger i_L, say 60 A, and a smaller w_1max, say 0.33, for i_2max = 20 A. In this case, one computes V_1min = 20.17 V. Figure 8 shows the values of w₂ as a function of i₂ (w₁i_L, 0 ≤ w₁ ≤ 0.33), for V₂ = 48 V, i_L = 60 A, and for various values of V₁. As in the previous case, an i_2max of 20 A can be obtained with 0 ≤ w₂ ≤ 1, as long as V₁ ≥ V_1min. In general, in order to decrease V_1min, it suffices to increase i_L. By using a lower w_1max, one increments the Boost effect that allows the use of a lower input voltage.

3.4. Reference Value for the Inductor Current

In the analysis presented in the previous section, a constant reference inductor current (i_L*) is considered. One issue is that for a low value of i₂*, a low value of w₁ is used. If one wishes to reduce v_C2*and i₂*, the value of w₁ will decrease and possibly saturate, slowing down the dynamic response. If one wishes to reverse the power flow, both i₂ and i_L need to be reversed. The value of i_L* should change from, say 60 to −60 A, which will take more time to realize than if the current variation was smaller.

Another option is to determine a value for i_L* as a function of i₂. One of the requirements of this converter is that since 0 ≤ w₁ ≤ 1 and i₂ = w₁ i_L, 0 ≤ |i₂| ≤ |i_L|. This can be based on a gain (k_i2L), where i_L* = k_i2L i₂, which is the inverse of the value of w₁ in steady state. To keep w₁ not too low, so that it can be decreased to regulate v_C2 without saturating at the minimum value, one can select k_i2L = 3, for w₁ = 0.33. This matches with the conditions shown in Figure 8, where for i_2max = 20 A, i_L = 60 A, and one can operate with V_1min = 24 V.

4. Obtaining u₁, u₂, and u₃ from w₁ and w₂

The control scheme developed in Section 3 for v_C2 and i_L considered control variables w₁ and w₂. For the actual implementation of the proposed multi-state PWM scheme, one needs the modulation signals u₁, u₂, and u₃ for generating the gating signals of the switches. As shown in Table 1, there are eight possible modes of operation for the non-isolated bi-directional Buck–Boost DC-DC converter. Only those that allow independent control of w₁ and w₂ can be used with the proposed control scheme. Based on the values they employ for u₁, u₂, and u₃, one can say that only modes 4–8 comply with this condition. Another aspect to be considered is whether they are compatible with the input and output voltages in the circuit. For instance, mode 4 (tri-state Buck) can only be used when v_C1 > v_C2, while mode 6 (tri-state Boost) can only be used when v_C1 < v_C2. Conversely, modes 5, 7, and 8 (tri-state Buck–Boost and quad-state) can be used for all cases.

Table 2. Multi-state modes of operation compatible with the proposed control scheme.

u1	u2	u3	Mode of Operation
0	w2	w1	4—Tri-state Buck with free-wheeling (fw)
1-w1	w2	1	5—Tri-state Buck–Boost with no fw
w2-w1	w2	w2	6—Tri-state Boost with fw
w2	w2	w2 + w1	7—Tri-state Buck–Boost with fw
c-w1	w2	c	8—Quad-state

shows how to obtain u₁, u₂, and u₃ from w₁ and w₂ for the five multi-state modes. For mode 5 (tri-state Buck–Boost with no free-wheeling), since 0 ≤ u₁ ≤ u₂, then 1 ≤ w₁ + w₂. For mode 6 (tri-state Boost with free-wheeling), w₁ ≤ w₂. Finally, for mode 8 (quad-state), one can use a constant large value for c, say 0.95, while ensuring that w₁ ≤ w₂ ≤ c.

5. Design Example

The following parameters were selected for a case study that concerns the interface of a supercapacitor rated at 48 V to a DC grid also rated at 48 V. It is assumed that the voltage in the supercapacitor (V₁) can vary between rated and half-rated (24 V ≤ V₁ ≤ 48 V). Concerning the DC-grid voltage (V₂), it is assumed that it can vary by +/− 5% around the rated value; thus, 45.6 V ≤ V₂ ≤ 50.4 V. The other converter parameters are R₁ = R₂ = 62.5 mΩ, C₁ = C₂ = 76.8 μF, L = 38.8 μH, f_sw = 250 kHz, −20 A ≤ i₂ ≤ 20 A.

Controller Design

Based on (11) and (15), one only needs the value of C₂ and L, respectively, to design the output voltage and inductor current controllers. Since in the actual switched converter, voltages and currents present switching harmonics, first-order Low-Pass Filters (LPFs) of 100 kHz are employed in both control loops. The bandwidth of the inductor current control loop (f_{x_iL}) is selected as 50 kHz (20% of f_sw) and its phase margin is selected as 60°. Since the output voltage of the converter (v_C2) is regulated based on the switching of the inductor current, the bandwidth of the voltage control loop is selected as 10 kHz (20% of f_{x_iL}) and its phase margin is selected as 60°. Instead of a conventional PI controller, a type 2 PI controller is used in this study. The additional pole further helps attenuating the switching harmonics in the sensed inductor current and output capacitor voltage. Their parameters are k_i = 13.63, τ_i = 106.16 μs, fp_i = 1668 kHz, and k_v = 2.46, τ_v = 193.43 μs, fp_v = 30.4 kHz.

6. Performance Verification and Discussion

There are several aspects of the proposed concept to be verified. First is the performance achieved with the proposed control scheme with PI controllers and feedback linearization for various operating conditions: forward and reverse power flows with V₁ < V₂ and V₁ > V₂. This can be done with the mathematical equations derived in Section 3 and employing control variables w₁ and w₂. Next is the performance of the actual switched converter. The goal is to demonstrate that one can convert control variables w₁ and w₂, which are obtained for the unified model, into u₁, u₂, and u₃ so that the converter operates in a desired mode. The dynamic response of the converter should be similar irrespectively of the selected mode of operation, as long as both w₁ and w₂ are used for obtaining u₁, u₂, and u₃.

The schematic diagram of the proposed control scheme is shown in Figure 9. There, one can see how the control laws for w₁ and w₂ are created as a function of the reference signals and sensed converter quantities. LPFs are included to attenuate the switching harmonics, and a limiter is used to prevent the magnitude of i_L, which is used in the denominator of (14), from becoming too small, during power flow reversals. Limiters are also used for keeping w₁ and w₂, which correspond to D₃ and D₁, within 0 and 1. Then, depending on the selected mode of operation, control variables, w₁ and w₂, are converted into the modulation signals u₁, u₂, and u₃, and used in the PWM modulator of Figure 3 to create the gating signals for the switched converter.

6.1. Control Scheme with PI Controllers and Feedback Linearization

The performance of the proposed unified controller for multi-state operation of the bi-directional Buck–Boost DC-DC converter is verified with the mathematical model in this section. The parameters of the SC interface (V₁ in Figure 1) are those shown in Section 5. An SC with a small capacitance (15 mF) is the base value used in this test so that one can observe the variations of the voltage in the storage unit (v₁) as well as the fast varying current and voltage waveforms in the same time frame. It is assumed that the magnitude of the DC bus voltage and feeder (v₂) and the value of the feeder impedance (R₂) are known. They are used in the calculation of the reference value for the output capacitor voltage (v_C2*) and inductor current (i_L*) as a function of the reference injected current (i₂*). As mentioned in Section 3.4, k_i2L = 3.

Key waveforms obtained with control laws for w₁ and w₂, (14) and (18), and the mathematical model of the converter, (8)–(10), are shown in Figure 10. On the top, one can see the waveform of the DC bus voltage (v₂), with an average value of 48 V and a +/−5% triangular ripple of 40 Hz along with the voltage across the SC (v₁). The latter varies as a function of the injected current (i₂), which is shown just below and concerns a staircase waveform of 20 Hz and a maximum current of 20 A. The voltage across the SC decreases when i₂ > 0 and increases when i₂ < 0. The injected power is shown in the screen at the bottom, presenting a zero average value. In such a case, the power losses across the two feeder resistors come from the SC, which tends to discharge over time, as seen in the top screen. For most of the time, v₁ < v₂, but v₁ > v₂ in a short segment. Therefore, the converter needs to operate in both the Boost (step-up) and Buck (step-down) modes. It is evident that there is good tracking of the reference values for the injected current (i₂), output capacitor of the converter (v_C2), and inductor current (i_L), with control signals, w₁ and w₂, which are shown just below those waveforms.

6.2. Simulation Results with the Switched Converter

Having shown that the proposed control scheme provides a fast and accurate control of the inductor and the output current of the converter using the developed mathematical model of the system, its feasibility is verified with the switched converter for the same test conditions. This is done by simulation with PSIM. In this case, the control variables (w₁ and w₂) are converted into the modulation signals of the PWM modulator (u₁, u₂, and u₃) so that the converter operates in a desired multi-state mode according to the expressions shown in Table 2. Since the converter is expected to operate in the Buck and Boost modes, only the tri-state Buck–Boost and quad-state modes, in Table 2, are considered at first.

Figure 11 shows key waveforms for the switched converter operating in the quad-state mode. The waveforms are essentially the same obtained for the mathematical model test. The main difference is the presence of switching harmonics. Index “con” indicates results obtained with the switching converter, while “mod” refers to the mathematical model.

In Figure 12, one can see a close view of the control parameters (w₁ and w₂), the modulation signals (u₁ ≤ u₂ ≤ u₃), and the gating signals of switches S₁ and S₃ for three switching cycles. There, one can observe all sorts of combinations of v_gS1 and v_gS3, meaning that all four switching states are employed in the quad-state mode.

Figure 13 shows the same waveforms for the switched converter operating in a tri-state Buck–Boost mode, mode 5 in Table 2. The waveforms are essentially the same as those for the quad-state mode. As shown in Figure 14, the free-wheeling mode, when S₂ and S₄ conduct together, is not present in this mode.

For testing the converter operating in the tri-state Buck and Boost modes, one should always have v₁ > v₂ and v₁ < v₂, respectively. In order to keep the same output voltage and current waveforms, one can change the initial voltage of the SC (v₁) so that the converter always operates in the desired mode. It is important to reduce the range of variation of v₁ by increasing the capacitance of the SC to 0.03 F. Figure 15 shows waveforms for the switched converter operating in the tri-state Buck mode, where v₁ is always bigger than v₂. Again, the unified controller allows a fast and close tracking of the reference quantities. The quantities obtained with the model, index “mod”, are shown along with those of the switched converter, index “con”, to show the similarities. The major difference is the presence of the switching harmonics that can be seen in some waveforms. The control parameters, modulation signals, and gating signals of switches are shown in Figure 16, where one can see that only three states are used, S₁₃–S₂₃–S₂₄, as seen in Table 1.

Finally, Figure 17 shows waveforms for the switched converter operating in the tri-state Boost mode, where v₁ is always smaller than v₂. It should be noted that the conventional class-C converter DC-DC converter could not be used in this case. The unified controller allows a good tracking of the reference quantities. In the bottom screen, the control variables obtained for the switching converter, index “con”, are similar to those obtained for mathematical model, index “mod”. The control parameters, modulation signals, and gating signals of switches, with only three states, are shown in Figure 18. In this particular case, u₂ = u₃, as determined in Table 2.

6.3. Comparison to a Conventional Technique

The performance of the proposed unified controller for multi-state operation will be compared with a conventional technique. In this case, it concerns the bi-directional DC-DC converter operating with dual state in the Buck–Boost mode (mode 2 in Table 1). The output current (i₂) is controlled with a single loop employing a standard PI controller. This is designed considering a model of the plant linearized around a certain operating point. The transfer function relating the injected current (i₂) to the duty cycle (D) of the converter is the following:

$$\frac{i_2\left(s\right)}{D\left(s\right)}=\frac{1}{R_{2}L{C}_2}\left[\frac{\left(V_{C1}+V_{C2}\right)-\left(I_{L}L\right)s}{s^2+\left(\frac{1}{R_2{C}_2}\right)s+\frac{{\left(1-D\right)}^2}{L{C}_2}}\right]$$

(23)

One can clearly see that the system presents a non-minimal phase when I_L > 0. A linear PI controller was designed for the current loop considering V_C1 = V_C2 = 48 V, D = 0.5, and I_L = 40 A. As a result of the RHP zero at 4.7 kHz, the bandwidth of the control loop should be about 20% of that; thus, it was selected as 1 kHz with a phase margin of 60°. A first-order LPF of 25 kHz was employed at the sensed output current signal. The parameters of the current loop PI controller are k_i = 5.1 m, τ_i = 918 μs, and f_pi = 5.8 kHz.

The conventional technique was used in the switched converter for the test case presented in Figure 12. The capacitance of the SC is 15 mF, and the DC-bus voltage (v₂) presents an average value of 48 V and a +/−5% triangular ripple of 40 Hz. Figure 19 shows the SC voltage and the DC-bus voltage on the top. The screen at the bottom shows the staircase reference output current along with the current obtained with the proposed, index “con”, and the traditional, index “trad” control schemes. In this case, the filtered (25 kHz) output current signals are shown, so that one can better observe the difference between the two responses. There, one can see that the response of the traditional PI controller presents longer settling times than the proposed scheme, and sometimes, it fails to track the reference current. This is because the PI controller was designed for operation close to a certain operating point. Figure 20 shows the response of the two controllers when both the input and the output voltages are 48 V, which are the values used in the model of the plant and design of the traditional PI controller. There, one can see that the proposed and the conventional control techniques perform equally well.

6.4. Discussion

The first test shown in Section 6.1 for the bi-directional DC-DC converter interfacing the SC to a DC grid considered the mathematical model of the converter and the control laws for control variables w₁ and w₂. The resulting waveforms show that the proposed control scheme with multi-variable control and feedback linearization provides a very good transient response to step variations in the reference currents i₂* and i_L*. In addition, the load disturbance, represented by the triangular voltage ripple on the DC bus, did not degrade the tracking of currents nor output voltage in steady state. The dynamic responses with fixed PI controllers were essentially the same, despite the large variation in the SC voltage, 28 to 50 V, and the reversal of the power flow direction. This is provided by the feedback linearization technique.

The second test shown in Section 6.2 considered the actual/switched bi-directional DC-DC converter, with the scheme for converting control variables w₁ and w₂, into PWM modulation signals u₁, u₂, and u₃, according to a selectable and desired multi-state mode of operation. The quad-state and tri-state Buck–Boost modes were used for the cases where the input voltage is smaller and larger than the output voltage. For the test with the tri-state Buck and tri-state Boost modes, the values of the SC and initial input voltage were adjusted so that the input voltage is always larger for the Buck and smaller for the Boost than the output or DC grid voltage, considering the same reference output current and DC bus voltage. It should be noted that taking into account all these tests, the input voltage varied between 28 and 58 V, but the dynamic response of the proposed control scheme with fixed PI controllers was essentially the same for all multi-state modes of operation.

The fact that the conventional current control scheme for a dual-state Buck–Boost DC-DC converter with a single PI controller works well when the operating conditions do not vary much was highlighted in Section 6.3. There, it was shown that in applications such as the interface of a SC, with a wide voltage variation, to an unregulated DC grid, the use of the proposed unified controller provides a better response in terms of dynamic response and tracking of the reference output current.

7. Conclusions

This paper presented a unified controller for multi-state operation of a bi-directional Buck–Boost DC-DC converter suitable for interfacing supercapacitors (SCs) to DC grids. According to the literature, the multi-state operation of DC-DC converters allows the elimination of Right Half Plane (RHP) zeros, switch power loss equalization, etc. The challenge of multi-variable control was tackled by employing feedback linearization and simple fixed PI controllers. A mathematical model for designing the controllers and obtaining the control laws for the generic parameters (w₁ and w₂) was developed. These can be converted to three modulation signals (u₁, u₂, and u₃) and used in a carrier-based PWM scheme for generating the gating signals for five different multi-state modes of operation. The technique was verified by means of simulations in the interface of a SC operating in a wide range of voltages to a DC bus with a +/−5% of voltage ripple. The same set of controllers provided fast and accurate regulation for step changes in the reference quantities under bi-directional power flow for the converter operating in quad-state, tri-state Buck–Boost, tri-state Buck, and tri-state Boost modes.