1. Introduction
As a natural platform for distributed renewable energy generation, DC microgrids have attracted a great amount of focus from scholars worldwide [
1,
2,
3,
4]. A typical DC microgrid structure consisting of parallel buck converters and different loads is shown in
Figure 1. As can be seen, parallel converters have been applied to increase the system stability and flexibility and reduce the power stress on each single converter. However, with the application of multiple converters, the suitable operation of the parallel converters needs to be solved [
5]. In the DC microgrid, many converter-linked loads are used, and they often operate in a tightly regulated point of load mode, which makes it so they can be redeemed as the constant power load (CPL). As opposed to the constant voltage load (CVL), which can usually be represented by a resistor, CPL always consumes a fixed amount of power regardless of the value of the input voltage. Therefore, the input current decreases while the input voltage increases, and vice-versa [
6,
7]. Although the high efficiency and controllability of the load can be ensured by tight regulation, this unwanted phenomenon creates negative incremental impedance, which makes the system poorly damped, introduces voltage oscillation, or even leads to instability issues [
8,
9].
In order to mitigate the instability problem caused by the CPLs in the DC microgrid, much research has been conducted by scholars. In the beginning, passive damping and linear-based active damping algorithms were proposed. In [
8,
10], the necessary capacitors, resistors, and LC filters are designed and added into the circuit to improve the damping of the system. Even though the oscillation of the system can be dissipated, the weight, cost, and power loss of the system also increase with the addition of those passive damping elements. For the linear-based active damping algorithm, the instability of the system can also be mitigated by reshaping the impedance of the source and load [
7,
11]. However, due to the adoption of small-signal-model-based linear techniques, these approaches can only ensure system stability around the operating points; a typical system behavior away from this point cannot be obtained [
7].
In order to guarantee the stability of the system in the large signal region, many nonlinear control methods are applied to mitigate the instability problem. In [
12], the instability problem caused by the CPL in the buck converter is solved by a boundary controller, even without knowing all the detailed parameters of the system. However, this control algorithm is designed on the hysteresis band; problems such as the variable switching frequency and degraded ripple effect are also introduced. In [
13], a wide operation range sliding mode controller with fixed switching frequency is proposed to ensure the stability of the system, but this algorithm needs to obtain the value of the capacitor current, and the measurement used leads to the degraded ripple filtering effect and the introduction of equivalent series resistance (ESR) of the capacitor. Moreover, this controller needs to operate in a very high frequency, which introduces the chattering problem. In [
14], a synergetic control method is applied to stabilize parallel DC-DC buck converters with CPL; the experiment shows that this method has good dynamics and fast response performance. However, this paper lacks a detailed analysis for CPL. In [
15], an adaptive passivity-based control (PBC) is proposed; in order to mitigate the instability caused by disturbance, PBC is combined with a nonlinear disturbance observer (NDO). However, the performance of this algorithm’s tuning relies on the parameters
R1d,
R2d,
λ1, and
λ2.
As a promising controller, more and more scholars use the model predictive control (MPC) to solve problems in power electronics [
16]. In general, MPC defines the control actions by minimizing a cost function that describes the desired system behavior. This cost function compares the reference with the predicted system output, which is computed from the system model [
17]. Therefore, an accurate predictive model and proper cost function are the corner stone of an effective MPC. However, the predictive model is usually impacted by disturbance, model uncertainty, and model mismatch. In order to overcome these defects, many modified MPC controllers have been proposed. In [
18], a PI-MPC is combined with a Luenberger observer to mitigate the static error caused by the model mismatch of the ignorance of the equivalent serial resistance (ESR). In this paper, a PI-MPC controller is used to track the reference, while the Luenberger observer is designed and set in the feedforward loop to compensate for the error caused by the model mismatch. In [
19], in order to eliminate the offset caused by the change of resistive load and the model mismatch of inductance, a continuous control set model predictive control (CCS-MPC) is combined with the Luenberger observer. The MPC methods proposed above have done a good job of eliminating the static error in output; however, they do not take into consideration the NIR caused by CPL, which has a much larger negative impact on stability. In [
20], a high-order sliding mode observer (HOSMO) is designed and combined with MPC to eliminate the output offset of a buck converter suppling CPL. With the help of HOSMO, the proposed method can track the reference both accurately and rapidly, regardless of the disturbance, as the variation of the input voltage, resistive load, and CPL. In [
21], an explicit model predictive control (eMPC) is used to ensure the stable operation of a boost converter supplying the CPL. By solving a multiparametric nonlinear problem over a simplex partition of state space, the optimal control law can be obtained. However, in order to mitigate the instability caused by CPL, both the output voltage and the output current of the converter are needed. A decentralized model predictive control (DMPC) is proposed to ensure the power sharing and stability of a multiboost converter system supplying the CPL [
22]. Supplied by the value of the power of the CPL, the proposed DMPC can correctly regulate the DC bus voltage. In [
23], the MPC is combined with an extended Kalman Filter to ensure the stability of a shipboard DC microgrid consisting of a time-varying CPL from the perspective of the secondary control.
In this paper, the instability problem caused by the disturbances (such as variation of input voltage, CVL, and CPL) will be addressed and mitigated. During this process of making the DC microgrid system more robust under disturbances, a composite adaptive CCS-MPC algorithm is proposed. In the controller, a CCS-MPC and capacitor dynamic-based voltage controller are designed and combined as a united double loop controller. Moreover, based on the dynamic characteristics of the circuit, a feedforward-based estimation algorithm is developed and set in the feedforward loop to compensate for the error made by disturbance. With the increment of the robustness by the proposed estimation algorithm, the proposed controller can track the reference both rapidly and accurately regardless of the disturbance.
This paper is organized as follows:
Section 2 presents the establishment of the symmetrical model of the circuit and the description of the problem.
Section 3 introduces the design of proposed adaptive CCS-MPC with the parallel feedforward estimation algorithm. The MATLAB simulation and hardware-in-the-loop (HIL) experiment results are presented in
Section 4 and
Section 5, respectively, while the conclusion is written in
Section 6.
2. System Equation Modeling and Problem Description
The system described in
Figure 1 can be represented by two symmetrical parallel source converters supplying composite loads (CVLs and CPLs) by a common load bus, which is depicted in
Figure 2 shown below. It is worth mentioning that the system is symmetrical which is beneficial for sharing the load of converter.
In this figure, the parameters (
E1,
E2), (
L1,
L2), and (
C1,
C2) represent the input voltage, inductance, and capacitance of the two buck converters, respectively. Meanwhile, the (
iL1,
iL2) and (
vC1,
vC2) are the inductor current and the capacitor voltage of two source converters. Assuming the system works in continuous current mode (CCM) and the output voltages for each converter here are equal, the dynamic model can then be derived as
where the
Ceq is the equivalent capacitance,
vC is the voltage of DC load bus, and (
μ1,
μ2) are the duty ratio of the converters.
According to the dynamic equations of the system, it can be known that the model mismatch of inductance, capacitance, and the equivalent serial resistance (ESR) of the inductor can lead to a static error of the output if their variations cannot be taken into consideration. The disturbances, such as the variation of the input voltage and resistance of the CVL and CPL, can also impact the stability of the system negatively. However, among all the destabilizing factors mentioned above, the last three have a more negative impact on the stability of system [
15,
18], and if these variations are involved, Equation (1) can be modified as follows:
where (Δ
E1, Δ
E2) represent the variation of input voltage, Δ
R represents the variation of resistance of the CVL, and Δ
P represents the variation of the CPL power. As can be seen from the equations above, if these disturbances cannot be taken into consideration by the controller, the stability of the system will be impacted, and the static error shown in the bracket will be introduced.
3. Design of the Proposed Method
In this section, the proposed adaptive CCS-MPC with feedforward estimation algorithm is constructed. The double loop structure is chosen as the basic form of the controller. In the voltage loop, a voltage controller which is based on the dynamic character of the capacitor is established. Meanwhile, a CCS-MPC is designed and applied as the current loop controller. Even though this double loop controller can track the reference in a nominal condition, a static error will be introduced when disturbances occur. In order to make this controller robust enough even when disturbances exist, a feedforward estimation algorithm derived from the dynamic equation of the circuit is developed. During the operation of the controller, the estimation algorithm works in parallel to supply the disturbance information to the double loop controller. Additionally, the load information estimated by the feedforward estimator can also ensure the load allocation between the parallel converters. Moreover, at the end of this section, in order to protect the system from the circulating current problem, the proposed controller is combined with the droop method to ensure that the output voltage of each converter is equal.
3.1. Design of Voltage Loop
For each buck converter, the change of capacitor voltage is directly related to the current flowing through it, as in the equation shown below:
According to the Euler Forward method, the above equation can be modified as the equation shown below. By doing so, the current flowing through the capacitor can be calculated from the change of the capacitor voltage, and
Ts in the equation below represents the switching period of the converter:
assuming the change of
vC follows a linear route during the period approaching the voltage reference
vref. Furthermore, the value of
iC cannot be arbitrarily large, since it is decided by the error between
vC and
vref, and a reference prediction horizon
N is introduced [
24,
25,
26]. As shown in
Figure 3,
vC arrives at its reference
vref after
N switching cycles.
As shown in
Figure 3, the voltage at moment
k vC(
k)
a (or
vC(
k)
b) is higher (or lower) than the reference voltage
vref, so it needs to follow the
av (or
bv) route to approach its final reference voltage
vref. Accordingly, this process also leads to the current step down to
iC(
k + 1) from
iC(
k)
a (or step up to
iC(
k + 1) from
iC(
k)
b). Therefore, Equation (4) can be modified as
After the value of
vC is obtained, the future inductor current can be also calculated with the help of the basic Kirchhoff’s current law: the inductor current at the next instant equals the current flowing through the capacitors, CVL and CPL. Assuming that the resistance of CVL and the power of CPL will not change suddenly, the inductor current at the next
k + 1 moment can be calculated by the equation below:
where
iL(
k + 1) is the inductor current in the next switching period and will be treated as the inductor current reference in the current loop. Therefore, for the parallel converters in this paper, the voltage loop controllers can be derived as
where [
iL1(
k + 1),
iL2(
k + 1)] and [
C1,
C2] are the inductor current at
k + 1 instant (reference inductor current) and the capacitor of each buck converter, respectively, and
a is the parameter concerning the power allocation between two buck converters. In order to realize the symmetrically equal current sharing,
a is set to 2.
3.2. Construction of Current Loop
In the inner current loop, a one-step horizon is selected as the current control method for both of the buck converters. Compared with other multistep horizon CCS-MPC methods, the mentioned method does not have the requirement of high computation ability [
19]. According to Equation (1), the change of the inductor current is directly related to the switch state, and its variation trend in one switching period can be depicted as
Figure 4.
Therefore, the current slope of
iL1 and
iL2 in one switching period can be represented by the following equation:
where
fi1,
fi2, and
fi3 are the current slope during the first turn-on period
t1, first turn-off period
t2, and second turn-on period
t3 in one switching period, respectively, and
Ln represents the inductance for the corresponding buck converter. As depicted in
Figure 4, the inductor currents at the
kTS +
t1,
kTS +
t1 +
t2 and
kTS +
t1 +
t2 +
t3 are
iS1,
iS2, and
iS3, and they can be calculated through the equation given below. It is worth mentioning that in order to reduce the calculation burden,
t1 and
t3 are forced to be equal:
where
iS0 is the inductor current at the beginning of
kTS, which can also be represented as
iL(
k). As one of the most important parts of the MPC method, cost function reflects the targets and constrains of the controller. Here, since the CCS-MPC works as the inner current loop, the only target of the cost function is to track the reference current as accurately as possible. Moreover, in order to make the inductor current as close as possible to the reference current, all of
iS1,
iS2, and
iS3 are formulated into the cost function and further compared with the reference current by calculating the error between them.
By substituting Equation (9) into Equation (10), the latter can be modified as
Replacing
t2,
t3 with
t1,
TS, calculate the derivation of Equation (11), and its extreme point, the optimal value of
t1 which allows the cost function obtain its minimum value, can be obtained:
After the optimal value t1 is calculated, the related optimal duty ratio can also be obtained through the relationships between t1, t2, t3, and Ts. Due to similar topology, the above procedure is applicable for both parallel converters.
3.3. Foundation of Feedforward Estimation Algorithm
During the construction of the double loop control algorithm shown above, the value of input voltage and load are regarded as constant. However, disturbances, such as the variation of input voltage, resistance, and CPL, significantly impact the stability of the DC microgrid. If the controller wishes to become robust, it must take those negative factors into consideration. From Equation (2), it can be seen that these disturbances can affect the derivatives of inductor current and capacitor voltage. Therefore, when the time-varying load is not involved, such as in Equations (13) and (14), it is clear that the Equations (6) and (8) cannot behave as planned. As can be seen from Equation (13), the variation of the load (CVL and CPL) can impact the voltage loop controller by creating a static error which is shown at the end of the equation:
where Δ
R and Δ
PCPL represent the variation of CVL and CPL, respectively, and
Cn stands for the capacitance of two buck converters. Similarly, when the change of the input voltage has been introduced from Equation (8), the current slope equations can be modified as
where Δ
E is the variation of the input voltage. In order to take these disturbances into consideration, an
LC dynamic-based feedforward estimation algorithm is developed. Based on Kirchhoff’s first law and Equation (4), the load current can be estimated by
In the equation above,
Ceq is the equivalent capacitance of the microgrid system, and
B is an estimation parameter and needs to be tuned. Similarly, based on the dynamic characteristics of the inductor and the Euler forward method, the input voltage can be obtained as follows for each buck converter:
Combine Equations (15) and (7), and the current loops for the buck converters are listed as
Similarly, substituting Equation (16) into Equation (8), the current slope during one switching period can be calculated as
Therefore, the load current which has taken the disturbance into consideration can be calculated by the feedforward estimation algorithm in Equation (15); then, the reference current for each buck converter subsystem is produced by the voltage loop controller as Equation (17). In this process, the load current is also allocated. Additionally, based on the other part of the estimation algorithm shown in Equation (16) and the CCS-MPC current control method proposed in Equations (18) and (12), the reference current can be tracked accurately, and the optimal duty ratio is calculated. However, all the controllers mentioned above assume that the output voltage of the converters is equal. The condition of unbalanced output voltage will be considered in the following.
3.4. I-V Droop Control
In order to reduce the load pressure of a single converter, to improve the reliability of the power supply, and to improve the load supplying capacity, the structure of two parallel buck converters and the symmetrical equal current sharing are adopted in this paper. However, if the two converters operate unbalanced, even 1% of the mismatch between their output voltages can significantly negatively impact the symmetrically equal current allocation [
27]. In the section above, we assumed that the output voltage of two converters is equal; when their output voltages are different, Equation (17) can be rewritten as
where
vC1 and
vC2 are the output voltage for the two buck converters, respectively. In order to fulfill equal current sharing, the proposed controller is combined with the
I-
V droop control which can eliminate the error between the output voltage from two converters without output deviation [
28]. By reversing the reference output from the traditional droop characteristic, the current reference produced by the
I-
V droop control can be given as
In Equation (20),
iDrefn (
n = 1, 2) is the current reference produced by the
I-
V droop control,
rn is the droop factor, which is derived by the virtual resistance method, and
vCn (
n = 1, 2) is the output voltage for the two converters, respectively. According to [
28], the product of the inductance current and virtual resistance of two systems should be equal.
Since the equal current allocation is chosen, the droop factor of the two buck converters is equal. Then, the current produced by the droop control is introduced into the voltage control algorithm in Equation (19), as shown below:
Therefore, the diagram of the proposed control algorithm can be described as
Figure 5. As it depicts, the voltage loop controller can be used to track the reference voltage and produce the reference current. Then, a CCS-MPC method is developed to track the reference current and calculate the optimal duty ratio, which will be sent to the buck converter later. Moreover, in the parallel feedforward loop, an estimation algorithm is used to calculate the disturbance information which will be sent to the controller mentioned above to ensure the fast and nonstatic-error operation of the system.