## 1. Introduction

There has been an increasing trend of more electric aircraft (MEA) in the aerospace industry [

1], towards better performance in efficiency, reliability and safety. The onboard power system is a typical microgrid (MG) system with distributed supplies and loads [

2]. The DC MG has been widely considered as a preferable solution for the future onboard MG and has garnered great attention [

3,

4,

5]. A typical paradigm of the DC MG is shown in

Figure 1. In such a system, most of the power supplies and loads, including the main generators, auxiliary power units, energy storage units and low-voltage (LV) DC loads, are connected to the high-voltage (HV) DC bus (typically 270 V) through power electronic converters.

This paper focuses on the AC–DC converters that interface variable-frequency generators with the MG. Currently, multi-pulse diode rectifiers with phase-shift transformers dominate in such applications [

2], because of their very high reliability. Yet, the bulky filter components and unidirectional power flow could not satisfy requirements of future MEAs. There are some studies that adopt voltage source converter (VSC) to replace diode rectifiers [

6,

7], which could achieve good power quality and smaller filter components. However, both the diode rectifiers and VSC do not have the capability to isolate DC bus faults, and have to work under low and narrow range of input voltages. A promising alternative to diode rectifiers and VSC is the current source converter (CSC), which is also known as buck rectifier or AC–DC matrix converter. Like VSC, CSC can also achieve high-quality input and output currents with small filter components. Moreover, CSC has the intrinsic capability to limit overcurrent and can work under a higher and wider range of input voltages. Therefore, it could achieve more efficient and compact design of the whole system. Many researchers have studied the usage of CSC in aircraft applications [

8,

9,

10,

11].

Modulation and control are important research topics of CSC, which have great achievements in civil applications. In [

12], the optimal modulation scheme is proposed, which could achieve the minimum switching loss and ripples, and thus has been widely adopted in later studies. Some new modulation and control techniques have also been developed for CSC regarding new topologies and applications [

13,

14,

15,

16].

Benefiting from the fast development of microcontrollers, predictive control has been suggested as an emerging control technique for power converters [

17,

18]. The basic principle of a predictive control scheme is to predict the system behavior using the discrete prediction models. There are typically two kinds of predictive control schemes: deadbeat predictive control (DBPC) and finite-set model predictive control (FS-MPC). DBPC generates the reference signals based on the prediction models, and a linear modulation scheme is usually used in the inner loop to operate the converter [

19,

20]. On the contrary, FS-MPC relies on the minimization of the defined cost function to select the optimal switching state for the converter, and eliminates the modulator. In addition to the simple concept and easy implementation, FS-MPC features fast dynamic response and multi-objective optimization. As the most popular predictive control scheme in recent years, FS-MPC has been applied to various power converters [

21,

22,

23,

24,

25,

26], including those used in naval MGs, which are similar to the aircraft MGs [

27,

28].

MPC has also attracted attentions in the area of CSC control. In [

29], FS-MPC is applied to control the input reactive power and output current of CSC. Yet, this scheme is unable to mitigate some input harmonics especially those around the filter resonant frequency [

30], resulting in the need of a very large filter. The input reactive power control can be replaced by direct source current control to achieve sinusoidal source currents [

31], which has been well acknowledged in recent studies [

32,

33]. However, reference source currents need to be appropriately generated, otherwise the output control performance could be degraded. In [

31,

32,

33], reference source currents are calculated according to the load model, but this kind of reference generation is only applicable to the situations with a known load model (e.g., passive R–L load).

For CSC used in the aircraft DC MG, an LC filter should be installed at the output circuit so as to smooth the load voltage. In this case, the load model is usually unknown and thus the above reference generation is not applicable. Besides, for optimal tradeoff between the input and output control performance, weighting factor in the cost function of the above FS-MPC scheme needs to be tuned properly, which is usually based on empirical knowledge and thus increases the implementation complexity. In [

34,

35], the proportional integral (PI) controller is used to generate the reference source currents and eliminate the weighting factor. Although this method could address the above two issues, it violates the fundamental principle of the predictive control and also increases the control complexity.

The contribution of this paper lies in the following: A novel hybrid predictive control scheme is proposed for the CSC with an output LC filter used in the aircraft DC MG. The proposed scheme has a hybrid and cascaded structure, which successfully eliminates both the PI controller and weighting factor, reducing the control complexity. DBPC is applied to control load voltage and output current, while FS-MPC is applied to control source currents. Reference source currents are generated from the output DBPC. In addition, the output DBPC has larger sampling time and the input FS-MPC is simplified by eliminating source current predictions, which reduces the computational burden.

This paper is organized as follows.

Section 2 presents the mathematical model of the CSC system for aircraft DC MG.

Section 3 elaborates the principle of the proposed hybrid predictive control scheme.

Section 4 shows the simulation and experimental verification.

Section 5 draws the conclusion.

## 2. Mathematical Model of the CSC System

Schematic of the CSC system for the aircraft DC MG is shown in

Figure 2. Power source connected to the input side of CSC could be the variable frequency main generators or the auxiliary power units. The input LC filter constituted by the inductor

L_{fi} and the capacitor

C_{fi} attenuates high-frequency harmonics in input currents of CSC. In practice, part or all of the filter inductance can be provided by the armature inductance of generators. CSC consists of six switches with reverse blocking capability. If bidirectional power flow is required, each switch can be composed of two transistors in common-source connection. Otherwise, one transistor and one diode in series could serve as each switch if unidirectional power flow is sufficient. One freewheeling diode can be installed at the output side of CSC to reduce the conduction loss, yet negative output voltage will be unavailable. An LC filter composed of the inductor

L_{fo} and capacitor

C_{fo} is installed at the output of CSC to provide smooth load voltage.

According to

Figure 2, continuous model of the input circuit of CSC can be expressed in the state-space form:

where

**u**_{s} and

**i**_{s} are the vectors representing source voltages and currents respectively;

**u**_{i} and

**i**_{i} are the vectors representing input voltages and currents of CSC respectively and matrices

A_{i} and

B_{i} are expressed as:

where

R_{fi} is the parasitic resistance of the filter inductor

L_{fi}.

For simplifying the derivation of prediction models for the output DBPC, differential equations rather than the state-space equation are adopted to model the output circuit:

where

u_{o} and

i_{o} are the output voltage and current of CSC respectively;

u_{L} and

i_{L} are the load voltage and current respectively and

R_{fo} is the parasitic resistance of the inductor

L_{fo}.

By controlling switches of CSC, appropriate input current vector

**i**_{i} and output voltage

u_{o} can be generated to obtain desired source current vector

**i**_{s}, output current

i_{o} and load voltage

u_{L}. According to the principle of CSC, there are nine valid switching states in total, as listed in

Table 1. The first six states generate active

**i**_{i} and non-zero

u_{o}. In

Table 1,

u_{xy} denotes the input line–line voltage between phase

x and phase

y, where

x,

y ∈ {A, B, C}. The last three states generate zero

**i**_{i} and

u_{o}. If a freewheeling diode is adopted, these zero states can be replaced by the on-state of the freewheeling diode.

## 3. Proposed Hybrid Predictive Control Scheme

#### 3.1. Control Block Diagram

Block diagram of the proposed hybrid predictive control scheme is shown in

Figure 3. It can be seen that the proposed scheme consists of two parts: (1) DBPC for the output circuit and (2) FS-MPC for the input circuit. Besides, it is clear that the proposed scheme has a cascaded structure. DBPC of the load voltage generates the reference output current

i_{o}*, while DBPC of the output current generates the reference output voltage

u_{o}*. Product of

i_{o}* and

u_{o}* is the reference source active power

p_{s}*. FS-MPC of source currents finally generates the optimal switching state for CSC.

The following parts of this section will present the principle of the output DBPC, input FS-MPC and the generation of reference source currents separately. It should be noted that sampling time for the input and output control are different in the proposed scheme, which are highlighted by different colors in

Figure 3. Setting different sampling time is necessary for the proposed scheme, for which the reason is discussed in Part 3.4.

#### 3.2. DBPC for the Output Circuit

According to the continuous model of the load voltage shown in Equation (4), the discrete model for predicting

u_{L} is obtained with the forward Euler method:

where

x[

n] denotes the value of variable

x at the beginning of the

nth sampling period. Note that the sampling time is

T_{so} for Equation (5). For the load voltage model, the only controllable variable is the output current

i_{o}. Therefore, if the load voltage

u_{L} reaches the desired value at the beginning of the (

n + 1)th sampling period, the reference value of

i_{o} can thus be obtained from Equation (5):

where the reference load voltage

u_{L}*[

n + 1] is a constant DC value. A constraint can be added at this step to limit the maximum output current.

Similarly, the discrete prediction model for the output current is obtained from Equation (3) with a forward Euler method:

Note the sampling time for the output current control is also

T_{so}. For the prediction model of the output current, the only controllable variable is the output voltage

u_{o}. Therefore, the reference output voltage can be obtained based on Equation (7):

It can be seen that, to obtain the reference output voltage

u_{o}*[

n], the reference output current

i_{o}*[

n + 1] at the beginning of the (

n + 1)th sampling period should be obtained first. However, according to Equation (6), only

i_{o}*[

n] is generated in the

nth sampling period, while

i_{o}*[

n + 1] is not available. Therefore, the following approximation needs to be applied:

Such approximation is reasonable because the output current is DC current and can be considered constant within one sampling period if the output filter inductor is big enough. With the substitution of Equation (9) into Equation (8), the reference output voltage is thus rewritten as:

#### 3.3. FS-MPC for the Input Circuit

The discrete prediction model for the input circuit is obtained from Equation (1):

where

y[

k] denotes the value of variable

y at the beginning of the

kth sampling period. Note that the sampling time is

T_{si} for the input control. Matrices in Equation (11) are expressed as:

and

The basic idea of FS-MPC is to find the optimal switching state by minimizing the defined cost function. For the digital control in practice, the switching state determined in the

kth sampling period has to be applied to the CSC in the (

k + 1)th sampling period, resulting in one sampling period delay. Performance of FS-MPC is sensitive to the control delay and thus delay compensation must be implemented. At the beginning of the

kth sampling period, the digital controller measures source voltage

**u**_{s}[

k], source current

**i**_{s}[

k] and input voltage

**u**_{i}[

k]. Besides, the input current

**i**_{i}[

k] can be looked-up from

Table 1 using the switching state

S[

k], which is determined in the previous sampling period. Therefore, source current

**i**_{s}[

k + 1] and input voltage

**u**_{i}[

k + 1] can thus be calculated based on Equation (11).

For determining the switching state

S[

k + 1] in the next sampling period, the prediction model for the source current vector and input voltage vector is obtained from Equation (11):

In FS-MPC, cost function is the only criterion to determine which switching state is the optimal one that should be applied to the converter. It should include prediction errors of all the desired control objectives. In this paper, to achieve sinusoidal source currents, prediction errors of source currents are included in the cost function:

where generation of the reference source current

**i**_{s}*[

k + 2] is presented in the next part. Note that there is no weighting factor adopted in the cost function, which saves the empirical adjustment and thus reduces the control complexity.

According to

Table 1, Equations (14) and (15), each valid switching state corresponds to a value of

**i**_{i}[

k + 1]. Therefore, there are seven possible values of the source current

**i**_{s}[

k + 2] in total, which generate seven values of

g. Zero

g means perfect tracking performance of source currents. Therefore, the switching state that generates the minimum value of

g should be applied to the CSC in the (

k + 1)th sampling period.

However, if the cost function shown in Equation (15) is directly evaluated to select the optimal switching state, the computational burden is relatively high. This is because every calculation of g requires the prediction of source currents using Equation (14), resulting in many multiplication operations. To reduce the computational burden, the FS-MPC can be simplified as presented below.

According to (14), the reference input current vector

**i**_{i}* is defined as:

With the substitution of Equations (14) and (16), (15) can be rewritten as:

According to the deduction, Equation (17) is completely equivalent to Equation (15). Therefore, if applied to the FS-MPC, Equation (17) could achieve exactly the same control performance with Equation (15). However, for each valid switching state, Equation (17) can be directly calculated with

**i**_{i} looked up from

Table 1, which saves the source current predictions, the computational burden is thus reduced significantly. In practice, the constant coefficient

_{i12}^{2} in Equation (17) can also be eliminated, since it does not affect the minimization of the cost function.

#### 3.4. Generation of Reference Source Currents

Ideally, the active power generated by the power source should be equal to those absorbed by the load. Since the reference output voltage and current are obtained using Equations (6) and (10), the reference source active power at the beginning of the (

k + 2)th sampling period can be calculated as:

where is the conversion efficiency from the supply to the load. The reference source current vector is further calculated based on the instantaneous power theory:

where

q_{s}* is the reference source reactive power, which can be set as zero for unity power factor operation, or other values for reactive power compensation. Theoretically, the source voltage

**u**_{s} at the beginning of (

k + 2)th sampling period should be used to calculate

**i**_{s}*[

k + 2], but it is unavailable at the

kth sampling period. Therefore,

**u**_{s}[

k + 2] is approximated by

**u**_{s}[

k] and can also be estimated using an interpolation algorithm in order to achieve higher control accuracy.

Note that, in Equation (18), the sampling time for variables at the left and right sides of the equal sign are different, which is necessary for the proposed control scheme. Although the predictive control has theoretically very fast dynamic response, it still takes some sampling periods for the actual source currents to reach the references, because of the practical voltage and current limit of the converter. If the input and output control have the same sampling time, the actual active power cannot respond to the load demand promptly, leading to the unstable operation of the control. In addition, as it can be deduced from Equations (6), (10) and (18), the smaller the sampling time of the output control is, the more high-frequency ripples of the reference source active power will contain, which deteriorate the input power quality. Therefore, to achieve sinusoidal source currents, the sampling time of the output control should be larger than that of the input control. In this study, the input sampling time T_{si} is 6.67 s, while the output sampling time T_{so} is 333 s or 667 s, which is 50 or 100 times of T_{si}. As a result, the reference source active power obtained from Equation (18) remains invariable within every 50 or 100 input sampling periods, which helps to reduce the low-frequency harmonics in source currents. Although T_{so} is much larger than T_{si}, it is still small enough to obtain fast dynamic response of the output control.

## 5. Conclusions

For the current source converter used in the aircraft DC MG, an output LC filter was necessary. In such an application, the proposed hybrid predictive control scheme fully utilized the technique of predictive control to operate the converter. By adopting the output deadbeat predictive control and input finite-set model predictive control, the usage of the PI controller and weighting factor was eliminated. The input control was also simplified by eliminating the source current predictions. Overall, the proposed scheme exhibited very low control complexity, which created a capability to work under high sampling frequency (up to 150 kHz). The steady-state and dynamic experimental results obtained on a SiC-MOSFET-based prototype demonstrated the superior performance of the proposed control scheme, even under very high and variable source frequency (360–800 Hz).

Realization of the power sharing in multi-supplies mode should be one of future work. For example, the typical droop control can be incorporated in the predictive control scheme. In this case, the steady-state error and low-frequency ripple could be important issues since they may cause the failure of droop control.