Finite Control Set MPC for Voltage and Frequency Stabilization in Islanded AC Microgrids with Line Impedance Consideration and Enhanced Power Sharing Control

In recent years, the depletion of nonrenewable sources and environmental pollution problems have increased in tandem with the rising demand for electricity. Therefore, developing microgrids based on distributed generation (DG) units, especially renewable energy sources (RES), has received significant interest for future modern power systems [1]. A typical microgrid is one of the foundational technologies of discrete power systems. Composed of distributed power sources, storage devices, power electronic converters, and controllable loads, it can operate autonomously (island mode) or in cooperation with a main grid (grid-connected mode) to provide an uninterrupted power supply for loads [2].

1.png

Figure 1. Microgrid configuration with several connected parallel DG units

In AC microgrids, the inverters are used to link these DG units to local loads or the grid, where these inverters are preferably operating in parallel, as shown in Figure 1. This offers the high dependability and system redundancy necessary for adaptable microgrid systems [3]. Several strategies for controlling parallel inverters, or more precisely, power sharing among different DG units, have been presented. They are divided into two categories based on whether or not they use control wire interconnections [4, 5]. In the first category, information signals must be exchanged among DG inverters, which leads to a complex structure of the microgrid and inhibits expansion, among the wired controls are master-slave control [6, 7], circular chain control [8], and average current sharing method [9, 10]. In the master-slave control, one of the inverters serves as the master, ensuring that the output voltage is produced with a constant frequency and amplitude, and the remaining inverters operate as slaves to follow the reference current specified by the master unit. Although this technique provides good output voltage and power sharing, the total dependence on the master and the communication among inverters reduce the reliability of the system. The second category of control is primarily founded on the droop control technique to address the issue of signal lines in wired interconnection control. The droop control technique utilizes only local measurements for each inverter. As a consequence, that offers higher dependability and flexibility in terms of the actual positioning of the units. This method entails dropping the frequency and amplitude of the output voltage in order to control the active and reactive power delivered by each inverter [11].

The droop control has several drawbacks that restrict its applicability range, including significant dependence on the inverter output impedance; amplitude and frequency deviations of the output voltage caused by the droop characteristics; and a slow dynamic response [12]. The conventional droop controller (P-f/Q-E) is most appropriate for high-voltage networks based on pure inductive line impedances. While in low-voltage distribution networks, the line impedances are mostly resistive [13], therefore, the conventional droop is ineffective, and in this case, the reverse droop controller (P-E/Q-f) must be implemented. The main difference between reverse droop control and conventional droop control is the direction in which the generator output is changed in response to frequency changes. Reverse droop control increases the output as the frequency increases, in contrast to conventional droop control, which decreases the output. In order to address the drawbacks of the conventional/ reverse droop technique, various improved droop approaches are proposed, including power angle droop control [14], a virtual flux-based technique [15], and a complex droop control technique [16]. In the study [14] high gain power angle enables better load sharing however it has a detrimental effect on overall stability. In the study [15] The power-sharing among DG units is achieved by decreasing the flux amplitude and adjusting the phase angle. Moreover, a direct flux control algorithm is presented to control the inverters and generate a defined flux from a droop controller. Hence, the control structure does not require multi-feedback loops or PWM modulators. Nonetheless, it's unclear how current limiting is applied. In the study [16], a new droop controller that takes into consideration the impact of complex impedance is presented. This controller can deliver superior power quality even when dealing with nonlinear loads and unbalanced line impedances.

In the presence of line impedance differences, the conventional/reverse droop technique is incapable of achieving accurate power distribution among parallel connected inverters. Several strategies, such as the virtual impedance technique, have been proposed to solve this issue [17, 18]. The virtual impedance technique compensates for these differences and ensures that power is shared equally among the inverters. This results in a stable and dependable power system with enhanced performance and reduced losses.

Over the past decade, the Model Predictive Control (MPC) method has gained widespread use in controlling power electronic converters thanks to its significant advantages over traditional linear control techniques. The MPC approach is based on predicting the future states of a controlled system using a modeled system [19]. One key benefit of this control technique is its ability to handle multi-objective optimization problems and easily manage system constraints [20]. However, it should be noted that MPC requires a higher number of calculations than linear control methods. Nonetheless, with the advent of fast microprocessors readily available on the market, implementing the MPC technique has become feasible.

Finite control set-model predictive control (FCS-MPC) represents a paradigm shift from traditional linear control techniques. Rather than designing loops for each controlled variable independently and then cascading them together, FCS-MPC utilizes the mathematical model of the power converter to predict the future behavior of variables and subsequently determines the optimal switching state based on a specified cost function [21].

This paper is structured as follows: In Section 2, an overview of power control using the droop control technique is provided. Section 3 offers the FCS-MPC strategy for a three-phase inverter with an LC filter. In addition, the proposed droop approach, double virtual impedances, and secondary control are presented. The simulation results of the scenario tests are shown in Section 4.

Figure 3 shows a comprehensive block diagram of the proposed power stage control of a DG unit with an LC filter connected in island mode, where the DG unit can consist of renewable and/or non-renewable generators. Furthermore, a variety of distributed loads are associated with an AC bus, including balanced, unbalanced, and nonlinear loads. The hierarchical control structure consists of two main levels of control. The first level is local control which is responsible for controlling active and reactive power and regulating the output voltage of the DG unit. The secondary level is a central control unit used to reduce voltage imbalance and harmonic distortion in the AC bus.

3.png

Figure 3. A comprehensive representation of the proposed control system of a DG unit within an island microgrid

The steps for implementing the proposed system can be outlined in the following manner:

• Calculate the active and reactive power supplied by the DG unit.
• Compensate the mismatch of the line impedance using the virtual impedance concept.
• Generate the reference voltage based on the proposed droop approach.
• Regulate the output voltage and current of the DG unit using FCS-MPC.
• Apply secondary control to maintain the voltage of the PCC at its nominal value (50 Hz, 311 Volts).

3.1 Primary control

3.1.1 Power calculation

The measurement of instantaneous active and reactive power supplied by each DG unit is crucial for the effective management of three-phase power systems. To achieve this, the output voltage (V_o-abc) and output currents (I_o-abc) of each DG unit are first measured in the a-b-c coordinates. These measurements are then transformed to the α-β coordinates using Clarke's transformation, as described in Eqns. (4) and (5). Once transformed to the α-β coordinates, the instantaneous active power (P_i) and reactive power (Q_i) are calculated using Eq. (6) [23].

$\left[\begin{array}{l}V_\alpha \\ V_\beta\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{crc}1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{l}V_a \\ V_b \\ V_c\end{array}\right]$               (4)

$\left[\begin{array}{l}I_\alpha \\ I_\beta\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{rrr}1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{l}I_a \\ I_b \\ I_c\end{array}\right]$               (5)

$\left[\begin{array}{l}P_i \\ Q_i\end{array}\right]=\left[\begin{array}{cc}V_\alpha & V_\beta \\ V_\beta & -V_\alpha\end{array}\right]\left[\begin{array}{l}I_\alpha \\ I_\beta\end{array}\right]$               (6)

Instantaneous measurements of both active and reactive power (P_i and Q_i) contain AC and DC components. The AC components are the oscillating parts caused by the harmonic and unbalanced components of the output voltage and current. On the other hand, the DC component represents the average value of active and reactive power and can be obtained using a low-pass filter with a 2 Hz cut-off frequency [24]. Hence, the average powers can be given as:

$\left\{\begin{array}{l}P=\frac{\omega_c}{1+\omega_c} P_i \\ Q=\frac{\omega_c}{1+\omega_c} Q_i\end{array}\right.$               (7)

where, $\omega_c$ is the low-pass filter's cut-off frequency.

3.1.2 Voltage reference generator

The effectiveness of using inductive output impedance as a control method to regulate the power delivered from DG units has been proposed in the study [12]. However, the sensitivity of the inductive output impedance to harmonics can significantly impact the power control system's performance. As a solution, a resistance output impedance is preferred over an inductive output impedance.

As mentioned previously, the droop control technique has an inherent connection between accuracy in power-sharing (P/Q) and amplitude/frequency control of the output voltage. From Eq. (1), it is obvious that the active power can also be controlled by varying the output resistance's value. In considering the fact that P decreases as R increases, we propose utilizing an adaptive output impedance that enhances the resistivity of each DG's output impedance, which is given by:

$R_d=R_d^{r e f}-k_R P$               (8)

where, $R_d^{r e f}$ is the reference output resistance, and k_R is the active power coefficient which modifies the resistive output impedance.

The significance of this proposed approach is that the control of active power can be achieved using the resistance output impedance while keeping the reference amplitude voltage constant, in contrast to the reverse droop method, which results in reference voltage deviation (see Eq. (2)). The reference voltage before applying the virtual impedance can be determined using the following equation:

$V_{\text {ref }}^{\text {old }}=\left\{\begin{array}{l}E_{p c c}^{r e f} \sin \left(2 \pi F_d t\right) \\ E_{p c c}^{r e f} \sin \left(2 \pi F_d t-2 \pi / 3\right) \\ E_{p c c}^{r e f} \sin \left(2 \pi F_d t+2 \pi / 3\right)\end{array}\right.$               (9)

where, $E_{p c c}^{r e f}$ is the reference amplitude of the PCC bus, and F_dis the frequency determined by Eq. (3) which is used to control the reactive power.

3.1.3 Double virtual impedances

The equivalent output impedance of the inverter plays a major role in achieving accurate and balanced power sharing, as the equivalent output impedance is affected by several variables, including feeder impedance, voltage/current control loop parameters, and filter parameters. Hence, any mismatch in these variables leads to inaccurate power sharing [25]. As a result, the virtual impedance concept has emerged as a required condition to overcome these problems. In general, the virtual output impedance is adjusted to overcome the dominant line impedance and make the equivalent output impedance identical for all inverters in the microgrid.

Voltage deviations. Although balanced power sharing can be accomplished with virtual impedance, its faulty design can result in significant voltage drops. As a result, this section will provide the virtual impedance design method that can minimize voltage drops using the summing approach as follows [3]:

$V_{\text {drop } 1}=\left(Z_{l 1}+Z_{v 1}\right) I_{l 1}=V_{\text {dropk }}=\left(Z_{l k}+Z_{v k}\right) I_{l k}$               (10)

where Z_lk and Z_vk are the line impedance and the virtual output impedance of the inverter (k), respectively.

The control rules used in the summation approach to adjust the virtual output impedances' optimization values of 1 ... k^th inverters are expressed by assuming that Z_l1 is larger than the others, Z_l1 > Z_lk, allowing Z_v1 = 0 to be chosen. Hens, Eq. (10) can be changed to the following form:

$Z_{v k}=\left(Z_{l 1}-Z_{l k}\right)$               (11)

The impedance information of each feeder must be known, either by obtaining the cable specifications or through the use of an online impedance estimation technique [26, 27].

The voltage drop caused by the virtual impedance of each DG unit is expressed by:

$\Delta V_{v k-a b c}=Z_{v k} I_{k o, a b c}$               (12)

where,

$Z_{v k}=\left(R_{v k}+j \omega L_{v k}\right)$               (13)

By applying the park transform to convert Eq. (12) to a dq reference frame, yielding Eq. (14):

$\left[\begin{array}{c}\Delta V_{v_{-} d} \\ \Delta V_{v_{-} q}\end{array}\right]=R_{v k}\left[\begin{array}{c}I_d \\ I_q\end{array}\right]+L_{v k} s\left[\begin{array}{c}I_d \\ I_q\end{array}\right]+\left[\begin{array}{cc}0 & -\omega L \\ \omega L & 0\end{array}\right]\left[\begin{array}{c}I_d \\ I_q\end{array}\right]$               (14)

To implement the virtual impedance with the proposed droop approach, the output current is used in fast negative feedback, resulting in a drop in the reference voltage. As a consequence, the reference voltage produced by each inverter is changed as follows:

$V_{r e f}^{\text {new }}=V_{r e f}^{\text {old }}-Z_{v k} I_{k o-a b c}$               (15)

Adaptive virtual impedance loop. The virtual resistance improves the damping of the oscillation system without introducing power loss or decreasing the efficiency of the system. In addition, the virtual resistance makes the output impedance of each DG unit more resistive, which leads to an enhancement in the decoupling between active and reactive power, improves the stability of the system, and reduces circulating currents and power oscillations [28]. As a result of these reasons, an adaptive virtual impedance was proposed by modifying Eq. (15) to Eq. (16). The detailed implementation of the dual virtual impedances is shown in Figure 4. The low-pass filter is used to reduce the impact of high-frequency oscillations caused by the derivative term.

$V_{r e f}^{n e w}=V_{r e f}^{o l d}-Z_{v k} I_{k o-a b c}-R_d I_{k o-a b c}$               (16)

4.png

Figure 4. Proposed droop control with double virtual impedances

3.1.4 Output voltage regulation

In the context of parallel inverters, there are two main strategies of MPC techniques that can be used, each with its own distinct features and advantages: continuous control set MPC (CCS-MPC) and finite control set MPC (FCS-MPC). CCS-MPC requires a modulator to supply the required voltage, leading to a fixed switching frequency. This limitation can result in higher switching losses and lower efficiency. In contrast, FCS-MPC utilizes the finite number of switching possibilities of the converter to solve optimization problems, offering greater flexibility in terms of the switching frequency. This results in improved efficiency and reduced switching losses. However, FCS-MPC implementation can be more complex than CCS-MPC, as an optimization problem must be solved at every sampling time [29].

In FCS-MPC, the mathematical model of the system is used to predict how potential variables will respond at the next sampling time (N = 1), and the cost function is used to obtain the desired control objectives. To apply this technique for VSI, both models of the three-phase inverter and the LC filter are required. There are eight voltage vectors available for a three-phase, two-level (3Ph-2L) inverter, including two zeros (0,7) and six active vectors (1-6), which are shown in Figure 5 and can be given by Eq. (17). The selection of the appropriate voltage vector (V_i) is based on the predicted output voltage and current errors at the next sampling time, which are determined by solving an optimization problem with an appropriate solver.

$V_i=\frac{2}{3} V_{d c} e^{i(j-1) \frac{\pi}{3}}, \quad j=0 ; \ldots ; 7$               (17)

where, V_dcis the DC link voltage.

5.png

Figure 5. Voltage vector options for inverter systems

The dynamic behavior of the inverter LC filer capacitor can be represented as:

$C \frac{d V_c}{d t}=I_f-I_o$               (18)

The mathematical model for filter inductance is as follows:

$L \frac{d I_f}{d t}=V_i-V_c$               (19)

By combining Eq. (18) and Eq. (19), the continuous-time state-space model of the system can be expressed by:

$\frac{d x}{d t}=A x+B V_i+B_d I_0$               (20)

where,

$x=\left[\begin{array}{l}I_f \\ V_c\end{array}\right], A=\left[\begin{array}{cc}0 & -1 / L \\ 1 / C & 0\end{array}\right], B=\left[\begin{array}{c}1 / L \\ 0\end{array}\right], B_d=\left[\begin{array}{c}0 \\ -1 / C\end{array}\right]$

The discrete-time LC filter model for sampling time T_s is required for the implementation of the proposed control digital algorithm, which is given by:

$x(k+1)=A_q x(k)+B_q V_i(k)+B_{d q} I_o(k)$               (21)

where,

$\left\{\begin{array}{l}B_q=\int_0^{T_s} e^{A \tau} B d \tau \\ A_q=e^{A T_s} \\ B_{d q}=\int_0^{T_s} e^{A \tau} B_d d \tau\end{array}\right.$               (22)

Generally, the sampling time T_s is quite small; therefore, the approximate of the first-order derivative calculation by forward Euler is given as:

$\frac{d x}{d t} \approx \frac{x(k+1)-x(k)}{T_s}$               (23)

By substituting Eq. (23) into Eq. (20), the future behavior values of the capacitor voltage and the inductance current are given by:

$\left\{\begin{array}{l}V_c(k+1)=V_c(k)+\frac{T_s}{C}\left(I_f(k)-I_o(k)\right) \\ I_f(k+1)=I_f(k)+\frac{T_s}{C}\left(V_i(k)-V_c(k)\right)\end{array}\right.$               (24)

Consequently, the prediction model uses the actual values of V_c(k), I_f (k), and I_o(k) to determine the predicted capacitor voltage and the inductance current at the next sampling time (k + 1).

Two cost functions have been established to deal with the required control objectives. The first one aims to minimize the difference between the reference voltage and the capacitor voltage. While the second cost function is employed to track the reference inductor current, which leads to regulating the load current when a fault occurs. By combining these two cost functions, the total cost function can be defined as:

$g_t=\lambda_v g_v+\lambda_i g_i$               (25)

where, λ_v and λ_i are the weighting factors, and the expressions of g_v and g_i are given as follows:

$\left\{\begin{array}{l}g_v=\left(V_{\text {ref }}^{\text {new }}-V_c(k+1)\right)^2 \\ g_i=\left(I_{\text {ref }}^{\text {new }}-I_f(k+1)\right)^2\end{array}\right.$               (26)

The reference inductance current is determined by applying Kirchhoff's Law, as shown below:

$I_{r e f}^{\text {new }}=j C \omega V_{r e f}^{\text {new }}+I_o$               (27)

To determine the optimal voltage vector V_i  for the inverter, a systematic process is followed. Initially, predictions are generated for future values of V_c(k + 1) and I_f (k + 1), resulting in eight possible predictions. These predictions are compared to their respective references using a proper cost function (Eq. (25)). By minimizing this cost function, the optimal voltage vector V_i is selected for application by the inverter at the next sampling instant. To gain a better understanding of the process, a flowchart presented in Figure 6 has been provided to summarize the steps outlined below.

6.png

Figure 6. Switch selection algorithm for VSI using MPC

7.png

Figure 7. N=2: (a) voltage vectors are varied at each sampling time. (b) The same voltage vector is used for two consecutive sampling times

In power generation, the efficiency of the system is significantly influenced by the converter's switching frequency, as a high switching frequency leads to increased losses. Therefore, in order to overcome this problem and achieve a low switching frequency, an improved MPC with a two-step prediction horizon (N=2) is proposed in [30, 31]. Whenever two steps are considered for the prediction, two voltage vectors are evaluated. For the first sample period, one of these vectors is selected, and for the second sample period, another vector is used. Hence, 49 different voltage vector combinations are feasible and can be obtained, as shown in Figure 7(a). However, the experimental implementation of this algorithm might be exceedingly challenging due to the enormous number of computations required. For this reason, a modified algorithm is employed, which is based on the same voltage vector used in both two sampling times, as depicted in Figure 7(b). Hence, at both sampling periods, just seven voltage vectors rather than 49 are considered. This results in a decrease in switching frequency, a lowering of voltage ripples, and an improved waveform quality.

3.2 Secondary control

The secondary control (SC) is added to the primary control to achieve high controllability of the microgrid, which is developed to restore the frequency and amplitude voltage deviations at the AC bus. SC techniques may be employed in either centralized or decentralized form, as shown in Figure 8. A centralized approach typically involves estimating the voltage (E_pcc) and frequency (f_pcc) of the AC bus which are then compared to their respective reference values. The resulting error signals are processed by a PI controller, which subsequently transmits corrective signals to all DG units, as expressed in Eq. (28) and shown in Figure 8(a). In contrast, a decentralized control strategy, as depicted in Figure 8(b), equips each DG unit with SC capabilities to independently correct voltage and frequency deviations.

$\left\{\begin{array}{l}f_{\mathrm{sec}}=k_{p f}\left(f_{p c c}^{r e f}-f_{p c c}\right)+k_{i f} \int\left(f_{p c c}^{r e f}-f_{p c c}\right) d t \\ E_{\mathrm{sec}}=k_{p v}\left(E_{p c c}^{r e f}-E_{p c c}\right)+k_{i v} \int\left(E_{p c c}^{r e f}-E_{p c c}\right) d t\end{array}\right.$               (28)

where, k_pf, k_if, k_pv and k_iv are the secondary control parameters of the frequency and amplitude compensator.

The accurate tuning of proportional-integral (PI) gains is a critical and challenging task, particularly with respect to guaranteeing improved system performance and power quality during the operation of DG units and load changes. To address this issue, an improved SC mechanism is proposed [32]. This control strategy employs an optimized PI controller based on a combination of genetic algorithms (GA) and artificial neural networks (ANNs). By leveraging this advanced computational technique, the proposed control mechanism is able to dynamically adjust the PI controller gains, resulting in more effective regulation of the microgrid's frequency and voltage levels.

8.png

Figure 8. Secondary control: (a) Centralized (b) Decentralized